LOW DIMENSIONAL SYSTEMS AND NANOSTRUCTURES dimensional... · •Dimensionality definitions II: Length scales (More macroscopic scheme) (Top-down) •Start from solid state physics

1. Length scales and low dimensionality

Nerea Zabala

Fall 2007

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LOW DIMENSIONAL SYSTEMS AND NANOSTRUCTURES

•Contents:

• Introduction: Nanoscience and Mesoscopic Physics.

• Dimensionality definitions.

• Relevant length scales.

• Examples of low dimensional systems.

• Fabrication and exploring tools.

• New phenomena and new applications.

LOW DIMENSIONAL SYSTEMS AND NANOSTRUCTURES

1. Length scales and low dimensionality

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• MESO- In between an atom and bulk solids. Size below which a solid does no longer behave bulk-like.

• Mesoscopic Physics...Physics of small condensed objects (a collection of atoms)

• Often in the nanometer-size regime ! discipline of “Nanoscience”

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Nanoscience and Nanotechnology

Why increasing interest for the nanoscale?

1 nm = 0,000000001 m

•(Motivation in previous course : Nanoscience: a historical perspective)

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ADN

Diameter of human hair

Diameter of red blood cell

Visible light wavelengths

Intel’s newest transistor

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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0D: Molecular P4Se3 1D: crystalline SiSe2

2D: crystalline Ge4Se3 3D: amorphous SiO27


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) and in course

“Soft matter and nanostructured materials (polymers, gels, colloids,...)

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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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0D: quantum dot

L0 = ! , characteristic length

1D: quantum wire2D: quantum well

L0 > Li, i = 1, n ! (3" n)D system

Lx, Ly, Lz < L0

Lx, Ly < L0 Lx < L0

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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 chapter).

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• Relevant length scales

•A few relevant scale lengths:

12S. Datta, “Electronic transport in mesoscopic systems”,1995

proccess transistor

(Texas Instruments).


•Some characteristic lengths:

•De Broglie wave length, Fermi wavelength:!, !F

•Mean free path:Lm

Related to kinetic energy of electrons

Initial momentum of electrons is destroyed

! =2"!p

=2"

k

Fermi gas: characteristic momentum kF ! !F =2

kF

One single filled band in 2DEG: ! =!

2"/ns , ns : sheet density

Boltzamann gas: p =!

2mkT

Length between collisions with impurities or phonons

Lm = v!t

typical

velocity

transport

relaxation time

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•Phase-relaxation length : L!

•Thermal dephasing length : LT

Initial phase of electrons is destroyed

•Some characteristic lengths:

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Quantun mechanical: phase of the electron wave function

L! =!

D!!

difusion

constant

typical

time of

elastic

collisions

D = (1/d)vLm

dimensionality of

electron gas

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

•For example: Transport through a constriction, 3 different regimes:


•Wire dimensions: W,L

•Mean free path: LmLmLm << W,L

Lm >> W,L

W < Lm < L

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•For example:conductance quantization in a quantum point contact

T. Heinzal, “Mesoscopic electronics in solid state nanostructures”, WileyAFM 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.

By applying voltages to

t h e p l a n a r g a t e

electrode, the width of

the wire is tuned.

At low T conductance is

quantized in units of

2e2h.

Ballistic regime


•For example:mesoscopic ring used to study Abraronov-Bohm effect

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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).

G. Fraser, “The New Physics for the 21th century”, Y. Imry, ch. 12

L! ! 100µm

(low T)

A significant fraction of

electrons traverse the

ring phase coherently

•Summary of conditions required for a mesoscopic device

T. Heinzal, “Mesoscopic electronics in solid state nanostructures”, Wiley



•For example:Kondo mirage

D. Eigler et al., IBM Almaden

http://www.almaden.ibm.com/almaden/media/image_mirage.html

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”)

(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

E. Sheer et al., Phys. rev. Lett. 78, 3535 (1997)


•Peroskite-like high temperature superconductors


Superconductivity related to 2D character due lo weakly connected 2D sheets of Cu and O


•Some quasi-one-dimensional materials:


(More in lecture 4)

(More in lecture 3)

MCBJ technique to produce metallic nanowires



CovalentCovalent

C-C bonds withinC-C bonds within

'molecule''molecule'

Variable spVariable sp

hybridisationhybridisation

spsp22

spsp22

pure sppure sp22

pure sppure sp33

!!"" ++

--

Carbon in all dimensions

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•Semiconductor nanostructures starting from GaAs-AlGaAs heterostructures

•Diminishing dimensions...

•2D electron gas

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E. Corcoran, TRENDS IN MATERIALS: DIMINISHING DIMENSIONS; November, 1990


•Semiconductor nanostructures starting from GaAs-AlGaAs heterostrcutures

QUANTUM

WIRE

•Squeezing 2D electron gas...

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•Also with Si (MOSFET)

Si technology


•Why GaAs?

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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


•OD systems, quantum dots or “artificial atoms”

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•Synthetic nanocrystals :CdS, CdSe in glassy matrix, CuCl in

NaCl crystals, Si, Ge...

•Self-assembled QD’s

•QD’s produced from heterostructures and lithographic etching. applications in nanoelectronics and optoelectronics

•Clusters of metallic atoms grown from vapour-phase condensation in

creasing size

•Fullerenes

, Size control (~1nm-> ~200nm)

• 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....


•OD systems, quantum dots or “artificial atoms”

• 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....

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(Described in previous course “Nanoscience: a historical perspective” ?

Also later in :”Fundamentals of nanoscale characterization”, “Experimental techniques”)

• 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....

•quantum Hall effect, Coulomb blockade, breakdown of Ohm’s law, quantum size effects...

•New phenomena

•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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•Scalability regimes:

Simulations of breaking of Na

nanowires

Eduardo Ogando, Thesis 2004


•Fermi surface topology for 3D (sphere), 2D (cylinder) and 1D (planes) electron gas

•General trends or signatures of low dimensionality

Fermi surface of a

quasi-one -dimensional

electron gas.

(More details in next lecture)

Wavy planes due to

weak coupling in real

systems or



•Density of states

(More details in next lecture) Eduardo Ogando, Thesis 2004

• New phenomena and new applications• New phenomena and new applications


•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)


•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

•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.

•Why interest in GaAs? Compare properties of GaAs vs. Si

(Be very concise)

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•Find examples of systems behaving as 0D

Documents

LOW DIMENSIONAL SYSTEMS AND NANOSTRUCTURES dimensional... · •Dimensionality definitions II: Length scales (More macroscopic scheme) (Top-down) •Start from solid state physics