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Quantum Confinement

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Quantum Wells = 2 dimensionalstructuresQuantum Wires = 1 dimensionalstructuresQuantum Dots = 0 dimensionalstructures!!

For many years, quantum confinement has been a fast growing field in

both theory & experiment! It is at the forefront of current research!

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Quantum Confinement in Nanostructures: OverviewElectrons Conf ined in 1 Direction:

Quantum Wells(thin films):

Electrons can easily move in2 Dimensions!

Electrons Conf ined in 2 Directions:Quantum Wires:

Electrons can easily move in1 Dimension!

Electrons Conf ined in 3 Directions:Quantum Dots:

Electrons can easily move in0 Dimensions!

Each fur ther conf inement dir ection changes a continuous k component

to a discrete component character ized by a quantum number n.

kx

nz

ny

ny

nz

nx

kxky

nz1 Dimensional

Quantization!

2 DimensionalQuantization!

3 DimensionalQuantization!

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Quantum Well QW= A single layer of material A (layer thickness L), sandwiched between 2macroscopically large layers of material B. Usually, the bandgaps satisfy:

EgA < EgB

Multiple Quantum Well MQW=Alternating layers of materials A(thickness L) & B(thickness L). In this case:

L >> LSo, the e- & e+ in one Alayer are independent of those in otherAlayers.

Superlattice SL=Alternating layers of materials A& B with similar layer thicknesses.

Quantum Confinement Terminology

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Some Basic Physics Density of states (DoS)

in 3D:

Structure Degree of

Confinement

Bulk Material 0D

Quantum Well 1D 1

Quantum Wire 2D

Quantum Dot 3D d(E)

dE

dN

E

E1/

dE

dk

dk

dN

dE

dN

DoS

Vk

kN

3

3

)2(34

statepervol

volspacek)(

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QM Review: The 1d (infinite) Potential Well(particle in a box) In all QM texts!!

We want to solve the Schrdinger Equation for:

x < 0, V ; 0 < x < L, V = 0; x > L, V -[2/(2mo)](d2 /dx2) = E Boundary Conditions:

= 0at x = 0 & x = L (V there) Energies:

En = (n)2/(2moL2), n = 1,2,3Wavefunctions:

n(x) = (2/L)sin(nx/L) (a standing wave!)

Qualitative Effects of Quantum Conf inement:Energies are quantized & changes from a

traveling wave to a standing wave.

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In 3Dimensions

For the 3D infinite potential well:

R

Real Quantum Structures arent this simple!!

In Superlattices & Quantum Wells,the potential barrier is

obviously not infinite! In Quantum Dots, there is usually ~ spherical confinement,

not rectangular.

The simple problem only considers a single electron. But, inreal

structures, there are many electrons& alsoholes! Also, there is oftenan effective mass mismatchat the boundaries.

That isthe boundary conditions weve used are too simple!

integerqm,n,,)sin()sin()sin(~),,( zyx L

zq

L

ym

Lxnzyx

2

22

2

22

2

22

888levelsEnergy

zyx mL

hq

mL

hm

mL

hn

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QM Review: The 1d (finite) Rectangular Potential WellIn most QM texts!! Analogous to a Quantum Well

We want to solve the Schrdinger Equation for:

[-{2/(2mo)}(d2/dx2) + V] = (E)

V = 0, -(b/2) < x < (b/2); V = Vo otherwise

We want bound

states: < Vo

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Solve the Schrdinger Equation:

[-{2/(2mo)}(d2/dx2) + V] =

(E) V = 0, -(b/2) < x < (b/2)V = Vo otherwise

Bound states are in Region I I

Region II:(x) is oscillatory

Regions I & III:(x) is decaying

-()b()b

Vo

V= 0

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The 1d (finite) rectangular potential wellA brief math summary!

Define: 2 (2mo)/(2); 2 [2mo( - Vo)]/(2)The Schrdinger Equation becomes:

(d2/dx2) + 2 = 0, -()b < x < ()b

(d2/dx2) - 2 = 0, otherwise.

Solutions:

= C exp(ix) + D exp(-ix), -()b < x < ()b

= A exp(x), x < -()b

= A exp(-x), x > ()b

Boundary Conditions:

& d/dx are continuous SO:

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Algebra (2 pages!)leads to:

(/Vo) = (22)/(2moVo)

, , are related to each other by transcendental equations.For example:

tan(b) = (2)/(2- 2) Solve graphically or numerically.

Get:Discrete Energy Levelsin the well(a finite number of finite well levels!)

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Even eigenfunctionsolutions (a finite number):

Circle,2 + 2 = 2, crosses = tan()

Vo

o

o

b

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Odd eigenfunction solutions:

Circle,2 + 2 = 2, crosses = - cot()

|E2| < |E1|

b

b

o

oVo

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Quantum Confinement in Nanostructures

Confined in:

1 Direction: Quantum well (thin film)Two-dimensional electrons

2 Directions: Quantum wire

One-dimensional electrons

3 Directions: Quantum dot

Zero-dimensional electrons

Each confinement direction converts a continuous k in a discrete quantum number n.

kx

nz

ny

ny

nz

nx

kx

ky

nz

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N atomic layers with the spacing a = d/n

N quantized states with kn n/d (n=1,,N)

Quantization in a Thin Crystal

An energy band with continuous k

is quantized into N discrete points kn

in a thin film with N atomic layers.

n = 2d/n

kn = 2

/

n = n/d

d

E

0 /a/d

EFermi

EVacuum

Photoemission

Inverse

Photoemission

ElectronScattering

k= zone

boundary

Q i i i Thi G hi Fil

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N atomic layers with spacing a = d/n :

N quantized states with kn N /d

Quantization in Thin Graphite Films

E

0 /a/d

EFermi

EVacuum

Photoemission

Lect.7b,

Slide11

k

1 layer =graphene

2 layers

3 layers

4 layers

layers= graphite

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Quantum Well States

in Thin Films

discretefor small N

becomingcontinuousfor N

Paggel et al.

Science 283, 1709 (1999)

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10

16

16

16

16

16

16

13

14

14

11.5

13

14

13

14

h(eV)Ag/Fe(100)

Binding Energy (eV)

012

PhotoemissionInten

sity(arb.units)

1

2

3

4

5

6

7

8

9

10

11

13

14

15

12

N

1

3

2

4

10

16

16

16

16

16

16

13

14

14

11.5

13

14

13

14

h(eV)Ag/Fe(100)

Binding Energy (eV)

012

PhotoemissionInten

sity(arb.units)

1

2

3

4

5

6

7

8

9

10

11

13

14

15

12

N

1

3

2

4

Periodic Fermi level crossing

of quantum well states with

increasing thickness

Counting Quantum Well States

Number of monolayers N

n

BindingEnergy(eV

)

0

1

2

1 2 3 4 5

6

7

8

(a) Quantum Well States for Ag/Fe(100)

n

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Kawakami et al.

Nature 398, 132 (1999)

Himpsel

Science 283, 1655 (1999)

Quantum Well Oscillations in Electron Interferometers

Fabry-Perot interferometer model: Interfaces act like mirrors for electrons. Sinceelectrons have so short wavelengths, the interfaces need to be atomically precise.

n

12

34

56

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The Important Electrons in a Metal

Energy EFermi

Energy Spread 3.5 kBT

Transport (conductivity, magnetoresistance, screening length, ...)

Width of the Fermi function:

FWHM 3.5 kBT

Phase transitions (superconductivity, magnetism, ...)Superconducting gap:

Eg 3.5 kBTc (Tc= critical temperature)

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Energy Bands of Ferromagnets

States near the Fermi level cause

the energy splitting betweenmajority and minority spin bands

in a ferromagnet (red and green).-10

-8

-6

-4

-2

0

2

4

XK

Ni

EnergyRelativetoEF

[eV]

0.7 0.9 1.1

k|| along [011] [-1 ]

Calculation Photoemission data

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(Qiu, et al.PR B 92)

Quantum Well States and Magnetic Coupling

The magnetic coupling between layers plays a key role in giant magnetoresistance(GMR), the Nobel prize winning technology used for reading heads of hard disks.

This coupling oscillates in sync with the density of states at the Fermi level.

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Filtering mechanisms

Interface: Spin-dependent Reflectivity Quantum Well States

Bulk: Spin-dependent Mean Free Path Magnetic Doping

Parallel Spin Filters Resistance Low

OpposingSpin Filters Resistance High

Giant Magnetoresistance and Spin - Dependent Scattering

M t l t i

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Giant Magnetoresistance (GMR):(Metal spacer, here Cu)

Tunnel Magnetoresistance (TMR):(Insulating spacer, MgO)

Magnetoelectronics

Spin currents instead of charge currents

Magnetoresistance = Change ofthe resistance in a magnetic field

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Quantum Wells, Nanowires, and Nanodots

ELEC 7970 Special Topics on

Nanoscale Science and Technology

Summer 2003

Y. Tzeng

ECE

Auburn University

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Quantum confinement

Trap particles and restrict their motion

Quantum confinement produces new materialbehavior/phenomena

Engineer confinement- control for specificapplications

Structures

(Scientific American)

Quantum dots (0-D) onlyconfined states, and no freely

moving onesNanowires (1-D) particles travelonly along the wire

Quantum wells (2-D) confinesparticles within a thin layer

http://www.me.berkeley.edu/nti/englander1.ppthttp://phys.educ.ksu.edu/vqm/index.html

http://phys.educ.ksu.edu/vqm/index.htmlhttp://phys.educ.ksu.edu/vqm/index.html

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Figure 11: Energy-band profile of a structure containing three quantumwells, showing the confined states in each well. The structure consists ofGaAs wells of thickness 11, 8, and 5 nm in Al0.4 Ga0.6 As barrier layers.

The gaps in the lines indicating the confined state energies show the

locations of nodes of the corresponding wavefunctions.

Quantum well heterostructures are key components of manyoptoelectronic devices, because they can increase the strength of electro-

optical interactions by confining the carriers to small regions. They are

also used to confine electrons in 2-D conduction sheets where electron

scattering by impurities is minimized to achieve high electron mobility

and therefore high speed electronic operation.

http://www.utdallas.edu/~frensley/technical/hetphys/node11.html#SECTION00050000000000000000

http://www.utdallas.edu/~frensley/technical/hetphys/hetphys.html

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http://www.eps12.kfki.hu/files/WoggonEPSp.pdf

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http://www.evidenttech.com/pdf/wp_biothreat.pdf

http://www.evidenttech.com/why_nano/why_nano.php

February 2003

http://www.evidenttech.com/pdf/wp_biothreat.pdfhttp://www.evidenttech.com/why_nano/why_nano.phphttp://www.evidenttech.com/why_nano/why_nano.phphttp://www.evidenttech.com/pdf/wp_biothreat.pdf

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February 2003The Industrial Physicist Magazine

Quantum Dots for SaleNearly 20 years after their discovery,

semiconductor quantum dots are emerging as a bona fideindustry with a few start-up companies poised to introduceproducts this year. Initially targeted at biotechnologyapplications, such as biological reagents and cellular imaging,quantum dots are being eyed by producers for eventual use inlight-emitting diodes (LEDs), lasers, and telecommunication

devices such as optical amplifiers and waveguides. The strongcommercial interest has renewed fundamental research anddirected it to achieving better control of quantum dot self-assembly in hopes of one day using these unique materials forquantum computing.Semiconductor quantum dots combine many of the properties

of atoms, such as discrete energy spectra, with the capability ofbeing easily embedded in solid-state systems. "Everywhere yousee semiconductors used today, you could use semiconductingquantum dots," says Clint Ballinger, chief executive officer ofEvident Technologies, a small start-up company based in Troy,New York...

http://www.evidenttech.com/news/news.php

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EviArrayCapitalizing on the distinctive properties of

EviDots, we have devised a unique and

patented microarray assembly. TheEviArray is fabricated with nanocrystal

tagged oligonucleotideprobes that are also

attached to a fixed

substrate in such away that the

nanocrystals can

only fluoresce when

the DNA probecouples with the

corresponding target

genetic sequence.

http://www.evidenttech.com/why_nano/docs.php

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EviDots - Semiconductor nanocrystalsEviFluors- Biologically functionalized EviDotsEviProbes- Oligonucleotides with EviDotsEviArrays- EviProbe-based assay system

Optical Transistor- All optical 1 picosecond performanceTelecommunications- Optical Switching based on EviDotsEnergy and Lighting- Tunable bandgap semiconductor

http://www.evidenttech.com/products/core_evidots/overview.phphttp://www.evidenttech.com/products/evifluors.phphttp://www.evidenttech.com/applications/eviprobe.phphttp://www.evidenttech.com/applications/eviarray.phphttp://www.evidenttech.com/applications/eviarray.phphttp://www.evidenttech.com/applications/eviprobe.phphttp://www.evidenttech.com/products/evifluors.phphttp://www.evidenttech.com/products/core_evidots/overview.php

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Why nanowires?

They represent the smallest dimension forefficient transport of electrons and excitons, andthus will be used as interconnects and criticaldevices in nanoelectronics and nano-

optoelectronics. (CM Lieber, Harvard)

General attributes & desired properties

Diameter 10s of nanometers

Single crystal formation -- common crystallographic orientation alongthe nanowire axis

Minimal defects within wire

Minimal irregularities within nanowire arrays

http://www.me.berkeley.edu/nti/englander1.ppt

N i f b i ti

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Nanowire fabrication

Challenging!

Template assistance

Electrochemical depositionEnsures fabrication of electrically continuous wires

since only takes place on conductive surfaces

Applicable to a wide range of materials

High pressure injectionLimited to elements and heterogeneously-melting

compounds with low melting points

Does not ensure continuous wires

Does not work well for diameters < 30-40 nm

CVD

Laser assisted techniques http://www.me.berkeley.edu/nti/englander1.ppt

M ti i

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Magnetic nanowires

Important for storage device applications

Cobalt, gold, copper and cobalt-coppernanowire arrays have been fabricated

Electrochemical deposition is prevalent

fabrication technique

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Silicon nanowire CVD growth techniques

With Fe/SiO2 gel template (Liu et al,

2001)Mixture of 10 sccm SiH4 & 100 sccm

helium, 5000C, 360 Torr and depositiontime of 2h

Straight wires w/ diameter ~ 20nm and

length ~ 1mm With Au-Pd islands (Liu et al, 2001)

Mixture of 10 sccm SiH4 & 100 sccmhelium, 8000C, 150 Torr and depositiontime of 1h

Amorphous Si nanowiresDecreasing catalyst size seems to

improve nanowire alignmentBifurcation is common

30-40 nm diameter and length ~ 2mmhttp://www.me.berkeley.edu/nti/englander1.ppt

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Template assisted nanowire growth

Create a template for nanowires to grow within

Based on aluminums unique property of selforganized pore arrays as a result of anodization to

form alumina (Al2O3)Very high aspect ratios may be achieved

Pore diameter and pore packing densities are afunction of acid strength and voltage inanodization step

Pore filling nanowire formation via variousphysical and chemical deposition methods

http://www.me.berkeley.edu/nti/englander1.ppt

Al O t l t ti

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Anodization of aluminum

Start with uniform layer of ~1mm Al

Al serves as the anode, Pt may serve as the cathode, and0.3M oxalic acid is the electrolytic solution

Low temperature process (2-50C)

40V is applied

Anodization time is a function of sample size and distancebetween anode and cathode

Key Attributes of the process (per M. Sander)

Pore ordering increases with template thickness pores are

more ordered on bottom of template Process always results in nearly uniform diameter pore, but

not always ordered pore arrangement

Aspect ratios are reduced when process is performedwhen in contact with substrate (template is ~0.3-3 mm

thick)

Al2O3 template preparation

http://www.me.berkeley.edu/nti/englander1.ppt

Th l i (Al O ) t l t

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(T. Sands/ HEMI group http://www.mse.berkeley.edu/groups/Sands/HEMI/nanoTE.html)

The alumina (Al2O3) template

100nmSi substrate

alumina template

(M. Sander)

http://www.me.berkeley.edu/nti/englander1.ppt

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Works well with thermoelectric materials and

metals Process allows to remove/dissolve oxide barrier

layer so that pores are in contact with substrate

Filling rates of up to 90% have been achieved

(T. Sands/ HEMI group http://www.mse.berkeley.edu/groups/Sands/HEMI/nanoTE.html

Bi2Te3 nanowire

unfilled pore

alumina template

Electrochemical deposition

http://www.me.berkeley.edu/nti/englander1.ppt

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Template-assisted, Au nucleated Si nanowires

Gold evaporated (Au nanodots) into thin

~200nm alumina template on silicon substrate Ideally reaction with silane will yield desired

results

Need to identify equipment that will support this

process contamination, temp and press issuesAdditional concerns include Au thickness, Au on

alumina surface, template intact vs removed

100nm

1m

Au dots

template (top)

Au

(M. Sander)http://www.me.berkeley.edu/nti/englander1.ppt

N b lli l d

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Nanometer gap between metallic electrodes

Electromigration caused by electrical current flowing througha gold nanowire yields two stable metallic electrodes

separated by about 1nm with high efficiency. The gold

nanowire was fabricated by electron-beam lithography and

shadow evaporation.

Before breaking

After breaking

http://www.lassp.cornell.edu/lassp_data/mceuen/homepage/Publications/EMPaper.pdf

SET with a 5nm CdSe nanocrystal

Quantum and localization of nanowire conductance

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Nanoscale size exhibits the following properties different from thosefound in the bulk:

quantized conductance in point contacts and narrow channelswhose characteristics (transverse) dimensions approach theelectronic wave length

Localization phenomena in low dimensional systems

Mechanical properties characterized by a reduced propensity forcreation and propagation of dislocations in small metallic samples.

Conductance of nanowires depend on

the length,

lateral dimensions,

state and degree of disorder and

elongation mechanism of the wire.

Q

http://dochost.rz.hu-berlin.de/conferences/conf1/PDF/Pascual.pdf

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Conductance during elongation ofshort wires exhibits periodicquantization steps with characteristicdips, correlating with the order-disorder states of layers of atoms inthe wire.

The resistance of long wires, as

long as 100-400 A exhibitslocalization characterization withln R(L) ~ L2

Short nanowire Long nanowire

http://dochost.rz.hu-berlin.de/conferences/conf1/PDF/Pascual.pdf

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Electron localization

At low temperatures, the resistivity of a metal is dominated by the

elastic scattering of electrons by impurities in the system. If we treat the

electrons as classical particles, we would expect their trajectories to

resemble random walks after many collisions, i.e., their motion is diffusive

when observed over length scales much greater than the mean free path.

This diffusion becomes slower with increasing disorder, and can be

measured directly as a decrease in the electrical conductance.

When the scattering is so frequent that the distance travelled by

the electron between collisions is comparable to its wavelength, quantum

interference becomes important. Quantum interference between different

scattering paths has a drastic effect on electronic motion: the electron

wavefunctions are localizedinside the sample so that the system becomesan insulator. This mechanism (Anderson localization) is quite differentfrom that of a band insulator for which the absence of conduction is due to

the lack of any electronic states at the Fermi level.

http://www.cmth.ph.ic.ac.uk/derek/research/loc.html

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Resistivity of ErSi2 Nanowires on Silicon

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ErSi2 nanowires on a clean surfaceof Si(001). Resistance of nanowire vs its length.

ErSi2 nanowire self-assembled along a axis of the Si(001)

substrate, having sizes of 1-5nm, 1-2nm and

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Last stages of the

contact breakage during

the formation of

nanocontacts.

Electronic conductance through nanometer-sized systems is quantized when its

constriction varies, being the quantum of conductance, Go=2 e2/h,

where e is the electron charge and h is the Planck constant, due to the changeof the number of electronic levels in the constriction.

The contact of two gold wire can form a small contact resulting in a relative

low number of eigenstates through which the electronic ballistictransport takes place.

Conductance current during the

breakage of a nanocontact.Voltage difference between

electrodes is 90.4 mV

http://physics.arizona.edu/~stafford/costa-kraemer.pdf

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Setup for conductance quantization

studies in liquid metals. A micrometric

screw is used to control the tip

displacement.

Evolution of the current and conductance at the

first stages of the formation of a liquid metal

contact. The contact forms between a copper

wire and (a) mercury (at RT) and (b) liquid tin

(at 300C). The applied bias voltage between tip

and the metallic liquid reservoir is 90.4 mV.http://physics.arizona.edu/~stafford/costa-kraemer.pdf

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Conductance transitions due to

mechanical instabilities for goldnanocontacts in UHV at RT:

Transition from nine to five and to

seven quantum channels.

Conductance transitions due to

mechanical instabilities for goldnanocontacts in UHV at RT: (a)

between 0 and 1 quantum channel. (b)

between 0 and 2 quantum channels.

http://physics arizona edu/ stafford/costa kraemer pdf