Quantum ConfinementBW, Chs. 15-18, YC, Ch. 9; S, Ch. 14; outside sources

Overview of Quantum ConfinementHistory: In 1970 Esaki & Tsu proposed fabrication of an artificial

structure, which would consist of alternating layers of 2 different semiconductors with

Layer Thickness

1 nm = 10 Å = 10-9 m SUPERLATTICE• PHYSICS: The main idea was that introduction of an artificial

periodicity will “fold” the Brillouin Zones into smaller BZ’s “mini-zones”.

The idea was that this would raise the conduction band

minima, which was needed for some device applications.

• Modern growth techniques (starting in the 1980’s), especially MBE & MOCVD, make fabrication of such structures possible!

• For the same reason, it is also possible to fabricate many other kinds of artificial structures on the scale of nm

(nanometers) “Nanostructures”

Superlattices = “2 dimensional” structures Quantum Wells = “2 dimensional” structures Quantum Wires = “1 dimensional” structures Quantum Dots = “0 dimensional” structures!!

• Clearly, it is not only the electronic properties of materials which can be drastically altered in this way. Also, vibrational properties (phonons). Here, only electronic properties & only an overview!

• For many years, quantum confinement has been a fast growing field in both theory & experiment! It is at the forefront of current research!

• Note that I am not an expert on it!

Quantum Confinement in Nanostructures: OverviewElectrons Confined in 1 Direction: Quantum Wells (thin films): Electrons can easily move in 2 Dimensions!

Electrons Confined in 2 Directions: Quantum Wires: Electrons can easily move in 1 Dimension!

Electrons Confined in 3 Directions: Quantum Dots: Electrons can easily move in 0 Dimensions!

Each further confinement direction changes a continuous k component to a discrete component characterized by a quantum number n.

kx

nz

ny

ny

nz

nx

kx

ky

nz1 Dimensional Quantization!

2 Dimensional Quantization!

3 Dimensional Quantization!

• PHYSICS: Back to the bandstructure chapter:

– Consider the 1st Brillouin Zone for the infinite crystal. The maximum wavevectors are of the order

km (/a)

a = lattice constant. The potential V is periodic with period a. In the almost free e- approximation, the bands are free e- like except near the Brillouin Zone edge. That is, they are of the form:

E (k)2/(2mo) So, the energy at the Brillouin Zone edge has the form:

Em (km)2/(2mo) or

Em ()2/(2moa2)

PHYSICS

• SUPERLATTICES Alternating layers of material. Periodic, with periodicity L (layer thickness). Let kz = wavevector perpendicular to the layers.

• In a superlattice, the potential V has a new periodicity in the z direction with periodicity L >> a

In the z direction, the Brillouin Zone is much smaller than that for an infinite

crystal. The maximum wavevectors are of the order: ks (/L)

At the BZ edge in the z direction, the energy has the form:

Es ()2/(2moL2) + E2(k)

E2(k) = the 2 dimensional energy for k in the x,y plane.

Note that: ()2/(2moL2) << ()2/(2moa2)

• Consider electrons confined along 1 direction (say, z) to a layer of width L:

Energies• The energy bands are quantized (instead of continuous) in kz &

shifted upward. So kz is quantized:

kz = kn = [(n)/L], n = 1, 2, 3

• So, in the effective mass approximation (m*), the bottom of the conduction band is quantized (like a particle in a 1 d box) & shifted:

En = (n)2/(2m*L2)

• Energies are quantized! Also, the wavefunctions are 2 dimensional Bloch functions (traveling waves) for k in the x,y plane & standing waves in the z direction.

Primary Qualitative Effects of Quantum Confinement

Quantum Well QW = A single layer of material A (layer thickness L), sandwiched between 2 macroscopically 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 >> L So, the e- & e+ in one A layer are independent of those in other A layers.

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

Quantum Confinement Terminology

• Quantum Mechanics of a Free Electron:

– The energies are continuous: E = (k)2/(2mo) (1d, 2d, or 3d)

– The wavefunctions are traveling waves:

ψk(x) = A eikx (1d) ψk(r) = A eikr (2d or 3d)

• Solid State Physics: Quantum Mechanics of an Electron in a Periodic Potential in an infinite crystal :

– The energy bands are (approximately) continuous: E= Enk

– At the bottom of the conduction band or the top of the valence band, in the effective mass approximation, the bands can be written:

Enk (k)2/(2m*)

– The wavefunctions are Bloch Functions = traveling waves:

Ψnk(r) = eikr unk(r); unk(r) = unk(r+R)

Brief Elementary Quantum Mechanics & Solid State Physics Review

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 (E)

dE

dN

E

E1/

dE

dk

dk

dN

dE

dNDoS

V

k

kN

3

3

)2(

34

stateper vol

volspacek )(

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

• We want to solve the Schrödinger Equation for:

x < 0, V ; 0 < x < L, V = 0; x > L, V

-[2/(2mo)](d2 ψ/dx2) = Eψ• Boundary Conditions:

ψ = 0 at x = 0 & x = L (V there)

• Energies:

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

Wavefunctions:

ψn(x) = (2/L)½sin(nx/L) (a standing wave!) Qualitative Effects of Quantum Confinement:

Energies are quantized & ψ changes from a traveling wave to a standing wave.

In 3Dimensions…• For the 3D infinite potential well:

R

Real Quantum Structures aren’t 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, in real structures, there are many electrons & also holes!

• Also, there is often an effective mass mismatch at the boundaries. That is the boundary conditions we’ve used are too simple!

integer qm,n, ,)sin()sin()sin(~),,( zyx Lzq

Lym

Lxnzyx

2

22

2

22

2

22

888levelsEnergy

zyx mL

hq

mLhm

mLhn

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

• We want to solve the Schrödinger Equation for:

[-{ħ2/(2mo)}(d2/dx2) + V]ψ = εψ (ε E)

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

We want bound states: ε < Vo

Solve the Schrödinger Equation:

[-{ħ2/(2mo)}(d2/dx2) + V]ψ = εψ

(ε E) V = 0, -(b/2) < x < (b/2)

V = Vo otherwise

Bound states are in Region II

Region II:

ψ(x) is oscillatory

Regions I & III:

ψ(x) is decaying

-(½)b (½)b

Vo

V = 0

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

Define: α2 (2moε)/(ħ2); β2 [2mo(ε - Vo)]/(ħ2)

The Schrödinger Equation becomes:

(d2/dx2) ψ + α2ψ = 0, -(½)b < x < (½)b

(d2/dx2) ψ - β2ψ = 0, otherwise.

Solutions:

ψ = C exp(iαx) + D exp(-iαx), -(½)b < x < (½)b

ψ = A exp(βx), x < -(½)b

ψ = A exp(-βx), x > (½)b

Boundary Conditions:

ψ & dψ/dx are continuous SO:

• Algebra (2 pages!) leads to:

(ε/Vo) = (ħ2α2)/(2moVo)

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

tan(αb) = (2αβ)/(α 2- β2)• Solve graphically or numerically.

• Get: Discrete Energy Levels in the well (a finite number of finite well levels!)

• Even eigenfunction solutions (a finite number):

Circle, ξ2 + η2 = ρ2, crosses η = ξ tan(ξ)

Vo

o

o

b

• Odd eigenfunction solutions:

Circle, ξ2 + η2 = ρ2, crosses η = -ξ cot(ξ)

|E2| < |E1|

b

b

o

oVo

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

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

Electron Scattering

k= zone

boundary

N atomic layers with spacing a = d/n :

N quantized states with kn ≈ N /d

Quantization in Thin Graphite FilmsE

0 /a/d

EFermi

EVacuum

Photoemission

Lect. 7b, Slide 11

k

1 layer = graphene

2 layers

3 layers

4 layers

layers = graphite

Quantum Well States

in Thin Films

discrete for small N

becoming continuous for N

Paggel et al.Science 283, 1709 (1999)

10

16

16

16

16

16

16

13

14

14

11.5

13

1413

14

h (eV)Ag/Fe(100)

Binding Energy (eV)

012

Photo

emiss

ion In

tensity

(arb.

units)

1

2

3

4

5

6

7

8

9

10

11

13

14

15

12

N

Temperature (K)

100 200 300

Line W

idth (e

V)

0.1

0.2

0.3

(N, n')

(3, 1)(7, 2)(12, 3)(13, 3)

(2, 1)

1

3

2

4

10

16

16

16

16

16

16

13

14

14

11.5

13

1413

14

h (eV)Ag/Fe(100)

Binding Energy (eV)

012

Photo

emiss

ion In

tensity

(arb.

units)

1

2

3

4

5

6

7

8

9

10

11

13

14

15

12

N

Temperature (K)

100 200 300

Line W

idth (e

V)

0.1

0.2

0.3

(N, n')

(3, 1)(7, 2)(12, 3)(13, 3)

(2, 1)

1

3

2

4

Periodic Fermi level crossing of quantum well states with

increasing thickness

Counting Quantum Well States

Number of monolayers N

n

Thickness N (ML)0510152025Wo

rk F

un

ctio

n (

eV)

4.34.4

Bin

din

g E

ner

gy

(eV

) 01212345678(a) Quantum Well States for Ag/Fe(100)

(b)n

Kawakami et al.Nature 398, 132 (1999)

HimpselScience 283, 1655 (1999)

Quantum Well Oscillations in Electron Interferometers

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

n

12

34

56

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)

Energy Bands of Ferromagnets

States near the Fermi level cause

the energy splitting between

majority and minority spin bands

in a ferromagnet (red and green).-10

-8

-6

-4

-2

0

2

4

XK

Ni

En

erg

y R

ela

tiv

e t

o E

F [

eV

]

0.7 0.9 1.1

k|| along [011] [Å-1 ]

Calculation Photoemission data

(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.

Minority spins discrete,Majority spins continuous

Magnetic interfaces reflect the two spins differently, causing a spin polarization.

Spin-Polarized Quantum Well States

Filtering mechanisms

• Interface: Spin-dependent Reflectivity Quantum Well States

• Bulk: Spin-dependent Mean Free Path Magnetic “Doping”

Parallel Spin Filters Resistance Low

Opposing Spin Filters Resistance High

Giant Magnetoresistance and Spin - Dependent Scattering

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

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

Magnetoelectronics

Spin currents instead of charge currents

Magnetoresistance = Change of the resistance in a magnetic field

Quantum Wells, Nanowires, and Nanodots

ELEC 7970 Special Topics on Nanoscale Science and Technology

Summer 2003

Y. Tzeng

ECE

Auburn University

Quantum confinementTrap particles and restrict their motionQuantum confinement produces new

material behavior/phenomena “Engineer confinement”- control for

specific applicationsStructures

(Scientific American)

Quantum dots (0-D) only confined states, and no freely moving onesNanowires (1-D) particles travel only along the wireQuantum wells (2-D) confines particles within a thin layerhttp://www.me.berkeley.edu/nti/englander1.ppt

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

Figure 11: Energy-band profile of a structure containing three quantum wells, showing the confined states in each well. The structure consists of GaAs 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 many optoelectronic 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

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

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

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

February 2003The Industrial Physicist Magazine

Quantum Dots for Sale                      Nearly 20 years after their discovery, semiconductor quantum dots are emerging as a bona fide industry with a few start-up companies poised to introduce products this year. Initially targeted at biotechnology applications, such as biological reagents and cellular imaging, quantum dots are being eyed by producers for eventual use in light-emitting diodes (LEDs), lasers, and telecommunication devices such as optical amplifiers and waveguides. The strong commercial interest has renewed fundamental research and directed it to achieving better control of quantum dot self-assembly in hopes of one day using these unique materials for quantum computing. Semiconductor quantum dots combine many of the properties of atoms, such as discrete energy spectra, with the capability of being easily embedded in solid-state systems. "Everywhere you see semiconductors used today, you could use semiconducting quantum dots," says Clint Ballinger, chief executive officer of Evident Technologies, a small start-up company based in Troy, New York...

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

Quantum Dots for SaleThe Industrial Physicist reports that quantum dots are emerging as a bona fide industry.

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

Evident NanocrystalsEvident's nanocrystals can be separated from the solvent to form self-assembled thin films or combined with polymers and cast into films for use in solid-state device applications. Evident's semiconductor nanocrystals can be coupled to secondary molecules including proteins or nucleic

acids for biological assays or other applications.

Emission Peak[nm] 535±10 560±10 585±10 610±10 640±10

Typical FWHM [nm] <30 <30 <30 <30 <40

1st Exciton Peak[nm - nominal]

522 547 572 597 627

Crystal Diameter[nm - nominal]

2.8 3.4 4.0 4.7 5.6

Part Number (4ml)SG-CdSe-Na-TOL

05-535-04 05-560-04 05-585-04 05-610-04 05-640-04

Part Number (8ml)SG-CdSe-Na-TOL

05-535-08 05-560-08 05-585-08 05-610-08 05-640-08

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

EviArrayCapitalizing on the distinctive properties of EviDots™, we have devised a unique and patented microarray assembly. The EviArray™ is fabricated with nanocrystal tagged oligonucleotideprobes that are also attached to a fixed substrate in such a way that the nanocrystals can only fluoresce when the DNA probe couples with the corresponding target genetic sequence.

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

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

Why nanowires?

“They represent the smallest dimension for efficient transport of electrons and excitons, and thus will be used as interconnects and critical devices in nanoelectronics and nano-optoelectronics.” (CM Lieber, Harvard)

General attributes & desired properties Diameter – 10s of nanometers Single crystal formation -- common crystallographic orientation

along the nanowire axis Minimal defects within wire

Minimal irregularities within nanowire arrays

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

Nanowire fabrication

Challenging! Template assistance Electrochemical deposition

Ensures fabrication of electrically continuous wires since only takes place on conductive surfaces

Applicable to a wide range of materials High pressure injection

Limited to elements and heterogeneously-melting compounds with low melting points

Does not ensure continuous wiresDoes not work well for diameters < 30-40 nm

CVD Laser assisted techniques

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

Magnetic nanowires

Important for storage device applications

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

Electrochemical deposition is prevalent fabrication technique

<20 nm diameter nanowire arrays have been fabricated

Cobalt nanowires on Si substrate(UMass Amherst, 2000)

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

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 deposition time of 2h

Straight wires w/ diameter ~ 20nm and length ~ 1m

With Au-Pd islands (Liu et al, 2001)Mixture of 10 sccm SiH4 & 100

sccm helium, 8000C, 150 Torr and deposition time of 1h

Amorphous Si nanowiresDecreasing catalyst size seems to

improve nanowire alignmentBifurcation is common30-40 nm diameter and length ~

2m http://www.me.berkeley.edu/nti/englander1.ppt

Template assisted nanowire growth

Create a template for nanowires to grow within

Based on aluminum’s unique property of self organized pore arrays as a result of anodization to form alumina (Al2O3)

Very high aspect ratios may be achievedPore diameter and pore packing densities

are a function of acid strength and voltage in anodization step

Pore filling – nanowire formation via various physical and chemical deposition methodshttp://www.me.berkeley.edu/nti/englander1.ppt

Anodization of aluminum Start with uniform layer of ~1m Al Al serves as the anode, Pt may serve as the

cathode, and 0.3M oxalic acid is the electrolytic solution

Low temperature process (2-50C) 40V is applied Anodization time is a function of sample size and

distance between 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 performed when in contact with substrate (template is ~0.3-3 m thick)

Al2O3 template preparation

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

(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

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 nanowireunfilled pore

alumina template

Electrochemical deposition

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

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 issues

Additional concerns include Au thickness, Au on alumina surface, template intact vs removed

100nm1µm

Au dots

template (top)

Au

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

Nanometer gap between metallic electrodes

Electromigration caused by electrical current flowing through a 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

Nanoscale size exhibits the following properties different from those found in the bulk:

quantized conductance in point contacts and narrow channels whose characteristics (transverse) dimensions approach the electronic wave length

Localization phenomena in low dimensional systems

Mechanical properties characterized by a reduced propensity for creation 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.

Quantum and localization of nanowire conductance

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

Conductance during elongation of short wires exhibits periodic quantization steps with characteristic dips, correlating with the order-disorder states of layers of atoms in the wire.

The resistance of “long” wires, as long as 100-400 A exhibits localization characterization with ln R(L) ~ L2

Short nanowire “Long” nanowire

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

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 localized inside the sample so that the system becomes an insulator. This mechanism (Anderson localization) is quite different from 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

Molecular nanowire with negative differential resistance at room temperature

http://research.chem.psu.edu/mallouk/articles/b203047k.pdf

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

ErSi2 nanowire self-assembled along a <110> axis of the Si(001) substrate, having sizes of 1-5nm, 1-2nm and <1000nm, in width, height, and length, respectively. The resistance per unit length is 1.2M/nm

along the ErSi2 nanowire. The resistivity is around 1cm, which is 4 orders of magnitude larger than that for known resistivity of bulk ErSi2, i.e., 35 cm. One of the reasons may be due to an elastically-elongated lattice spacing along the ErSi2 nanowire as a result of lattice mismatch between the ErSi2 and Si(001) substrate.

http://www.riken.go.jp/lab-www/surf-inter/tanaka/gyouseki/ICSTM01.pdf

Resistivity of ErSi2 Nanowires on Silicon

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 change of 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 ballistic transport 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

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

Conductance transitions due to mechanical instabilities for gold nanocontacts in UHV at RT: Transition from nine to five and to seven quantum channels.

Conductance transitions due to mechanical instabilities for gold nanocontacts 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