Quantum Confinement
BW, Chs. 15-18, YC, Ch. 9; S, Ch. 14; outside sources
Overview of Quantum Confinement
History: 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: Overview
Electrons Confined in 1 Direction:
Quantum Wells (thin films):
 Electrons can easily move in
2 Dimensions!
ky
kx
nz
Electrons Confined in 2 Directions:
Quantum Wires:
ny
 Electrons can easily move in
1 Dimension!
kx
2 Dimensional nz
Quantization!
Electrons Confined in 3 Directions:
Quantum Dots:
 Electrons can easily move in
0 Dimensions!
1 Dimensional
Quantization!
nz
nx
3 Dimensional
Quantization!
ny
Each further confinement direction changes a continuous k component
to a discrete component characterized by a quantum number n.
• 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)
Primary Qualitative Effects of Quantum Confinement
• 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.
Quantum Confinement Terminology
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.
Brief Elementary Quantum Mechanics &
Solid State Physics Review
• 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 eikr (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) = eikr unk(r); unk(r) = unk(r+R)
Some Basic Physics
• Density of states (DoS)
dN dN dk
DoS 

dE dk dE
in 3D:
k space vol
N (k ) 
vol per state
4 3k 3

(2 )3 V
dN
dE
Structure
Degree of
Confinement
Bulk Material
0D
E
Quantum Well
1D
1
Quantum Wire
2D
1/ E
Quantum Dot
3D
d(E)
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(nx/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:
 ( x, y, z ) ~ sin(
nx
Lx
) sin(
Energy levels 
my
Ly
) sin(
n 2 hR2
8 mLx 2

qz
Lz
m2h 2
8 mLy 2
), n, m, q  integer
q 2h2
 8mL 2
z
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!
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:
We want bound
states: ε < Vo
[-{ħ2/(2mo)}(d2/dx2) + V]ψ = εψ (ε  E)
V = 0, -(b/2) < x < (b/2);
V = Vo otherwise
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
-(½)b
(½)b
Vo
V= 0
Region II:
ψ(x) is oscillatory
Regions I & III:
ψ(x) is decaying
The 1d (finite) rectangular potential well
A 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)
Vo
b o
o
b
|E2| < |E1|
Quantum Confinement in Nanostructures
Confined in:
1 Direction: Quantum well (thin film)
Two-dimensional electrons
ky
kx
nz
2 Directions: Quantum wire
ny
One-dimensional electrons
kx
nz
nz
3 Directions: Quantum dot
Zero-dimensional electrons
ny
nx
Each confinement direction converts a continuous k in a discrete quantum number n.
Quantization in a Thin Crystal
E
An energy band with continuous k
is quantized into N discrete points kn
in a thin film with N atomic layers.
Electron
Scattering
EVacuum
Inverse
Photoemission
EFermi
Photoemission
0 /d
k
/a
d
= zone
boundary
N atomic layers with the spacing a = d/n
n = 2d / n
N quantized states with kn ≈ n  /d ( n = 1,…,N )
kn = 2 / n = n  /d
Quantization in Thin Graphite Films
E
Lect. 7b,
Slide 11
1 layer =
graphene
2 layers
EVacuum
EFermi
3 layers
Photoemission
0 /d
k
/a
4 layers
N atomic layers with spacing a = d/n :
 N quantized states with kn ≈ N  /d
 layers
= graphite
Quantum Well States
in Thin Films
becoming
continuous
for N  
discrete
for small N
Paggel et al.
Science 283, 1709 (1999)
n
hAg/Fe(100)
 (eV)
4
13
3
0.3
14
14
13
13
12
2
14
16
11
10
9
13
8
16
16
16
7
1
14
0.2
6
5
4
16
16
10
3
0
Binding Energy (eV)
1
2
3
4
5
6
1
7
8
(b)
ction (eV)
1
0
2
0.1
2
1
Periodic Fermi level crossing
of quantum well states with
increasing thickness
(a) Quantum Well States for Ag/Fe(100)
Binding Energy (eV)
14
2
(N, n')
(2, 1)
(3, 1)
(7, 2)
(12, 3)
(13, 3)
N
15
Line Width (eV)
Photoemission Intensity (arb. units)
11.5
Counting Quantum Well States
100 4.4 200
Number of monolayers N
300
Temperature (K)
n
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
6
2
3
4
5
1
Himpsel
Science 283, 1655 (1999)
Kawakami et al.
Nature 398, 132 (1999)
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
Calculation
Ni
Energy Relative to EF [eV]
4
Photoemission data
2
0
-2
0.7
0.9
k||
along [011]
1.1
[Å-1
]
-4
States near the Fermi level cause
-6
the energy splitting between
majority and minority spin bands
-8
in a ferromagnet (red and green).
-10

K
X
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.
(Qiu, et al.
PR B ‘92)
Spin-Polarized Quantum Well States
Magnetic interfaces reflect the two spins differently, causing a spin polarization.
Minority spins discrete,
Majority spins continuous
Giant Magnetoresistance and Spin - Dependent Scattering
Parallel Spin Filters  Resistance Low
Opposing Spin Filters  Resistance High
Filtering mechanisms
• Interface: Spin-dependent Reflectivity  Quantum Well States
• Bulk: Spin-dependent Mean Free Path  Magnetic “Doping”
Magnetoelectronics
Spin currents instead of charge currents
Magnetoresistance = Change of
the resistance in a magnetic field
Giant Magnetoresistance (GMR):
(Metal spacer, here Cu)
Tunnel Magnetoresistance (TMR):
(Insulating spacer, MgO)
ELEC 7970 Special Topics on
Nanoscale Science and Technology
Quantum Wells, Nanowires, and Nanodots
Summer 2003
Y. Tzeng
ECE
Auburn University
Quantum confinement
 Trap particles and restrict their motion
 Quantum confinement produces new material
behavior/phenomena
 “Engineer confinement”- control for specific
applications
 Structures
Quantum dots (0-D) only
confined states, and no freely
moving ones
Nanowires (1-D) particles travel
only along the wire
Quantum wells (2-D) confines
particles within a thin layer
http://www.me.berkeley.edu/nti/englander1.ppt
http://phys.educ.ksu.edu/vqm/index.html
(Scientific American)
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 electrooptical 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 2003
The 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 selfassembly 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 Sale
The Industrial Physicist
reports that quantum dots
are emerging as a bona
fide industry.
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
Evident Nanocrystals
Evident'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.
http://www.evidenttech.com/why_nano/docs.php
http://www.evidenttech.com/index.php
EviArray
Capitalizing on the distinctive properties of
EviDots™, we have devised a unique and
patented microarray assembly. The
EviArray™ is fabricated with nanocrystal
tagged oligonucleotide
probes 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 nanocrystals
EviFluors- Biologically functionalized EviDots
EviProbes- Oligonucleotides with EviDots
EviArrays- EviProbe-based assay system
Optical Transistor- All optical 1 picosecond performance
Telecommunications- Optical Switching based on EviDots
Energy 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 nanooptoelectronics.” (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 wires
 Does 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 ~ 1mm
 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 nanowires
 Decreasing catalyst size seems to
improve nanowire alignment
 Bifurcation is common
 30-40 nm diameter and length ~ 2mm
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 achieved
 Pore 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 methods
http://www.me.berkeley.edu/nti/englander1.ppt
Al2O3 template preparation
 Anodization of aluminum
 Start with uniform layer of ~1mm 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 mm
http://www.me.berkeley.edu/nti/englander1.ppt
thick)
The alumina (Al2O3) template
(T. Sands/ HEMI group http://www.mse.berkeley.edu/groups/Sands/HEMI/nanoTE.html)
alumina template
Si substrate
100nm
http://www.me.berkeley.edu/nti/englander1.ppt
(M. Sander)
Electrochemical deposition
 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
Bi2Te3 nanowire
unfilled pore
alumina template
http://www.me.berkeley.edu/nti/englander1.ppt
(T. Sands/ HEMI group http://www.mse.berkeley.edu/groups/Sands/HEMI/nanoTE.html
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
Au dots
Au
100nm
1µm
http://www.me.berkeley.edu/nti/englander1.ppt
(M. Sander)
template (top)
Nanometer gap between metallic electrodes
Before breaking
After breaking
SET with a 5nm CdSe nanocrystal
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.
http://www.lassp.cornell.edu/lassp_data/mceuen/homepage/Publications/EMPaper.pdf
Quantum and localization of nanowire conductance
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.
http://dochost.rz.hu-berlin.de/conferences/conf1/PDF/Pascual.pdf
Short nanowire
Conductance during elongation of
short wires exhibits periodic
quantization steps with characteristic
dips, correlating with the orderdisorder states of layers of atoms in
the wire.
“Long” nanowire
The resistance of “long” wires, as
long as 100-400 A exhibits
localization characterization with
ln R(L) ~ L2
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
Resistivity of ErSi2 Nanowires on Silicon
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 1cm, which is 4
orders of magnitude larger than that for known resistivity of bulk
ErSi2, i.e., 35 mcm. One of the reasons may be due to an elasticallyelongated 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
Last stages of the
contact breakage during
the formation of
nanocontacts.
Conductance current during the
breakage of a nanocontact.
Voltage difference between
electrodes is 90.4 mV
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.
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.
http://physics.arizona.edu/~stafford/costa-kraemer.pdf
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.
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
Conductance transitions due to
mechanical instabilities for gold
nanocontacts in UHV at RT:
Transition from nine to five and to
seven quantum channels.