```Lecture 19
Sensors of Structure
•
•
•
•
•
Matter Waves and the deBroglie wavelength
Heisenberg uncertainty principle
Electron diffraction
Transmission electron microscopy
Atomic-resolution sensors
Count Loius de Broglie
• Postulated that all objects have a
wavelength given by
λ=h/p
• λ=wavelength
• h=Planck’s constant
• p=momentum of object
• In practice, only really small objects
have a sensible wavelength
Wave-Particle duality
• A consequence of the
deBroglie hypothesis is
that all objects can be
thought of as “wavicles”:
both particles and waves
• This has troubled many
philosophically-minded
scientists over the years.
• Inescapable if we want
to build atomicresolution sensors.
Heisenberg’s “Uncertantity
Principle”
• Cannot simultaneously measure an
object’s momentum and position to
a better accuracy than ħ/2
Δpx Δx≥ħ/2
• Direct consequence of waveparticle duality
• Places limitations on sensor
accuracy
Electron Diffraction
• Accelerated electrons have wavelength of
order 1 Angstrom=1x10-10 m
• Same order as atomic spacing
• Electrons undergo Bragg diffraction at
atomic surfaces if the atoms are lined up in
planes, (i.e. in a crystal)
Bragg reflection
• Constructive
interference when the
path length difference
is a integer multiple of
the wavelengths
• nλ=2d sinθ
• Detailed description
requires heavy
(mathematical)
Quantum Mechanics.
Diffraction Patterns
• Only certain angles of reflection are allowed.
• The diffracted electrons form patterns.
• In polycrystalline material, these are rings
X-rays on left,
electrons
on right.
Davisson-Germer
experiment
• Application of diffraction to
measure atomic spacing
• Single crystal Ni target
• Proved deBroglie hypothesis
that λ=h/p
Proof that λ=h/p
Accelerated electrons have energy eV:
eV= ½ mv2 => v = (2Ve/m)1/2
de Broglie said:
λ=h/p=h/(mv)=h/(2mVe)1/2=1.67 Å
Davisson-Germer found lattice spacing:
λ=dsinθ=1.65 Å
Excellent agreement between theory and
experiment!
Application: Pressure sensing
• Atomic spacing changes with pressure:
Pressure=E(ΔL/L)
Where E=Young’s modulus (N/m2)
• As d (spacing between atomic planes)
changes, the angle of diffraction changes
• Diffraction rings move apart or closer
together
STM and AFM
• Electron diffraction can probe atomic
lengthscales, but
– Targets need to be crystalline
– Need accelerated electrons=>bulky and
expensive apparatus.
– Need alternatives!
http://www.personal.psu.edu/users/m/m/mmt163/E%20SC%20497E_files/Quantum_Corral.htm
Atomic level imaging and manipulation
• Scanning Tunnelling Microscopy
• Atomic Force Microscopy
Quantum Corral
• Image shows ‘Quantum corral’ of 48 Fe
atoms on a Cu surface
• Low-temp STM used for assembly and
imaging
• Can see Schrodinger standing waves
• Colors artificial
Quantum Mechanics
• STM and AFM inherently quantummechanical in operation
• Need to understand the electron
wavefunction to understand their operation
• We need some QM first
The wavefunction
• The electrons of an atom are described by
their wavefunction:
– Ψ= Ψ0ei/ħ (px-Et)
– Contains all information about electron
• Eg probability of electron being in a certain
region is P(x)=∫ Ψ*Ψdx
Schrodinger’s Eqn
• -ħ/2m d2Ψ/dx2 + U(x)Ψ=iħ dΨ/dt
• All ‘waveicles’ must obey this eqn
• U(x) is the potential well
– In the case of atoms, it can be approximated by
a square well
The square well
• Solve Schrodinger’s eqn for a potential
– U(x)=0 between x=0 and x=L
– U(x)=U0 everywhere else.
• Assume that the solutions do not vary with
time (stationary states)
– Ψ= Ψ(x)
Solutions for a square well
• Ψ(x)=Asin(n*pi*x/L) inside the well
– These are simply standing waves in a cavity,
with n denoting the mode number
• Same as solution from classical physics
Wavefunction trails
• Ψ(x)=Ae±ax outside
the well
– This is the important
bit for STM and AFM
– Means that the
wavefunction extends
beyond the atomic
surface
Tunnelling phenomena
• If another atom is brought
close enough to the first,
the wavefunction from the
first atom can overlap into
the second
• Means electron has
probablilty of being found
in second atom
• Electron has tunnelled
through the potential
barrier
In STM, a tip is brought in very close proximity to a
surface to be analysed: the electrons can tunnel from tip
to surface (or vice versa).
STM operation
Tunnelling current
very sensitive
function of
separation
Keep tip current
constant, and
measure variations
in height with a
piezoelectric
crystal
Lecture 20
Sensors of Structure
•
•
•
•
•
Matter Waves and the deBroglie wavelength
Heisenberg uncertainty principle
Electron diffraction
Transmission electron microscopy
Atomic-resolution sensors
deBroglie
• Postulated that all objects have a
wavelength given by
– λ=h/p
• λ=wavelength
• h=Planck’s constant
• p=momentum of object
• In practice, only really small objects have a
sensible wavelength
Wave-Particle duality
• A consequence of the deBroglie hypothesis is
that all objects can be thought of as
“wavicles”: both particles and waves
• This has troubled many philosophicallyminded scientists over the years.
• Inescapable if we want to build atomicresolution sensors.
Heisenberg Uncertantity Principle
• Cannot simultaneously measure an object’s
momentum and position to a better accuracy
than ħ/2
– Δpx Δx≥ħ/2
• Direct consequence of wave-particle
duality
• Places limitations on sensor accuracy
Electron Diffraction
• Accelerated electrons have wavelength of
order 1 Angstrom=1e-10m
• Same order as atomic spacing
• Electrons undergo Bragg diffraction at
atomic surfaces if the atoms are lined up in
planes, ie a crystal
Bragg reflection
• Constructive
interference when the
path length difference
is a integer multiple of
the wavelengths
• nλ=2d sinθ
• Detailed description
requires heavy
(mathematical) QM.
Diffraction Patterns
• Only certain angles of
reflection are allowed.
• The diffracted
electrons form
patterns.
• In polycrystalline
material, these are
rings
X-rays on left, electrons on right.
Davisson-Germer experiment
• Application of diffraction to measure atomic spacing
• Single crystal Ni target
• Proved deBroglie hypothesis that λ=h/p
Proof that λ=h/p
Accelerated electrons have energy eV:
eV= ½ mv2 => v = (2Ve/m)1/2
de Broglie said:
λ=h/p=h/(mv)=h/(2mVe)1/2=1.67 Å
Davisson-Germer found lattice spacing:
λ=dsinθ=1.65 Å
Excellent agreement between theory and
experiment!
Pressure sensing
• Atomic spacing changes with pressure:
– Pressure=E(ΔL/L)
• E=Youngs modulus (N/m2)
• As d changes, angle of diffraction changes
• Rings move apart or closer together
STM and AFM
• Electron diffraction can probe atomic
lengthscales, but
– Targets need to be crystalline
– Need accelerated electrons=>bulky and
expensive apparatus.
– Need alternatives!
http://www.personal.psu.edu/users/m/m/mmt163/E%20SC%20497E_files/Quantum_Corral.htm
Atomic level imaging and manipulation
• Scanning Tunnelling Microscopy
• Atomic Force Microscopy
Quantum Corral
• Image shows ‘Quantum corral’ of 48 Fe
atoms on a Cu surface
• Low-temp STM used for assembly and
imaging
• Can see Schrodinger standing waves
• Colors artificial
Quantum Mechanics
• STM and AFM inherently quantummechanical in operation
• Need to understand the electron
wavefunction to understand their operation
• We need some QM first
The wavefunction
• The electrons of an atom are described by
their wavefunction:
– Ψ= Ψ0ei/ħ (px-Et)
– Contains all information about electron
• Eg probability of electron being in a certain
region is P(x)=∫ Ψ*Ψdx
Schrodinger’s Eqn
• -ħ/2m d2Ψ/dx2 + U(x)Ψ=iħ dΨ/dt
• All ‘waveicles’ must obey this eqn
• U(x) is the potential well
– In the case of atoms, it can be approximated by
a square well
The square well
• Solve Schrodinger’s eqn for a potential
– U(x)=0 between x=0 and x=L
– U(x)=U0 everywhere else.
• Assume that the solutions do not vary with
time (stationary states)
– Ψ= Ψ(x)
Solutions for a square well
• Ψ(x)=Asin(n*pi*x/L) inside the well
– These are simply standing waves in a cavity,
with n denoting the mode number
• Same as solution from classical physics
Wavefunction trails
• Ψ(x)=Ae±ax outside
the well
– This is the important
bit for STM and AFM
– Means that the
wavefunction extends
beyond the atomic
surface
Atomic level imaging and
manipulation
• Scanning Tunnelling
Microscopy
• Atomic Force
Microscopy
T(E)exp(-2L)
Tunnelling phenomena
• If another atom is brought
close enough to the first,
the wavefunction from the
first atom can overlap into
the second
• Means the electron has
probability of being found
in second atom
• Electron has tunnelled
through the potential
barrier
Transmitted intensity
proportional to e-2s
Incident
Where  
2m(U  E )

Transmitted
Reflected
Characteristic scale of tunnelling is δ=1/
If U-E =4 eV (typical value of metal work function),
then for an electron:
δ=(1.05*10-34)/[(2(9.1*10-31)(4*1.6*10-19))]1/2
STM Principles
• “Scanning Tunnelling
Microscope”
• Tunnelling current depends
exponentially on distance from
surface
– Move tip across surface, and
the current changes as the tip
“feels” the “bumps” caused by
valence electron wave
functions
• Image shows individual atoms
of a sample of Highly Oriented
Pyrolytic Graphite.
In STM, a tip is brought in very close proximity to a
surface to be analysed: the electrons can tunnel from tip
to surface (or vice versa).
STM operation
Tunnelling current
very sensitive
function of
separation
Keep tip current
constant, and
measure variations
in height with a
piezoelectric
crystal
STM and piezos
• Piezoelectric bar
• Application of bias
causes expansion
contraction of
crystal
• Tube scanner – 2-D
scanning
0V
+V
-V
STM Operating
Modes
• Const. current mode:
– Move tip over surface
and measure changes in
height with piezo.
• Const. height mode:
– Keep tip-surface separation
const, and measure changes
in current.
– Need very flat samples to
avoid tip crash!
Raster Scanning over
area from .1X.1mm to
10X10 nm
Scan rates can be quite
fast
Resolution/scan size/scan
Scanning issues
Vacuum Operation
• Needs Ultra-High
vacuum
– Prevents unwanted
surface
– Lots of turbo pumps
and stainless steel
– Bakeout and surgical
handling procedures
Atomic manipulation using STM
• Can move or desorb atoms
as well as image.
• Desorb=unstick from
surface
• Absorb=diffuse into bulk
• Put high voltage on tip to
draw current and “arc weld
surface”
• Use small bias to pick up
atoms and assemble them
into cheesy logo
Some gratuitous STM images
• Great for grant applications
and press releases!
Nanoscale
Lithography
• Selective oxidation of
semiconductor
surfaces
• Positioning of single
atoms
The STM image
• The STM image is a file of
(x,y,height) co-ords
• It can be manipulated to
produce all sorts of images
– Fantastic colour schemes of
dubious taste
– Animation and fly-by videos
• Quantum corrals; can image ewavefunction
– See the ripples
– Spiles are Fe d-orbitals
– Yellow atoms are Cu
The Atomic Force Microscope
• Atomic Force Microscope
(AFM)
• STM measures tunelling
current; but AFM
measures van der waals
forces directly
• Van der Vaals force
attractive with
FVDW1/s7
The AFM
• Detect minute movements in cantilever by
bouncing laser off and using interferometry
(remember laser sensors)
• Photodetector measures the difference in
light intensities between the upper and
lower photodetectors, and then converts to
voltage.
• Feedback from photodiode signals, enables
the tip to maintain either a constant force or
constant height above the sample.
• Atomic resolution
• Sample need not be electrically conductive
The AFM cantilever
• Most critical component.
• Low spring constant for detection of small forces
(Hookes law F=-kx)
• High resonant frequency to minimise sensitivity to
mechanical vibrations (ωo2=k/mc)
• Small radius of curvature for good spatial
resolution
• High aspect ratio (for deep structures), can use
nanotubes
AFM
• Can get atomic scale
resolution, just like STM.
• Still needs UHV and
vibration isolation for
atomic scale resolution.
• Different Modes:
– Contact
– Non-contact (resonant
response of cantilever
monitored)
Contact mode
• Responds to short range interatomic forces
– Variable deflection imaging
• scan with no feedback, measure force changes across surface
– Constant Force imaging
• Force and cantilever deflection kept constant to image surface
topography
• Caution is required to ensure cantilever doesn’t
damage surface
Non-Contact mode
• Responds to long range
interatomic forces  greater
sensitivity required
• Instead of monitoring quasistatic cantilever deflections
measure changes in resonant
response of cantilever
• Cantilever connected to
piezoelectric element – bends
with applied potential
• Lower probability of inducing
damage to surface
piezo
ac
cantilever
• Cantilever driven close to resonant
frequency, ωo
• If cantilever has spring const, ko in absence of
surface interactions
• Then in presence of force gradient, F’=dFz/Dz
Keff=ko-F’
• This causes shift in resonant frequency i.e
ωeff2=keff/mc=(ko-F’) /mc=(ko /mc)(1- F’/ko)
ωeff=ωo (1- F’/ko)1/2
• If F’ small ωeff ~ωo (1- F’/2ko), hence a force
gradient F will shift the resonant frequency
Near field Scanning Optical Microscope
(NSOM) Combined with AFM
•Optical resolution
determined by
NSOM/AFM Probe
diffraction limit (~λ)
•Illuminating a sample
with the "near-field"
of a small light source.
• Can construct optical
images with resolution
well beyond usual
"diffraction limit",
(typically ~50 nm.)
SEM - 70nm aperture
NSOM Setup
Transmission
Ideal for thin films or coatings
which are several hundred nm
thick on transparent substrates
(e.g., a round, glass cover slip).
Photoluminescence
emsission
Sample locally illuminated with
SNOM, spectrally resolved global
photemission measured.
Lecture 21
The STM Tip
– Tunnelling current very
sensitive function of
separation
– Keep tip current
constant, and measure
variations in height
with a piezoelectric
crystal.
Tip Atoms
Tunnelling electrons
Surface Atoms
STM Principles
• “Scanning Tunnelling
Microscope”
• Tunnelling current depends
exponentially on distance from
surface
– Move tip across surface, and
the current changes as the tip
“feels” the “bumps” caused by
valence electron wave
functions
• Image shows individual atoms
of a sample of Highly Oriented
Pyrolytic Graphite.
STM Operating Modes
• Const. current mode:
– Move tip over surface and
measure changes in height
with piezo.
• Const. height mode:
– Keep tip-surface separation const,
and measure changes in current.
– Need very flat samples to avoid
tip crash!
Scanning
Raster Scanning over
area from .1X.1mm to
10X10 nm
Scan rates can be quite
fast
Resolution/scan
Vacuum Sucks
• Needs Ultra-High
vacuum
– Otherwise unwanted
surface
– Lots of turbo pumps
and stainless steel
– Bakeout and surgical
handling procedures
The STM
• Can move or desorb atoms
as well as image.
• Desorb=unstick from surface
• Absorb=diffuse into bulk
– Put high voltage on tip to
draw current and “arc weld
surface”
– Use small bias to pick up
atoms and assemble them into
cheesy logo
The STM image
• The STM image is a file of
(x,y,height) co-ords
• It can be manipulated to
produce all sorts of images
– Fantastic colour schemes of
dubious taste
– Animation and fly-by videos
• Quantum corrals; can image
e- wavefunction
– See the ripples
– Spiles are Fe d-orbitals
– Red atoms are Cu
Gratuitous STM images
• Great for grant applications and press releases
The AFM
• Atomic Force
Microscope
• STM measures
tunelling current; but
AFM measures van
der waals forces
directly
The AFM
• Detect minute movements
in cantilever by bouncing
laser off and using
interferometry (remember
laser sensors)
• Atomic resolution
• Sample need not be
electrically conductive
AFM
• Can get atomic scale
resolution, just like STM.
• Still needs UHV and
vibration isolation for
atomic scale resolution.
• Different Modes:
– Contact
– Non-contact
– Tapping
Lecture 22
Lecture 23
Remember Dielectric Materials?
• Many molecules and crystals have a non-zero Electric
dipole moment.
• When placed in an external electric field these align with
external field.
• The effect is to reduce the
strength of the electric field within
the material.
• To incorporate this, we define a
new vector Field,the electric
displacement, D
Diamagnetism
• Polarisation of dielectrics involves the creation of
induced electric dipoles by an external electric
field
• The equivalent effect for magnetic materials is
diamagnetism
• The atoms in a diamagnet alter their electron
orbits in order to oppose the field
• Diamagnet acts like a bar magnet and repels
external field
• Water is a diamagnet-hence the levitating frog!
Levitating Frog
• Huge (20 T)
magnet induces
strong
diamagnetic
field in the
water.
• Force is strong
enough to
levitate frog
Paramagnetism
• In some materials, there are permanent magnetic
moments.
• When an external magnetic field is applied, these
line up with the field to reinforce it
• This is a much stronger effect than diamagnetism,
and in the opposite effect
• Ferromagnetism is a form of paramagnetism
• Where do these permanent magnetic moments
originate from?
Magnetisation
Some atoms have unpaired electrons.
Each unpaired electron possess a magnetic
dipole moment, μ,which is an integer
multiple of
μB=eħ/2me (The Bohr Magneton)
In essence, consider the atoms as tiny
magnets. The magnetic moments arise
from both the spin and orbital motion of
the electrons.)
e
Magnetisation
Thus some materials have a
magnetisation
M= (nμ) /volume
Where n is the number of
dipoles present
M may also depend on
external factors like
temperature or magnetic
field
So some matter produces its
own magnetic field
Bm= μ0M (Where μ0 is the
permeability of vacuum)
More magnetisation
• B=B0+Bm
• Introduce magnetic field strength
H=(B /μo) – M
• Thus B= μo(H+M)
• Total field=External field + Field
due to material
• Consider current through coils with
n loops/metre
• Empty coil: B= μ0H so H=nI
• Coil with material inside (ie a core)
H=nI still, but enhanced B if
contribution (M) from material in
core
Dia-, Para-, Ferro-,…
• Magnetisation M depends on magnetic field
strength H via M=χH
χ =magnetic susceptibility
• χ >0 =>paramagnetic (boosts applied field)
• χ <0 =>diamagnetic (reduces applied field)
Ferromagnetism can be thought of as an extreme
case of paramagnetism with a variable χ
(typically χ>>0 for Ferromagnetic material).
Remember these Ferromagnetism
demos?
Atomic level view (I)
• Ferromagnetism
– Atoms have associated dipole moments
– If dipoles aligned  net B-field.
– At higher Temperatures thermal agitation reduces
alignment
– Presence of external field  alignment of dipoles,
Removal of field  alignment remains
– Thus explain hysterisis
• Paramagnetism
– Atoms have associated dipole moments but only
interact weakly with each other
– In the absence of external field no net magnetisation
– In presence of external field some alignment but must
compete with thermal agitation
Atomic level view (II)
• For a Paramagnetic material
M=CT(B/T) where C is the Curie constant
– Hence for paramagnetic material M increases at low T
• Diamagnetism
– Atoms have no associated dipole moments (even
number of outer shell electrons)
– Presence of external field  weak dipole moment
which opposes applied field (Lenz’s Law).
– Removal of field  alignment disappears
Superconductivity:
A Historical introduction
• 1908: He first
liquefied ( by Onnes)
• 1911:Observed that
resistance of liquid Hg
plummets at 4.2 K
• This ultra-low
resistance termed
superconductivity
• 1933: Meissner
– Magnetic field is expelled from superconductor
– Superconductivity ceases at B >Bc
• 1962: Josephson Junction
– Two superconductors separated by a thin
insulating oxide barrier
• 1986: High Tc (150 K) Superconductors
Types of superconductors
• There are two main classes of superconductor:
– Type I - (e.g. Hg) have a critical field and a critical
temperature above which they abruptly cease
superconducting
– Type II - have an intermediate phase, which extends
to higher temperatures and fields.
Type I superconductors
• Mostly single element metals
• Superconductivity destroyed for T>Tc
or B>Bc
• Variation of Bc with temperature can
be expressed as
  T 2 
Bc  Bc (0 K ) 1    
  Tc  
• Low value of Bc (typically < 0.1 T)
means generating strong magnetic
fields using type I superconductors not
possible
Examples of type I superconductors
Meissner Effect
• Inside the superconductor
– J=σE for all conductors, but superconductors
have σ infinite, so electric field is zero even
when a current flows
where φ=magnetic flux
• E=0 => φ=const
• B= φ/Area so B becomes “trapped” inside.
Meissner effect
• If we put a perfect conductor in
an external magnetic field at
T>Tc, then cool to T<Tc, then
remove field, we expect some
field trapped inside.
• In fact, B=0 always inside a
superconductor at T<Tc, B<Bc.
• Superconductor is a perfect
diamagnet, and hence repels
permanent magnets.
Type II superconductors
• Mostly alloys and whizzy
ceramics
• At fields between Bc1 and
Bc2 some flux lines
penetrate
• At B> Bc2 these flux lines
overlap and
superconductivity dies
• Bc2 very large (<30 T), so
these materials suitable for
superconducting magnets
Some type II superconductors
Worked example
• A solenoid is to be constructed using Nb3Al (Bc2=32
T). The wire has a radius of 1.0 mm, and the
solenoid is to be wound on a cylinder of diameter
8.00 cm and length 90.0 cm. There are 150 turns per
cm of length. How much current is required to get a
field of 5.0 T at the centre?
• B=μ0nI
– N=150 turns/cm=1.5x104 turns/m;
– B= 5.0 T
– I=B/nμ0=265 Amps
Useful properties
• Persistent currents
B
– Once a current has been set up in the
loop, it keeps going ‘forever’ (R < 1e26 Ωm)
• Flux trapping
– Consider superconducting loop, T>Tc
– Apply B field, then cool to T<Tc
– Field is trapped inside the loop, but
expelled from the material
– Trapped flux is quantised in units of
h/2e, “magnetic flux quantum”,
o=2.0678x10-15 T.m2
T>Tc
T<Tc
Applications of
superconductivity
(Magnetic Resonance Imaging)
• Used to detect presence of H
atoms
• Magnetic resonance imager has
1.5 Tesla superconducting magnet
• Once current is caused to flow in
the coil it will continue to flow as
long as the coil is kept at liquid
helium temperatures.
• losses of the order of one part per
million of the main magnetic field
per year.
http://www.cis.rit.edu/htbooks/mri/
Mechanism of Superconductivity
• How can an electron
possibly travel through a
material without being
scattered?
• Clue: Isotope effect
showed dependence on
atomic number of lattice
atoms
• Normal conductivity
independent of lattice
(unless defects are
present)
Cooper Pairs
Cooper Pairs
• e- with opposite momentums
and spins form single particle
• Cooper pairs are bosons, e- are
fermions
• Pairs no longer have to obey
Pauli exclusion principle
• All Cooper pairs are in the
same state!
• Individual e- not free to
scatter off lattice impurities
• They are all in phase and locked in with each other over
the entire sample volume
• All the Cooper pairs form a coherent state
• If we provide enough energy,
we can break up the pair,
destroying superconductivity
• This energy is Eg=3.53 kBTc
• Leads to a “band gap”,
analagous to semiconductors
Josephson Junctions
• 2 pieces of superconductor,
separated by thin (~10 Å)
insulating layer
• Tunnelling of bound pair
through insulating layer
possible
Superconducting
material
Insulating
Layer
(oxide)
• Unique effects associated with
pair tunnelling
Ψ1
Ψ2
sc2
sc1
oxide
The DC Josephson effect
• At zero voltage there exists a
superconducting current given by
Is=Imaxsin(φ1-φ2)=Imaxsin(δ)
Dependant on thickness of
insulating layer and junction
area
Phase difference between the
wavefunctions in the two
superconductors
• Imax is the maximum current across the
junction under zero bias conditions
The AC Josephson effect
• A DC voltage across a Josephson junction
generates an alternating current given by
Is=Imaxsin(δ+2πft)
where f=2eV/h
• Hence Josephson current oscillates at frequency
proportional to applied voltage
• Is used as method of determining voltage standard
– microwave current passed through junction
– Stable operation only possible for voltages satisfying
V=nhf/2e (n is interger)
SQUIDs
• “Superconducting QUantum
Interference Devices”
• Consist of 2 or more
Josephson junctions in
parallel
• Extremely sensitive to
changes in magnetic field
Superconductor
Junctions
I
SQUIDs
• Interference caused by difference in
phase of the of the arrival of the current
through two different paths
• Analogous to optical interference
(Young’s slits)
• Difference in phase Δδ given by
Δδ=δa-δb=(2q/ħ)
where  is the flux passing through the
loop
• Hence the phase difference is dependant
on the flux passing through the loop
Ib
Ia
Junctions
I
SQUIDs
• The maximum current through the loop can be shown
to be Imax=Iocos(q/ħ)
• Hence current maxima observed for
= nπħ/q
• The Interference between two junctions can be used to
make very sensitive measurement of magnetic field
I
B
Applications of SQUIDs
• Biomagnetism
– Some processes in animals/humans produce very small
magnetic fields. The only type of detector sensitive
enough to measure such a field is a SQUID.
– Magnetoencephalography (MEG) the imaging of the
human brain. Involves measuring the magnetic field
produced by the currents due to neural activity.
– Advantages other methods which only image the
structure of the brain
• Non Destructive testing
– Potential uses in monitoring internal faults or wear in
metal containing structures.
Lecture 24
History of Superconductivity
• 1908: Liquid He first
discovered (Onnes)
• 1911:Resistance of
liquid Hg plummets at
4.2 K
• Superconductors have
zero resistance!
History of Superconductivity
• 1933: Meissner
– Magnetic field is expelled from superconductor
– Superconductivity ceases at B >Bc
• 1962: Josephson Junction
– Two superconductors separated by a thin
insulating oxide barrier
• 1986: High Tc (150 K) Superconductors
Types of superconductors
• There are two main types of superconductor:
– Type I - (eg Hg) have a critical field and a critical
temperature above which they stop superconducting
– Type II - have an intermediate phase, which extends to
higher temperatures and fields.
Type I superconductors
• Mostly single element metals
• Superconductivity destroyed for T>Tc
or B>Bc
• Variation of Bc with temperature can
be expressed as
  T 2 
Bc  Bc (0 K ) 1    
  Tc  
• The low value of Bc (typicall <0.1T)
 generating strong magnetic fields
using type I superconductors not
possible
Type I superconductors
Meissner Effect
• Inside the superconductor
– J=σE for all conductors, but superconductors
have σ infinite, so electric field is zero even
when a current flows
• φ=magnetic flux
• E=0 => φ=const
• B= φ/Area so B becomes “trapped” inside.
Meissner effect
• If we put a perfect conductor in
a field at T>Tc, then cool to
T<Tc, then remove field, expect
field trapped inside.
• In fact, B=0 always inside a
superconductor at T<Tc, B<Bc.
• Superconductor is a perfect
diamagnet, and hence repels
permanent magnets.
Type II superconductors
• Mostly alloys and whizzy
ceramics
• At fields between Bc1 and
Bc2 some flux lines
penetrate
• At B> Bc2 these flux lines
overlap and
superconductivity dies
• Bc2 very large (<30 T), so
these materials suitable for
superconducting magnets
Type II superconductors
Worked example
• A solenoid is to be constructed using Nb3Al (Bc2=32
T). The wire has a radius of 1.0 mm, and the
solenoid is to be wound on a cylinder of diameter
8.00 cm and length 90.0 cm. There are 150 turns per
cm of length. How much current is required to get a
field of 5.0 T at the centre?
• B=μ0nI
– N=150 turns/cm=1.5e4 turns/m;
– B= 5.0 T
– I=B/nμ0=265 Amps
Useful properties
• Persistent currents
B
– Once a current has been set up in the
loop, it keeps going ‘forever’ (R < 1e26 Ωm)
• Flux trapping
– Consider superconducting loop, T>Tc
– Apply B field, then cool to T<Tc
– Field is trapped inside the loop, but
expelled from the material
– Trapped flux is quantised in units of
h/2e, “magnetic flux quantum”,
o=2.0678x10-15 T.m2
T>Tc
T<Tc
Applications of superconductivity
(Magnetic Resonance Imaging – MRI)
• Used to detect presence of H
atoms
• Magnetic resonance imager has
1.5 Tesla superconducting magnet
• Once current is caused to flow in
the coil it will continue to flow as
long as the coil is kept at liquid
helium temperatures.
• losses of the order of a ppm of the
main magnetic field per year.
http://www.cis.rit.edu/htbooks/mri/
Devices based on
superconductors
• Josephson Junctions
• Superconducting QUantum Interference
Devices (SQUIDS)
– Highly sensitive magnetic field detectors
• To understand these, need to understand
mechanisms responsible for
superconductivity
Mechanism of Superconductivity
• For normal conductor behaviour
the resistivity is due to scattering
from lattice ions or impurities
• How can an e- possible travel
through a material without being
scattered?
• Clue: Isotope effect showed
dependence on atomic number
lattice atoms
• Normal conductivity independent
of lattice (unless defects)
Mechanism of Superconductivity
• The features of superconductiviy can be explained using
theory developed by Bardeen, Cooper and Schrieffer
(BCS theory)
• Main Features of BCS Theory
– 2 electrons in a superconductor can form a bound pair via an
attractive interaction
– known as a cooper pair
• How can negatively charged electrons form bound pair?
– Net attraction possible if electron interact via motion of crystal
lattice
Cooper Pairs
• Lattice momentarily deformed by passing electron
(electron causes nearby lattice ions to move inwards)
• This results in a slight increase in positive charge in this
region
• 2nd electron sees the positive charge of the distorted region
and is attracted towards it
Cooper Pairs
• Paired electrons possess opposite
momentums and spins form single
particle
• Cooper pairs are bosons, electrons
are fermions
• Pairs no longer have to obey Pauli
exclusion principle
• All Cooper pairs are in the same
state! (all or nothing behaviour of
superconductivity)
• Individual electrons not free to
scatter off lattice impurities
More Cooper pairs
• All the cooper pairs form a coherent state
• They are all in phase and locked in with each other
over the entire sample volume
• If we provide enough energy, we can break up the
pair, destroying superconductivity
• This energy is Eg=3.53kBTc
• Leads to a “band gap”, analagous to
semiconductors
Josephson Junctions
• 2 pieces of superconductor,
separated by thin (~10 Å)
insulating layer
• Tunnelling of bound pair
through insulating layer
possible
Superconducting
material
Insulating
Layer
(oxide)
• Unique effects associated with
pair tunnelling
Ψ1
Ψ2
sc2
sc1
oxide
The DC Josephson effect
• At zero voltage there exists a
superconducting current given by
Is=Imaxsin(φ1-φ2)=Imaxsin(δ)
Dependant on thickness of
insulating layer and junction
area
Phase difference between the
wavefunctions in the two
superconductors
• Imax is the maximum current across the
junction under zero bias conditions
The AC Josephson effect
• A DC voltage across a Josephson junction
generates an alternating current given by
Is=Imaxsin(δ+2πft)
where f=2eV/h
• Hence Josephson current oscillates at frequency
proportional to applied voltage
• Is used as method of determining voltage standard
– microwave current passed through junction
– Stable operation only possible for voltages satisfying
V=nhf/2e (n is interger)
SQUIDs
• “Superconducting QUantum
Interference Devices”
• Consist of 2 or more
Josephson junctions in
parallel
• Extremely sensitive to
changes in magnetic field
Superconductor
Junctions
I
SQUIDs
• Interference caused by difference in
phase of the of the arrival of the current
through two different paths
• Analogous to optical interference
(Young’s slits)
• Difference in phase Δδ given by
Δδ=δa-δb=(2q/ħ)
where  is the flux passing through the
loop
• Hence the phase difference is dependant
on the flux passing through the loop
Ib
Ia
Junctions
I
SQUIDs
• The maximum current through the loop can be shown
to be Imax=Iocos(q/ħ)
• Hence current maxima observed for
= nπħ/q
• The Interference between two junctions can be used to
make very sensitive measurement of magnetic field
I
B
Applications of SQUIDs
• Biomagnetism
– Some processes in animals/humans produce very small
magnetic fields. The only type of detector sensitive
enough to measure such a field is a SQUID.
– Magnetoencephalography (MEG) the imaging of the
human brain. Involves measuring the magnetic field
produced by the currents due to neural activity.
– Advantages other methods which only image the
structure of the brain
• Non Destructive testing
– Potential uses in monitoring internal faults or wear in
metal containing structures.
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