Transcript Phonons in Superconductivity

Phonons
Packets of sound found present in
the lattice as it vibrates … but the
lattice vibration cannot be heard.
Unlike static lattice model , which deals with average positions of atoms in a crystal,
lattice dynamics extends the concept of crystal lattice to an array of atoms with finite
masses that are capable of motion. This motion is not random but is a superposition of
vibrations of atoms around their equilibrium sites due to the interaction with neighbor
atoms. A collective vibration of atoms in the crystal forms a wave of allowed
wavelength and amplitude.
Just as light is a wave motion that is considered as composed of particles called
photons, we can think of the normal modes of vibration in a solid as being particle-like.
Quantum of lattice vibration is called the phonon.
Uniform Solid Material
Considering the regular
lattice of atoms in a uniform
solid material, you would
expect there to be energy
associated with the
vibrations of these atoms.
But they are tied together
with bonds, so they can't
vibrate independently. The
vibrations take the form of
collective modes which
propagate through the
material.
(X-1)
(X)
(X+1)
Phonon:
A Lump of Vibrational Energy
Propagating lattice vibrations can be considered
to be sound waves, and their propagation speed
is the speed of sound in the material.
Phonon:
Sound
Wavepackets
If N atoms make up the lattice, and the spring constant
between the atoms is C, then we can write an equation for
the force when the atoms are displaced:
d 2x
F  ma  m 2  Cx
dt
or
d 2 xs
m 2  C  xs 1  xs 1  2 xs 
dt
Using a position/time dependence
of exp(iwt) and solving, we get the
relationship between wave number
k and frequency w:
 C
w   2 1  cos( ka)
 m
where
2
n
k
, n  1,2,...N
Na
• It is usually convenient to consider phonon
wave vectors k which have the smallest
magnitude (|k|) in their "family". The set of
all such wave vectors defines the first
Brillouin zone. Additional Brillouin zones
may be defined as copies of the first zone,
shifted by some reciprocal lattice vector.
There are Acoustic and Optical
Phonons
• Acoustic phonons occur when wave numbers are small
(i.e. long wavelengths) and correspond to sound
transmission in crystals. Acoustic phonons vary
depending on whether they are longitudinal or transverse
• "Optical phonons," which arise in crystals that have more
than one atom in the unit cell. They are called "optical"
because in ionic crystals are excited very easily by light
(by infrared radiation in NaCl). The positive and negative
ions vibrate to create a time-varying dipole moment.
Optical phonons that interact in this way with light are
called infrared active.
Solid is a periodic array of mass points, there are
constraints on both the minimum and maximum
wavelength associated with a vibrational mode.
Electron Motion in a Crystal at
Normal State
Phonons in Superconductivity
Cooper Pairs
Nucleus
Exchange of phonons
holds electrons together.
By pairing off two electrons
pass through more easily.
More on Cooper Pairs
• The electrons in the superconducting state are
like an array of rapidly moving vehicles. Vacuum
regions between cars locks them all into an
ordered array as does the condensation of
electrons into a macroscopic, quantum ground
state.
• Random gusts of wind across the road can be
envisioned to induce collisions, as thermally
excited phonons break pairs. With each collision
one or two lanes are closed to traffic flow, as a
number of single-particle quantum states are
eliminated from the macroscopic, many-particle
ground state.
“Zero Resistance”
5-Minute Break
Wave-Particle Duality
Photons
In trying to explain black body radiation and thermoelectric effect, Max Planck and Albert Einstein developed theories
that when put together led to the following principles.
1. Light is made up of particles called photons.
2. The energy of a photon is dependent only upon its frequency.
E = hn = hc/l
Where h is Planck’s constant (h = 6.626 x 10-34 J-s).
What this means is that if an object is giving off (or absorbing) light it is actually emitting (absorbing) photons. The
energy of each photon is dependent only upon its frequency (or wavelength or color).
In 1925, Louis DeBroglie hypothesized that if light, which everyone thought for so long was a wave, is a particle, then
perhaps particles like the electron, proton, and neutron might have wave-like behaviors.
In the same way that waves are described by their wavelength, particles can be described by their momentum, p
p = mv
where m is the mass of the particle and v is its velocity.
We can relate the velocity of a wave-particle with its wavelength by equating Planck’s relationship for the energy of a
photon with Einstein’s Law of Relativity:
E = hn = hc/l
E = mc2
If we equate these two equations we get a relationship between momentum (a particle property) and wavelength (a
wave property)
hc/l = mc2 = pc
p = h/l
Or by substituting mv for momentum we can wavelength of any object to its velocity and mass.
l= h/mv
Element spectral lines were empirically
described using the integer values of ‘n’
The only problem with
these models is that they
do not account for line
splitting
Bohr Model of the Atom
1. Electrons are in stable circular orbits
about the nucleus and do not decay
2. Electrons move to higher orbit by gaining
energy (absorbing a photon of energy
hn), or drop to a lower orbit, emitting a
photon
3. Electron angular momentum, pq is an
integer multiple of n, or pq  nh/2
• What was correct about Bohr’s Model:
– Electrons reside in quantized energy levels.
– The Bohr model accurately and quantitatively
predicts the energy levels of one electron
atoms.
• What was incorrect about Bohr’s Model:
– Electrons don’t orbit the nucleus in well
defined circular orbits.
– Fails to accurately predict the energy levels in
multielectron atoms.
Heisenberg Uncertainty Principle
Uncertainty Principle → It is not possible to
precisely determine the momentum (hence the
energy) and the position of a particle
simultaneously. This is quantified in the
mathematical expression:
x p = h/4
x m v = h/4
where x represents the uncertainty in the position
of the particle and p represents the uncertainty
in the momentum of the particle (p = mv).
Results of Uncertainty Principle
We cannot say where a particle (electron) is,
only the probability of finding it at that
particular place
Quantum mechanics defines the functional
relationships we can use to solve for
properties such as position, momentum
and energy, using a wave function, y
Math, Math and more Math
In one dimension, we can describe the energy of a particle
in terms of its wave function as:
p2
h d 2y ( x, t )
h dy ( x, t )
V  E 
 V ( x)y ( x, t )  
2
2
2m
4i
dx
2i
dt
Writing y(x,t) as two separate variables, y(x)f(t), we get
independent equations in terms of position and time:
d 2y 4m
 2  2 E  V ( x) y ( x)  0
dx
h
and
d 2f 4i
 2 Ef (t )  0
2
dx
h
The Hydrogen Atom
We can solve the wave
function for the
Hydrogen atom by using
spherical coordinates
(r,q,f) and solving the
independent wave
equations using
separation of variables:
Y(r,q,f) = yrQqFf
This leads to three of the four
quantum numbers: n,l,m and
s
The n,l and m quantum numbers
Principle Quantum Number (n)
Possible Values = 1, 2, 3, …
The principle quantum number defines the size of the
orbital. As n increases we see the following trends:
The orbital becomes larger
The energy of the electron increases
The electron is less tightly bound to the nucleus
Azimuthal Quantum Number (l)
Possible Values = 0 to n-1
The azimuthal quantum number defines the shape of the
orbital. We will commonly use a letter designation to
designate the value of l.
s = sphere
p = dumbbell
d = double dumbbell
Magnetic Quantum Number (ml)
Possible Values = l, …, -l
The magnetic quantum number describes the orientation
of the orbital.
These three quantum numbers completely define the size,
shape, and orientation of an orbital. However, there is a
final quantum number which is an intrinsic property of the
electron.
Potential energy
well for the Si
atom, and groundstate electron
configuration