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SOLID STATE PHYSICS (MIND MAP 3) - Coggle Diagram
SOLID STATE PHYSICS (MIND MAP 3)
LATTICE VIBRATION I
The static lattice models are valid only at zero temperatures
When the temperature zero, each atom has thermal energy
therefore, there is vibration around the equilibrium lattice position.
CRYSTAL DYNAMICS
energy they possess as a result of zero point known as zero point energy.
the amplitude of the motion increases as the atoms gain more thermal energy at higher temperature.
atomic motions are governed by the forces exerted on atoms when they are displaced from their equilibrium positions.
the small amplitude limit is known as harmonic limit.
restoring on each atom is proportional to its displacement (Hooke's Law)
HOOKE'S LAW
The extension of the spring is directly proportional to the pulling force
Formula; F=-kx
point which the Elastic Region ends call inelastic limit or the proportional limit.
inelastic limit
point at which permanent deformation occurs.
after the elastic limit, if force is taken off the sample, it will not return to its original size and shape.
deformation has occurred
proportional limit
point at which the deformation is no longer directly proportional to the applied force. (not longer obey Hooke's Law)
SIMPLE HARMONIC MOTION (SHM)
is periodic motion in the absence of friction and produced by a restoring force that is directly proportional to the displacement and oppositely directed.
formula ; F = -kx
SIMPLE PERIODIC MOTION (SPM)
motion in which a body moves back and forth over a fixed path, returning to each position and velocity after a definite interval of time.
period, T ; time for one complete oscillation. (second)
Frequency, f ; number of complete oscillation per second (hertz)
THERMAL ENERGY AND LATTICE VIBRATIONS
atoms vibrate about their equilibrium position.
they produce vibrational waves
this motion is increased as the temperature is raised.
HEAT CAPACITY FROM LATTICE VIBRATIONS
in metals -> from the conduction electrons
in magnetic materials -> from magneting ordering
Heat capacity - is the amount of heat needed to raise 1Kg of substance by 1 degree Celsius
Heat capacity at constant volume (Cv)
Heat capacity at constant pressure (Cp)
unit ; J⋅K−1⋅kg−1
Lattice heat capacity (Clat)
HEAT CAPACITY OF THE LATTICE
CLASSICAL GAS THEORY
only have translational kinetic energy (energy due to motion from one location to another)
formula;
THREE MODELS OF LATTICE VIBRATIONS
EINSTEIN MODEL (1906)
average thermal energy of an oscillator of frequency ;
HIGH TEMPERATURE LIMIT
yields the classical result of Dulong And Petit, Cv=3Nk
LOW TEMPERATURE LIMIT
Cv -> 0 as T-> 0.
SUMMARY
high temperature behavior is good since T -> 0
DEBYE MODEL (1912)
recovers the Dulong-Petit law at high temperature Cv;
BORN VON KARMAN MODEL (1912)
boundary conditions are periodic boundary which impose the restriction that a wave function must be periodic on a certain Bravais lattice.
LATTICE VIBRATION III
ID DIATOMIC CHAIN OF 2N ATOMS
there are N atoms of mass m & N atoms of mass M
Every mass m has two near neighbors of mass M & every mass M has two near neighbors of mass m.
The repeat distance is a, so the atoms are equally spaced with separation (1/2) a
EQUATION OF MOTION FOR MASS M (nth)
EQUATION OF MOTION FOR MASS m (n-1)th
OFFER A SOLUTION FOR THE MASS M
OFFER A SOLUTION FOR THE MASS m
ω-K RELATION FOR DIATOMIC CHAIN
NORMAL MODE
In the simplest approximation, we imagine that the atoms are attached to each other by a linear spring
Normal mode is a pattern of motion in which all parts of the system move in a sinusoidal fashion, with the same frequency.
NORMAL MODE FREQUENCIES OF A CHAIN OF TWO TYPES OF ATOMS
At A, the two atoms are oscillating in antiphase with their center of mass at rest
at B, the lighter mass m is oscillating and M is at rest
at C, M is oscillating and m is at rest
ACOUSTIC/OPTICAL BRANCHES
The optical branch is a higher energy vibration (the frequency is higher, and you need a certain amount of energy to excite this mode)
Transverse optical mode for diatomic chain
Transverse acoustical mode for diatomic chain
there is a band of frequencies between the two branches that cannot propagate
the width of this forbidden band depends on the difference of the masses
if the two masses are equal, the two branches join (become degenerate) at π/a
AT OPTICAL BRANCH
k=0, maximum frequency ω opmax ;
k= π/a, minimum frequency ω opmin ;
The branches are narrower for small m / M
while they are widened for m / M → 1.
AT ACOUSTICAL BRANCH
k=π/a, maximum frequency ω acmax ;
The difference ω opmin - ω acmax give the value of the frequency gap width between the two branches in k = π/a
This gap becomes narrower when both atoms are nearly identical and no longer exist for m = M
LATTICE VIBRATIONS IV
PHONON DISPERSION AND SCATTERING
quanta of lattice vibration are called phonons
lattice vibrations are responsible for transport of energy in many solids.
THE DISPERSION RELATION OF THE MONATOMIC 1D LATTICE
optical branch -> upper branch is due to the +ve sign of the root
acoustical branch -> lower branch is due to the -ve sign of the root
REAL CRYSTAL SYSTEM
In a 3-D crystal, the atoms vibrate in three dimensions with two vibration modes and three vibrational branches, one longitudinal and two transverse.
Force constant may vary the branches are not degenerate
longitudinal ;
transverse ;
In simplest 1D case with only nearest-neighbor interactions
In general 3D case, the monatomic Bravais lattice, in which each unit cell has a single atom.
The solution of this equation in 3D can be represented in terms of normal modes
The roots of this equation lead to three different dispersion relations, or three dispersion curves
SIMPLE CRYSTAL SYSTEMS (ω versus k curves)
TRANSVERSE MODE
in real crystals, these two transverse modes directions
each of these two transverse modes and the one longitudinal mode all have different frequencies.
THE NUMBER OF BRANCHES
Here the unit cell contains two or more atoms.
Of these, three branches are acoustic, and the remaining (3z − 3) are optical.
the number of allowed k-values in any single branch is N
for every allowed value of k, there are 3z different vibrational excitations
for the graph ω vs k there are 3z curves in any one k-direction
MODES THROUGHOUT THE BRILLOUIN ZONE
BRILLOUIN ZONES OF THE RECIPROCAL LATTICE
FIRST BRILLOUIN ZONE DEFINITION
Wigner-Seitz = FIRST BRILLOUIN ZONE
FIRST BRILLOUIN ZONE FOR 3D LATTICES
leads to a polyhedron whose planes bisect the lines connecting a reciprocal lattice point to its neighboring pint.
WIGNER SEITZ CELL CONSTRUCTION
unit cell is to connect each lattice point to all its neighboring point with a line segment and bisect each line segment with a perpendicular plane.
PHONONS
quantum of lattice vibration
energies of phonons are quantized
PHOTON
quantum of electromagnetic radiation
energies of photons are quantized
PHONON MOMENTUM
A phonon of wavevector k will interact with particles such as photons, neutrons and e
lectrons as if it had a momentum ħk.
If a phonon k is absorbed in the process, K’ = k +K +G.
PHONON GENERATION
PIEZOELECTRIC
EM waves (10 GHz) can generate an oscillating
electric field
The oscillating electric field the piezoelectric
transducer at the same frequency
Not suitable for frequencies > 10GHz
THERMAL EXCITATION
Current is flow through metal wires causing the
electron temperature to rise.
Probability to emit phonon > photon
Above the threshold frequency, only photons are
produced.
ELECTRON TUNNELLING
A thin layer of insulator is placed between two thin
layers of metal to form a barrier for the electrons
The electron is speed up with additional kinetic
energy eV
Phonon is emitted by hot electrons that losses
energy when returning to equilibrium
LATTICE VIBRATION II
When the lattice is at equilibrium each atom is positioned exactly at its lattice site.
So, now suppose that an atom displaced from its equilibrium site by a small amount.
due to force acting on this atom, it will tend to its equilibrium position.
MONOATOMIC CHAIN
the simplest crystal is the one dimensional chain of identical atoms with identical masses, M connected by identical springs of spring constant, K.
consists of very large number of identical atoms with identical masses.
atoms are separated by a distance "a"
atoms only move in a parallel direction to the chain.
force to the right ; K(Un+1 - Un)
force to the left ; K(Un - Un-1)
total force = (force to the right) - (force to the left)
total force ;
Equation of motion for nth atom ;
The dispersion relation of the monoatomic 1D lattice ;
the result is ;
ω-K relation : Dispersion Relation
transverse wave is particles are displaced
perpendicular to the direction the wave travels
longitudinal wave the particles are displaced parallel to the direction the wave travels
PHASE VELOCITY AND GROUP VELOCITY
longitudinal wave the particles are displaced parallel to the direction the wave travels ;
energy is transported by the wave at a generally slower speed the group velocity ;
