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)

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)

THERMAL ENERGY AND LATTICE VIBRATIONS

• atoms vibrate about their equilibrium position.
• they produce vibrational waves
• this motion is increased as the temperature is raised.

CLASSICAL GAS THEORY

• only have translational kinetic energy (energy due to motion from one location to another)
• formula;
  

• 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

• recovers the Dulong-Petit law at high temperature Cv;

• boundary conditions are periodic boundary which impose the restriction that a wave function must be periodic on a certain Bravais lattice.

• 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 ;
• 

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 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

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

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

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.
• 

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.
• 
• 
  

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