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MINDMAP3 SOLID STATE PHYSICS, MOHAMAD FARID IZUAN BIN AZMI A19SC0169 -…
MINDMAP3
SOLID STATE PHYSICS
Lattice Vibration 1
Heat Capacity
• Amount of heat (J or cal) needed to raise an unit amount of substance (g or mol) by an unit in temperature (°C or K)
• Substance with higher heat capacity need more heat supplied to raise its temperature
• If T drop to 0K, the heat capacity tends to zero.
• Units: Joules/Kelvin/mole, J/K/mole, J/°C /mole, J/°C /g
• Heat capacity can be found by differentiating the average phonon energy
•
Heat capacity at constant Volume (Cv):
slope of the plot of internal energy with temperature.
• Heat capacity at constant volume, Cv as a function of temperature for solid
• Heat capacity of a mole of many solid element:
• The low-T behavior can be explained by quantum theory
• Einstein -Solid as ensemble of independent quantum harmonic oscillators vibrating at frequency
• Debye -Advanced the theory by treating the quantum oscillators as collective modes in the solid (phonons)
•
Heat capacity at constant Pressure (Cp):
slope of the plot of enthalpy with temperature.
•
Heat capacity of lattice
• Heat capacity of insulator failed for lower temperature and it totally failed for metals
• Metal’s heat capacity at room temperature is similar to insulator
3 MODEL LATTICE VIBRATION
1.Einstein Model
• Method in solid thermodynamics that was invented originally by Einstein in 1907
• Treats the solids as many individual,non interacting quantum harmonic oscillators
• Not accurate at low temperature
2.Debye Model
• Method developed by Peter Debye to estimate phonon contribution to the specific heat of the solid
• Treat vibrations of the solid atomic lattice as phonons in a box
• Accurate at low temperature
3.Born Von Karman Model
• Basic model for describing the movements of atoms in a crystal lattice
• Born–von Karman boundary conditions are periodic boundary conditions which impose the restriction that a wave function must be periodic on a certain Bravais lattice
• This condition is often applied in solid state physics to model an ideal crystal.
Lattice Vibration 2
Monoatomic chain
• The simplest crystal is the 1D chain of identical atoms with identical masses M connected by spring constant K.
• Chain consists of a very large number of identical atoms with identical masses.
• Atoms move only in a direction parallel to the chain.
• Only nearest neighbours interact
• Force to the right
•Force to the left
•The total force = Force to the right – Force to the left
•For a wavelike solution in which all atoms oscillate with a same amplitude A and frequency ω:
Dispersion Relation
•The relationship between angular frequency,ω and and wave number
• K = 2λ/l
• lmin = 2a
• Kmax = π/a
• -π/a<K< π/a
• Atomic displacement for λ=2a,k= π/a
• Atomic displacement for a wave with λ=7a/4,k=8 π/7 identical to wave with λ=7a/3,k=6 π/7a,as given by broken curve
• Points A, B and C correspond to the same instantaneous atomic displacements as well as the same frequency.
• At B ,dω/dk > 0. Wave travelling to the right;
• At A & C both wave travelling to the left and equivalent.
• The k values of points A and C differ by 2π/a
• Adding any multiple of 2π/a to k does not alter the atomic displacements or the group velocity.
• The only range is -π/a < k π/a.
Phase Velocity
Group Velocity
• Linear dispersion
• Sound velocity for 1D lattice
• For small λ or frequency, lattice behaves as continuum and no dispersion take place
• Small q – close to the center of Brillouin zone
Lattice Vibration 3
1D diatomic chain of 2N atoms
• There are N atoms of mass m & N atoms of mass M.
• This is a model of a solid with two atoms per unit cell.
• Every mass (m/M) has two near neighbours of mass (M/m)
• Atoms are equally spaced with separation (1/2) a.
• The force constant coupling each atom to its nearest-neighbors is K.
• The total force = Force to the right – Force to the left
•Equation of motion for mass M (nth): mass x acceleration = restoring force
•For a wavelike solution
• Equation of motion for mass m (n-1)th:
•For a wavelike solution
•As
n
does not appear in above equation,our assumed solution may be rewritten in the form:
•
ω versus k relation for diatomic chain
• The dispersion relation have two branches (optical & acoustical)
• The dispersion relation is periodic in k with a period
2 π /a = 2 π /(unit cell length).
At optical branch
• k = 0, maximum frequency
ωop.max
• k = π/a, minimum frequency
ωop.min
so :
At acoustical branch
k = π/a, maximum frequency ωac.mak
• There is a band of frequencies between the two branches
• The width of this forbidden band depends on the difference of the masses.
• Two branches join at π/a when two masses are equal
Normal mode
• Pattern of motion in which all parts of the system move in a sinusoidal fashion, with the same frequency.
• All atoms move 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 centre 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.
•
The velocity of sound
•Max value of optical branch
• By substituting these values of ω in α (relative amplitude) equation & using cos(ka/2) ≈1 ,the corresponding values of α as;
• Min value of acoustical brach
Lattice Vibration 4
Number of branches
• For non-Bravais 3D lattice, unit cell contains two or more atoms
• For any
k
direction,3z dispersion curve. 3 acoustic,
(3z − 3) optical.
• The dispersion relation yields acoustic & optical branches:
For monatom(z = 1)
• Three degrees of freedom: 3 branches
• 1 LA, 2 TA.
•Example:
Plumbum
(FCC)
For diatoms(z=2)
• 3 degree of freedom: 6 branches
• doubly degenerate TA, doubly degenerate TO, 1 LA,1 LO
Example:
Sodium Chloride (FCC)
Real Crystal System
• In a 3-D crystal, atoms vibrate in three dimensions with two vibration modes and three vibrational branches, one longitudinal and two transverse
• For real crystals, Force constant may vary - the branches are not degenerate
• If polyatom – many optical branch
• In simplest 1D case
• In general 3D case
Solution:
• Three simultaneous equations involving Ax, Ay and Az coupled together and are equivalent to a 3 x 3 matrix equation
•The roots of this equation lead to three different dispersion relations
• All branches pass through origin (all the branches are acoustic)
• When k lies along a direction of
high symmetry
, two branches are transverse acoustic and one is longitudinal acoustic
• Both modes have different frequencies
• At
non-symmetry
directions, waves are not pure longitudinal transverse, but have a mixed character
1st Brillouin Zones of Reciprocal Lattice
• Also called as Weigner-Seitz cell
• 1st BZ is containing information of lattice vibration of solid
• k value in 1st BZ correspond to unique vibrational modes.
• k value outside zone equivalent to k1 inside 1st BZ
2-D lattice
: WS cell is the shaded rectangle bounded by the dashed planes
3-D lattices
: WS cell lead to polyhedron whose planes bisects the line connecting a reciprocal lattice to its neighboring point
1-D lattice
: WS cell is the line segment bounded by two dashed planes
Phonons VS Photons
Phonons
• Quantum of lattice vibrations
• Energies of phonons are quantized
Photons
• Quantum of electromagnetic radiation
• Energies of photons are quantized
Phonon Momentum
• In elastic scattering of a crystal is governed by the wavevector selection rule K’ = K +G,
• G is a vector in the reciprocal lattice
• K is the wavevector of the incident photon
• K’ is the wavevector of the scattered photon.
• If photons interact inelastically with lattice.
Phonon Generation
1.Piezoelectric
2.Thermal Excitation
3.Electron Tunnelling
MOHAMAD FARID IZUAN BIN AZMI
A19SC0169
