Specific Heat of Solids Quantum Size Effect on the Specific Heat Electrical and Thermal Conductivities of Solids Thermoelectricity Classical Size Effect on Conductivities and Quantum Conductance THERMAL PROPERTIES OF SOLIDS AMD THE SIZE EFFECT

Lattice Vibration in Solids: Phonon Gas Specific Heat of Solids  Atoms in solids Inter-atomic forces keep them in position only move around by vibrations near their equilibrium positions Specific heat of bulk solids: lattice vibration, free electron macroscopic behavior from microscopic point of view  Lattice Periodical array of atoms Lattice vibration contribute to thermal energy storage and heat conduction

 Einstein Specific Heat Model 3 vibrational degree of freedom high-temperature limit of the specific heat of elementary solids good prediction for specific heat of solid at high temperature  Dulong-Petit law  Einstein Specific Heat Model simple harmonic oscillator model each atom : independent oscillator All atoms vibrate at the same frequency. quantized energy level specific heat as a function of temperature

Quantized energy of atoms Vibrational partition function Einstein temperature

Internal energy Specific heat vibration along single axis vibration along three axes

Limitation of Einstein model Einstein specific heat is significantly lower than the experimental data in the intermediate range. Force-spring interaction must be considered.

Debye Specific Heat Model  Assumption Velocity of sound: same in all crystalline directions and for all frequencies a high-frequency cutoff and no vibration beyond this frequency (shortest wavelength of lattice wave should be on the order of lattice constants) Upper bound m, determined by the total number of vibration modes 3N ( N : number of atoms) Vibrations inside the whole crystal just like standing waves  Equilibrium distribution Phonon : Bose-Einstein statistics The total number of phonons is not conserved since it depends on temperature.

 BE statistics The energy levels are so closely spaced that it can be regarded as a continuous function : BE distribution function total number of phonons is not conserved

D( ) : density of states of phonons the number of quantum states per unit volume per unit frequency or energy ( h ) interval v a : weighted average speed  Degeneracy of phonons The number of quantum states per unit volume in the phase space Given volume V and within a spherical shell in the momentum space (from p to p + dp )

Relation between f( ) and f BE ( )

upper limit of the frequency (due to lattice constant) Total number of quantum states must be equal to 3N Debye temperature n a = N/V : number density of atoms

Vibration contribution to the internal energy Distribution function for phonons vibration contribution to internal energy

Let

Molar specific heat

When

Relative difference between calculated value and experimental data : about 5% When  Riemann zeta function n

agrees with experiments within a few percent When

Debye Model vs. Einstein Model

Free Electron Gas in Metals Free electron Translational motion of free electrons within the solid: largely responsible to the electrical and thermal conductivities of metals : Fermi energy  at 0 K  Fermi-Dirac distribution

due to positive and negative spins in terms of the electron speed v Degeneracy for electrons Distribution function in terms of the kinetic energy of the electron using

 Density of states for free electrons  Fermi Level, The number density of electrons as Since  –  < 0,

Since the difference between  (T) and  F is small,  Sommerfeld expansion Apply FD statistics to study free electrons in metals Resolve the difficulty in the classical theory for electron specific heat Number of electrons does not depends on temperature

Internal energy

= Electronic contribution + Lattice contribution Specific heat of free electrons Electronic contribution to the specific heat of solids is negligible except at very low temperatures (a few kelvins or less)

Quantum Size Effect on the Specific Heat For nanoscale structures such as 2-D thin film and supperlattices, 1-D nanowires and nanotubes, or 0-D quantum dots and nanocrystals, substitution of summation by integration is no longer appropriate. 2-D thin film: confined in one dimension 1-D nanowires: in two dimensions 0-D quantum dots: in three dimensions

1-D chain of N + 1 atoms in a solid with dimension L with end nodes being fixed in position N number of vibrational modes eigenfunctions

Periodic Boundary Conditions Born-von Kármán lattice model medium: an infinite extension with periodic boundary conditions standing wave solutions for a solid with dimensions of L x, L y, L z (see Appendix B.7) : lattice wavevector total number of modes = total number of atoms along the 1-D chain

General Expressions of Lattice Specific Heat lattice vibrational energy : static energy at T = 0 K : zero-point energy K : wavevector index, dispersion relation P : polarization index

specific heat in terms of density of states density of states : the number of states (or modes) per unit volume and per unit energy interval

3-D view volume of one quantum state Dimensionality: density of states 3-D reciprocal lattice space or k -space number of quantum states per unit volume :

For a linear dispersion relation For a single polarization Eq.(5-7)

2-D reciprocal lattice space or k -space 2-D projection volume of one unit cell number of quantum states per unit area : thin film, superlattice When

1-D reciprocal lattice space or k -space number of quantum states per unit length : nanowires, nanotubes When independent of frequency

Thin Films Including Quantum Wells Ex 5-4: specific heat of thin film thin film made of a monatomic solid film thickness L with q monatomic layers, L = qL 0 average acoustic speed v a independent of temperature......

molar specific heat total number of modes for q = 1, 3, 5, … for q = 2, 4, 6, … specific heat per unit volume

The lattice is infinitely extended in the x and y directions

k D : cutoff value determined by setting the total number of modes equal to the number of atoms per unit area total number of modes total number of atoms......

For a single layer for q = 1, 3, 5, … for q = 2, 4, 6, …

As In 3-D case, dispersion relation transformation

Electrical and Thermal Conductivities of Solids Electrical Conductivity e-e- A V L z r e : resistivity,  : conductivity A c : cross-sectional area, J : current density E : electric filed

3 Neweton’s 2 nd law Current density J : drift velocity  : relaxation time n e : electron number

 thermal conductivity (kinetic theory) Thermal Conductivity of Metals

 Wiedemann-Franz law Lz : Lorentz number

Derivation of Conductivities from the BTE  Distribution function local equilibrium, relaxation time approximation D(  ) : density of states

steady state, relaxation time approximation electric filed E along with temperature gradient in z direction

Assume under local equilibrium In the case of no Temperature gradient

: electron number density

Note that : Dirac delta fucrtion Assume that

Drude-Lorentz expression

 Thermal Conductivity In the case of no electric field, E = 0 for an open system of fixed volume, energy flux particle flux

where

same result as simple kinetic theory

kinetic theory Thermal Conductivity of Insulators under local equilibrium for isotropic distribution in k -space

for a large system with isotropic dispersion x m : corresponds to maximum frequency of each phonon polarization