non-continuum energy transfer: electrons

AME 60634 Int. Heat Trans.

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Non-Continuum Energy Transfer: Electrons


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The Crystal Lattice• The crystal lattice is the organization of atoms and/or molecules in

a solid

• The lattice constant ‘a’ is the distance between adjacent atoms in the basic structure (~ 4 Å)

• The organization of the atoms is due to bonds between the atoms– Van der Waals (~0.01 eV), hydrogen (~kBT), covalent (~1-10 eV), ionic

(~1-10 eV), metallic (~1-10 eV)

cst-www.nrl.navy.mil/lattice

NaCl Ga4Ni3

simple cubic body-centered cubic

tungsten carbide

hexagonal

a


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The Crystal Lattice• Each electron in an atom has a particular potential energy

– electrons inhabit quantized (discrete) energy states called orbitals– the potential energy V is related to the quantum state, charge, and

distance from the nucleus

• As the atoms come together to form a crystal structure, these potential energies overlap hybridize forming different, quantized energy levels bonds

• This bond is not rigid but more like a spring

potential energy


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The Crystal Lattice – Electron View• The electrons of a single isolated atom occupy atomic orbitals,

which form a discrete (quantized) set of energy levels• Electrons occupy quantized electronic states characterized by

four quantum numbers– energy state (principal) energy levels/orbitals– magnetic state (z-component of orbital angular momentum)– magnitude of orbital angular momentum– spin up or down (spin quantum number)

• Pauli exclusion principle: no 2 electrons can occupy the same exact energy level (i.e., have same set of quantum numbers)

• As atomic spacing decreases (hybridization) atoms begin to share electrons band overlap


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Electrons - Conductors• In the atomic structure, valence electrons are in the outer most

shells– loosely bonded to the nucleus free to move!

• In metals, there are fewer valence electrons occupying the outer shell more places within the shell to move

• When atoms of these types come together (sharing bands as discussed before) electrons can move from atom to atom– electrons in motion makes electricity! (must supply external force –

voltage, temperature, etc.)

• In metals the valence electrons are free to move electrons are the energy carrier

• In insulators the valence shells are fully occupied and there’s nowhere to move energy carriers are now the bond (spring) vibrations (phonons)


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Electrons – Free Electron Model

G. Chen

free electron

In metals, we treat these electrons as free, independent particles• free electron model, electron gas, Fermi gas• still governed by quantum mechanics and statistics

free electron gas


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Electrons – Energy and Momentum

wave function |ψ2| can be thought of as electron probability (or likelihood of an electron being there) Heisenberg uncertainty principle

eigenfunction of Shrödinger’s equationsenergy

momentum

The energy and momentum of a free electron is determined by Schrödinger’s equation for the electron wave function Ψ

We assume a form of the wave function

From here we determine the electron’s energy and momentum

k is again the wave vector


D. B. Go Slide 8

Electrons – Energy and MomentumRecall phonons: we sought a relationship between energy (frequency) and momentum (wave vector) ω = f(k) (dispersion relation)

dispersion relation for an acoustic phonon

dispersion relation for free electron

- we assumed form of the solution:

- we set up a governing equation:

Much like phonons, from the dispersion relation we can determine the density of states, which combined with the occupation will tell us the internal energy specific heat


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Electrons – Energy and k-spaceWe saw with phonons that only discrete values of k (wave vectors) can occur basically, only certain wavelengths can be supported by the atomic structure

real space k-space

Additionally, for electrons, because of the Pauli exclusion principle, each wave vector (k state) can only be occupied by 2 electrons (of opposite spin)

Recall in the analysis of electrons, the wave function was related to the wave vector

It can be shown, that the wave vector may take only certain discrete states (eigenvalues)


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Electrons – Energy and k-spaceWe saw with phonons that only discrete values of k (wave vectors) can occur basically, only certain wavelengths can be supported by the atomic structure

real space k-space

We can describe the allowable momentum states in k-space which takes the form of a circle (2D) or sphere (3D)


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Electrons - Density of States• The density of states (DOS) of a system describes the number of

states (N) at each energy level that are available to be occupied– simple view: think of an auditorium where each tier represents an

energy level

http://pcagreatperformances.org/info/merrill_seating_chart/

greater available seats (N states) in this energy level

fewer available seats (N states) in this energy level

The density of states does not describe if a state is occupied only if the state exists occupation is determined statistically

simple view: the density of states only describes the floorplan & number of seats not the number of tickets sold


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Electrons – Density of StatesDensity of States:

The number of states is determined by examining k-space

With some manipulation, it can be shown that the 3D density of states for electrons is

With some manipulation, it can be shown that the 3D density of states for phonons is


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Electrons – Fermi Levels• The number of possible electron states is simply the integral of the

density of states to the maximum possible energy level.– at T = 0 K this is the equivalent as determining the number of electrons

per unit volume– we put an electron in each state at each energy level and keep filling up

energy states until we run out

• However, the number of electrons in a solid can be determined by the atomic structure and lattice geometry known quantity

• We call this maximum possible energy level the Fermi energy and we can similarly define the Fermi momentum, and Fermi temperature


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Electrons - Occupation• The occupation of energy states for T > 0 K is determined by the

Fermi-Dirac distribution (electrons are fermions)

• Electrons near the Fermi level can be thermally excited to higher energy states

electron number density

ε F =

5 e

V

1000 K

300 K

εF = 5 eV


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Electrons – Specific Heat

total electron energy

specific heat

If we know how many electrons (statistics), how much energy for an electron, how many at each energy level (density of states) total energy stored by the electrons! SPECIFIC HEAT

For total specific heat, we combine the phonon and electron contributions

Basic relationships


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Electrons – Electrical & Thermal Transport• Thus far, we have determined electron energy and energy storage

by assuming a free electron model freely moving electrons

• We can also use the free electron approach to predict electrical and thermal transport limited applicability (what about the lattice!)– we attempt to correct for real structure by using an effective mass m*

(greater than real mass)

• We can quickly assess the electrical transport by a simple application of Newton’s law


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Electrons – Electrical TransportNewton’s 2nd Law

Coulombic force drag due to collisions

The steady-state solution gives the average electron “drift” velocity

The current density is the rate of charge transport per unit area (like heat flux)

compare to Ohm’s law!

Therefore, the electrical conductivity is simply

what is the relaxation time/mean free path?


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Electrons – Scattering Processes• Electrons will scatter off phonons and impurities but not the static

crystal ions

• For the time being, let’s assume we know these mean free paths. We can combine them using Matthiesen’s rule

• We now know the electrical conductivity

• What about the thermal conductivity? kinetic theory still applies!


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Electons – Thermal Conductivity• Recall from kinetic theory we can describe the heat flux as

• Leading to

Fourier’s Law

what is the mean time between collisions?


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Electrons – Thermal ConductivityLet’s assume we know the mean free path, for the time being …

these appear related!

Wiedemann-Franz Ratio

- if we know (measure) one we can find the other- based on the fundamental assumption that the electrical and thermal

mean free paths are equivalent- good at high and low T- based on the FREE ELECTRON MODEL


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Electrons – Free Electron Model• Limits of Free Electron Model

– poorly predicts some aspects of thermal/electrical transport– poorly predicts magnitude of specific heat– poorly predicts magnetic properties– does not explain difference between metal and insulator!

• To properly understand electrons we must account for their interactions with the lattice– we will not go into these details, but you should appreciate the

implications– this enables us to understand the different types of materials & why

computers, photovoltaics, etc. work!

Recall that free electron energy is parabolic!

- like phonons, there is also a Brillouin zone where the dispersion relation repeats


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Electrons – Effect of Lattice

band gap!


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Electrons – Effect of Lattice


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Electrons – Band Gaps

G. Chen


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Electrons – Material Types

G. Chen


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Electrons – What We’ve Learned• Electrons are particles with quantized energy states

– store and transport thermal and electrical energy– primary energy carriers in metals– usually approximate their behavior using the Free Electron Model

• energy• wavelength (wave vector)

• Electrons have a statistical occupation, quantized (discrete) energy, and only limited numbers at each energy level (density of states)

– we can derive the specific heat!

• We can treat electrons as particles and therefore determine the thermal conductivity based on kinetic theory

– Wiedemann Franz relates thermal conductivity to electrical conductivity

• In real materials, the free electron model is limited because it does not account for interactions with the lattice– energy band is not continuous– the filling of energy bands and band gaps determine whether a material

is a conductor, insulator, or semi-conductor

non-continuum energy transfer: electrons

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