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The Crystal Lattice

simple cubic

body-centered cubic

hexagonal

a

Ga4Ni3

tungsten carbide

NaCl

cst-www.nrl.navy.mil/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)

The Crystal Lattice

potential energy

- 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

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

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)

Electrons – Free Electron Model

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

G. Chen

free electron

free electron gas

Electrons – Energy and Momentum

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

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

We assume a form of the wave function

k is again the wave vector

From here we determine the electron’s energy and momentum

energy

eigenfunction of Shrödinger’s equations

momentum

Electrons – Energy and Momentum

- Recall phonons: we sought a relationship between energy (frequency) and momentum (wave vector) ω = f(k)(dispersion relation)

- we set up a governing equation:

- we assumed form of the solution:

dispersion relation for an acoustic phonon

dispersion relation for free electron

- 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

Electrons – Energy and k-space

We 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

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)

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

Electrons – Energy and k-space

We 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)

Electrons - Density of States

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 occupiedonly 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

- 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/

Electrons – Density of States

Density 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

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

Electrons - Occupation

electron number density

300 K

εF = 5 eV

εF = 5 eV

1000 K

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

Electrons – 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

total electron energy

specific heat

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

Basic relationships

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

Electrons – Electrical Transport

Newton’s 2nd Law

drag due to collisions

Coulombic force

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?

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!

Electons – Thermal Conductivity

Fourier’s Law

what is the mean time between collisions?

Recall from kinetic theory we can describe the heat flux as

Leading to

Electrons – Thermal Conductivity

Let’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

Electrons – Free Electron Model

Recall that free electron energy is parabolic!

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

- 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!

Electrons – Effect of Lattice

band gap!

Electrons – Band Gaps

G. Chen

Electrons – Material Types

G. Chen

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

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