What is the role of phonons and electron gas in heat transport in micro scales?

[hanson]

hi all! I am not a physics but enginereing major.
I am confusing about what's going on with heat tranport in micro- scales.
I encounter terms like phonon-electron interaction, electron gas, lattice etc..
What exactly is a phonon? does it posses mass? What is electron gas?

Could anyone explain in layman terms the energy tranport in micro scale?

 

[Gokul43201]

Heat is simply the total kinetic energy of the various parts of an object at rest. If your object is a container of gas or liquid, the "parts" are simply the atoms/molecules that are in random, Brownian motion. If the object is a solid, these atoms/molecules are held in certain fixed positions about which they may execute small oscillations. The kinetic energy of these oscillating atoms tells you how hot the solid is.

A standard analogy for picturing a solid is as a network (or lattice) of masses (balls, say) connected by springs. If you take a corner of this spring network and shake it a little, you will find that the oscillations are comminucated from this corner to the rest of the solid through the intervening springs, and with time, farther and farther parts of the spring-network start dancing. Given enough time, you'll notice that all parts of the network are oscillating with roughly the same amplitude.

These collective oscillations of all the atoms of a solid are what are known as phonons. They are things that transport a fixed amount of energy through the solid (and note that a single phonon is not the oscillation of a single atom; it is a tiny collective oscillation of all the atoms). If there are no other "parts" to the solid, then the energy in these phonon modes tells you how hot a solid is. Just like the springs, these phonons carry energy (heat) from one point on the solid to other points.

There are some solids which have "parts" other than the fixed atoms/molecules. These are metals. In a metal, you have these fixed atoms (or ions), but you also have a collection of electrons contributed by each of these atoms, that are essentially free to move about the entire solid. These nearly free electrons behave somewhat like the atoms of a gas (or liquid) - they move (nearly) freely until they hit something hard; then they do a sharp turn and move off in another direction, and so on. It is for this reason that the free electrons in a solid can be treated as an electrons gas (or an electron liquid). Now since the free electrons have kinetic energies that are separate from that of the fixed ions, they too can carry heat.

So, in a general solid, heat is carried by phonons and free electrons. The fewer the number of free electrons in the solid, the smaller is the capacity to hold and transfer heat using the electrons. It is for this reason that metals typically are good thermal conductors - they have a large number of free electrons to carry heat. And non-metals tend to have a lower thermal conductivity, unless their phonons are very good at carrying heat (eg: diamond, which has very strong covalent bonds, is a good thermal conductor even though it has essentially no free electrons).

The total thermal conductivity of a solid is hence, the sum of the electronic and phononic contributions. But these contributions are not independent of each other or of the temperature of the solid. The ability of electrons to transport heat, for instance, is hindered if they keep bumping into phonons (the vibrating lattice). It is for this reason that metals have a higher electrical resistivity at higher temperatures. In fact, in a pure metal, the electrical resistance is essentially nothing but a result of these electron-phonon interactions, which disrupt the momentum of the conducting electrons.

 

[hanson]

Thank you Gokul for your detailed and comprehensive explanation! I have a much better idea of what's going on after reading your passage.

But I still don't quite grasp the idea of "phonons" very well.
You said that phonons are collective oscillations of ALL the atoms of a solid. Say, IF, there are a cubic lattice, the top left corner was shaked a little bit, then the "springs" connecting the atoms at that particular corner are transferring the energy/oscillation to the remaining atoms of the entire lattice. Is this a phonon? (Does it violate the rule of ALL atoms?) Is a phonon belongs to the entire lattice or a small group of atoms?

I also hear of "lattice frequency", and the equation E=hv, which shall be the energy carried by the phonon, right? Would incoporating these concepts faciliate the explanation?

Please kindly help.

 

[Gokul43201]

A phonon is a packet of energy.

The total energy of the oscillating system can be incremented or decremented by only integer multiples of a certain fixed amount. This smallest difference is called a phonon and is treated like a "particle" becuase that allows us to study the changes in the energy of the system as the creation and destruction of these particles.

 

[hanson]

But what does it mean by the collision between phonons? The frequency of the phonon will be altered after a collision?

 

[Astronuc]

But what does it mean by the collision between phonons? The frequency of the phonon will be altered after a collision?

Collision of phonons is much like the collision of water waves (or waves in solids). The frequency of the phonons is dependent upon the properties of the material, e.g. bulk/elastic modulus, atomic mass.

This might help a little - mse.stanford.edu/faculty/clemens/Lect19.pdf

I'll see if I can find some other basic material. I expect that this subject is not taught at the undergraduate level, or at least is not very common.

There are many considerations with respect to thermal conductivity, such as chemical homogeneity. Gokul provided the example of diamond which is very or relatively pure carbon. So the nano-properties are relatively uniform. Comparable to this would be single crystal, chemically pure elements.

Polycrystalline structures and alloys complicate the phenomenon because atoms of different mass vibrate at different frequencies, and have different atomic binding energies. The mechanical state, or dislocation (defect) density is another factor - phonons scatter off defects (which really means that momentum/energy is transferred to atoms which are at some angle from the direction of the phonon velocity vector.

Ceramics, particularly insulators, have very different behavior of thermal conductivity than do metals.

Thermal conductivity in metals is dominated by the electron gas, which are the conduction electrons.

This might also help -
http://en.wikipedia.org/wiki/Phonon