Physical Metallurgy6th Lecture

MS&E 410

D.Ast

dast@ccmr.cornell.edu

255 4140

The rules are

1. Conductors have either partially filled bands Na) or full bands that overlap into the next empty band (Mg)

2. Insulators have full bands separated by a large gap from the next empty band

3. Semiconductors have full bands separated by a small gap (max about 1.6 eV) from the next empty band

Review: Monovalent, Alkali

Divalent, Alkali-Earth

Why the Hall Effect is positive in Earth Alkali

1. Only the electrons at the Fermi surface can move (moving means adding kinetic energy and the only ones that can “move up” are the one at the surface).

2. The Hall effect applies a magnetic field. The electrons, in a monovalent metal filled curve around an electron space. Hall effect negative

3. Electrons curving around an “empty space” act like holes. They can do so only if the Fermi surfaces are connected across zone

You have to “fold the second Brillouin zone back into the first one” which generates strange figures known as monster, coronet, cigars...

• The “effective mass” is large when E vs k is gently curved, is infinite at an inflection point, and is small when E vs k is sharply curved.

• Hence, the holes, making small orbits, have a small effective mass, accelerate fast in an electric field and hence have a high mobility

•The electrons doing big orbits around the belly are the reverse

• So they don’t count much, overall, and Rhall is positive

What about the electrons that spilled in the second zone ?

Well, I will leave this as a question for you to to ponder...

First zone

Energy, group velocity, and effective mass vs k in the

cosine band

Is a negative mass hocus pocus ? No !!!!!

• There are a number of EE devices, the most famous of which is the Esaki Diode that use the effect that an electron with a positive effective mass can be “converted” to one with a negative effective mass (I.e. a hole) by adding energy to it in an electric field !!!

I will skip the rest of the Baker drawings (they are in our folder) but say a few words about

Because it very strange that is occurs at all !!

• d electrons having the same spin avoid each other. This is the origin of Hunds rule: It keeps the apart in the electron cloud around the nucleus, minimizing the repulsive energy between them

• The avoidance is known as the exchange energy.

Density of electrons of the same spin, energy momentum around an electron

The exchange “hole” can be derived without QM using the correspondence principle and the image charge in electrostatics

Origin of the “exchange energy”

1st Fe

4 unpaired 3d spins in shell 3

Fe

4 unpaired 3d spins in shell 3

For the d electrons to bond (and they do, see last lecture: cohesive energy vs Nd) they must have opposite spins

No Net Moment !!

If the 3d would share entirely locally, the d shell would be complete --- they don’t but there is a covalent component

Feynman* picture:The electron sea does not have a random spin but tends to get polarized opposite to 3d shell net spin in the vicinity of an atom. Thus we have

Richard Feynman, Physicist extraordinare, Nobel Prize Winner, Bon Vivant, and the man that analyzed the Challenger Shuttle disaster. Perhaps you should Google him ?

Fe E-gas Fe

And the spins of the Fe are lined up !!!! The counter spin of the E-gas reduces the netspin from 4, to 2.2. (see lecture 5)

This model, discussed in Feynman’s book show that the exchange energy provides the force that lines up the Fe atom spins.

Now let’s proceed to a slightly more sophisticated view, based on “chemical” quantum theory.

In an atom the spins of electrons of the same energy like to be parallel. Hunds rule. Reason: They then avoid each other, and this lowers their mutual repulsion energy

In a metal, the electrons are delocalized as “Bloch waves” which you can consider to be “giant molecular orbitals”

Now, if there would be 5 Bloch waves of equal energy, Hunds’s rule says that they would have 5 electrons of parallel spin !

Continuing the “giant molecular orbit” picture:

• obeying Hunds rule, the electrons in the “5 giant molecular wave functions” would avoid each other, lowering electrostatic repulsion energy.

• UNFORTUNATELY, the wave functions “need to fit in a box” and don’t come from the solution of a spherically symmetric potential. Each state can hold only two electrons of opposite spin.

• If we insist on electrons with all the same spin, we “climb up” the energy ladder twice as fast as if we allow opposite spins. That increases the electron energy.

• There is a trade off between “exchange energy” and kinetic energy (climbing up E vs k in the free electron picture is gaining kinetic energy

1. The spin aligned d electrons forming a solid lower their energy by “delocalizing” combining into a giant wave function that fills the entire crystal.

2. In the case of magnetic transition metals, the d electrons go into states with parallel spins. Thus each of these giant wave functions can be filled with one electron only (normal would be two of opposite spin).

3. This forces these spin aligned electrons to take up higher energies than otherwise.

4. If the density of states is high, the energy penalty paid is small.

5. In the transition metals, the d band is very narrow in E and D(E) is very high. Thus the “climb up penalty is small.

6. Magnetism occurs when the gain in exchange energy (one spin per state) is larger than the gain in kinetic (delocalization) energy (two spins per state)

Insulators

An insulator is always in touch with metallic contacts

This introduces a second condition…..

A insulator needs to have

• A large bandgap (Eg> 1.6 eV) to ensure that the naturally present population of thermal carries (generated by bond breaking) is well below 1010/cm3

and it needs to have:

• A large number of traps in the bandgap to ensure that carriers injected by illumination, or from contacts get trapped immediately before they can pick up enough energy from the electric field to do impact ionization.

Left : Bubble raft Right: Schematic

Because a bubble raft is dominated by nearest neighbor forces it makes a good metal model

Terminology

An interstitial (and mobile) Fe+ ion and a substitutional B-

dopant atom in the Si lattice.

Fe-B pair*

The resulting Fe-B pair a) takes out the dopant action of B b) inflicts instability on solar cells

The Au(3)Au system is the father of all order-disorder studies

First investigated by x-rays by B. Warren it continues to occupy the physics community.. There must be thousands of papers on it.

ORDERED ALLOYS

Around 400 C the ordered structure transforms into a random solid solution of Cu and Au.

The experimental transition enthalpy is 12meV/atom.

What’s nice about it is

• it transforms before it melts !

• It makes decent x-ray reflection (I.e. crystal quality is good :-)

The mother of all intermetallic compounds is Ni3Al.

You would not be able to fly home without it !

Why ?

The phase field of Ni3Al is almost as narrow as that of good covalent compound such as GaAs!

And just as stable at high T !

HW-6.1

Google “purple plague” and “failure” and report

a) What is purple plague

b) What is formed (formula(s)) ?

c) under what conditions does if form ?

d) Why is the formation detrimental (one sentence)

Note: It besets both IC failure and optical failures

An optical efficiency problem was reported in the Naval Research Laboratory ATM flight instrument at Ball Brothers Research Corporation. A failure analysis review attended by experts from Bausch & Lomb, Ft. Belvoir, Virginia, Goddard Space Flight Center, Naval Research Laboratory, and MSC concluded that failure was due to the "Purple Plague," an aluminum coating overcoating on gold. An alert would be released by Goddard for dissemination throughout NASA and the Air Force to preclude further use of this particular method of coating optics.

"Weekly Progress and Problem Summaries for the Administrator Skylab Program," 27 May and 17 June 1971; MSFC, "Weekly Activity Reports," 26 May and 3 June 1971.

THE END

When World War II ended Dr. Feynman was invited by Dr. Bethe to teach at Cornell, and he accepted. Within four years he had completed the work for which he won the Nobel Prize, in quantum electrodynamics, or QED.