ECE 874:Physical Electronics

Prof. Virginia AyresElectrical & Computer EngineeringMichigan State Universityayresv@msu.edu

VM Ayres, ECE874, F12

Lecture 15, 18 Oct 12

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Example problem:(a) What are the allowed (normalized) energies and also the forbidden energy gaps for the 1st-3rd energy bands of the crystal system shown below?(b) What are the corresponding (energy, momentum) values? Take three equally spaced k values from each energy band.

k = 0

k = ± a + b

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0.5k = 0

k = ± a + b

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(a)

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(b)

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“Reduced zone” representation of allowed E-k states in a 1-D crystal

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k = 0

k = ± a + b

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(b)

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“Reduced zone” representation of allowed E-k states in a 1-D crystal

This gave you the same allowed energies paired with the same momentum values, in the opposite momentum vector direction.

Always remember that momentum is a vector with magnitude and direction. You can easily have the same magnitude and a different direction.

Energy is a scalar: single value.

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Can also show the same information as an “Extended zone representation” to compare the crystal results with the free carrier results.Assign a “next” k range when you move to a higher energy band.

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Example problem: There’s a band missing in this picture. Identify it and fill it in in the reduced zone representation and show with arrows where it goes in the extended zone representation.

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The missing band: Band 2

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Notice that upper energy levels are getting closer to the free energy values. Makes sense: the more energy an electron “has” the less it even notices the well and barrier regions of the periodic potential as it transports past them.

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Note that at 0 and ±/(a+b) the tangent to each curve is flat:dE/dk = 0

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A Brillouin zone is basically the allowed momentum range associated with each allowed energy band

Allowed energy levels: if these are closely spaced energy levels they are called “energy bands”Allowed k values are the Brillouin zones

Both (E, k) are created by the crystal situation U(x). The allowed energy levels are occupied – or not – by electrons

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VM Ayres, ECE874, F12

(b)

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What happens to the e- in response to the application of an external force: example: a Coulomb force F = qE (Pr. 3.5):

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(d)

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(d)

[100][100]<100> type6 of these

<111> type8 of these

Warning: you will see a lot of literature in which people get careless about <direction type> versus [specific direction]

SymmetricConduction energy bands

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(d) <111> and <100> type transport directions certainly have different values for aBlock spacings of atomic cores. The , X, and L labels are a generic way to deal with this.

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Two points before moving on to effective mass:

Kronig-Penney boundary conditions Crystal momentum, the Uncertainty Principle and

wavepackets

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Boundary conditions for Kronig-Penney model:

Can you write these blurry boundary conditions without looking them up?

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Locate the boundaries:

[transport direction p 56]

b aKP

aKP + b = aBlock

-b a0

-ba

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Locate the boundaries: into and out of the well.

[transport direction p 56]

b aKP

aKP + b = aBlock

-b a0

-ba

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Boundary conditions for Kronig-Penney model, p. 57:

Is the a in these equations aKP or aBl?

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Boundary conditions for Kronig-Penney model, p. 57:

Is the a in these equations aKP or aBl? It is aKP.

VM Ayres, ECE874, F12

Two points before moving on to effective mass:

Kronig-Penney boundary conditions Crystal momentum, the Uncertainty Principle and

wavepackets

VM Ayres, ECE874, F12

VM Ayres, ECE874, F12

Chp. 04: learn how to find the probability that an e- actually makes it into - “occupies” - a given energy level E.

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k2

k wavenumber Chp. 02

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Suppose U(x) is a Kronig-Penney model for a crystal.

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On E-axis: Allowed energy levels in a crystal, which an e- may occupy

hbark = crystal momentumhttp://en.wikipedia.org/wiki/Crystal_momentum

So a dispersion diagram is all about crystal stuff but there is an easy to understand connection between crystal energy levels E and e- ‘s occupying them.

The confusion with momentum is that an e-’s real momentum is a particle not a wave property.

Which brings us to the need for wavepackets.