ece 874: physical electronics

ECE 874:Physical Electronics

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

VM Ayres, ECE874, F12

Lecture 14, 16 Oct 12

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Three different “a”:- lattice constant- “unit cell” of a periodic potential, p.52- well width

I

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Lattice constant a of the Unit cellGaAs: 5.65 Ang

I

“unit cell” a of a 1-D periodic potentialBlock theorem

Well width aKronig-Penney model for a 1-D periodic potential

VM Ayres, ECE874, F12

Example problem: An electron is moving along the [110] direction in GaAs, lattice constant = 5.65 Ang.

+z

+x

+y

(a) Write down both versions of the Block theorem explicitly solving for the “unit cell” of the periodic potential in terms of the lattice constant.

(b) Draw a model of the transport environment using the Kronig-Penney model where the well width is 20% of the “unit cell” of a periodic potential. Write the dimensions in terms of the lattice constant.

VM Ayres, ECE874, F12

Example problem: An electron is moving along the [110] direction in GaAs, lattice constant = 5.65 Ang.

+z

+x

+y

[110]

Face diagonal distance = ✔2 a

Distance between atoms = 2 a/2✔

VM Ayres, ECE874, F12

VM Ayres, ECE874, F12

+z

+x

+y

+z

+x

+y

[110]

[110]

Rotate [110] to go “straight”

(b)

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b + a = aBl = 2 a✔ LC/2 = 3.995 Ang

[110]

b = 0.8 (aBl = 3.995 Ang)

= 3.196 Ang

aKP = 0.2 (aBl = 3.995 Ang)

= 0.799 Ang

b aKP

(b)

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Finite Well boundary conditions, Chp. 02:

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Finite Well allowed energy levels, Chp. 02:

Graphical solution for number and values of energy levels E1, E2,…in eV.

a is the finite well width.

VM Ayres, ECE874, F12

Similar for Kronig-Penney model but new periodicity requirements:

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Kronig-Penney model allowed energy levels, Chp. 03:

Graphical solution for number and values of energy levels E1, E2,…in eV.

a = width of well, b = width of barrier, a + b = Block periodicity aBl

VM Ayres, ECE874, F12

Kronig-Penney model allowed energy levels, Chp. 03:

Graphical solution for number and values of energy levels E1, E2,…in eV.Also have values for k from RHS.

VM Ayres, ECE874, F12

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.

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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