ece 874: physical electronics
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ECE 874: Physical Electronics. Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University [email protected]. Lecture 14, 16 Oct 12. Three different “a”: - lattice constant - “unit cell” of a periodic potential, p.52 - well width. I. I. - PowerPoint PPT PresentationTRANSCRIPT
ECE 874:Physical Electronics
Prof. Virginia AyresElectrical & Computer EngineeringMichigan State [email protected]
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
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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.
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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✔
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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.
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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
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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.Also have values for k from RHS.
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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.
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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