ECE 875: Electronic Devices

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ECE 875Electronic Devices

• Prof. Virginia Ayres
• Electrical Computer Engineering
• Michigan State University
• ayresv_at_msu.edu

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Course Content Core Part I Semiconductor
Physics Chapter 01 Physics and Properties of
Semiconductors a Review Part II Device
Building Blocks Chapter 02 p-n Junctions Chapter
03 Metal-Semiconductor Contacts Chapter
04 Metal-Insulator-Semiconductor Capacitors
Part III Transistors Chapter 06 MOSFETs
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Course Content Beyond core
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Lecture 02, 10 Jan 14
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Crystal Structures Motivation
Electronics Transport e-s moving in an
environment Correct e- wave function in a
crystal environment Block function Y(R)
expik.a y(R) Y(R a) Correct E-k energy
levels versus direction of the environment
minimum Egap Correct concentrations of
carriers n and p Correct current and current
density J moving carriers I-V measurement J
Vext direction versus internal E-k Egap
direction Fixed e-s and holes C-V measurement
(KE PE) Y EY
x Probability f0 that energy level is occupied
q n, p velocity Area
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Unit cells
A Unit cell is a convenient but not minimal
volume that contains an atomic arrangement that
shows the important symmetries of the crystal
Why are Unit cells like these not good
enough? Compare Sze Pr. 01(a) for fcc versus
Pr. 03
Non-cubic
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fcc lattice, to match Pr. 03
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14 atoms needed
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Crystal Structures MotivationElectronics
Transport e-s moving in an environment
Correct e- wave function in a crystal
environment Block function Y(R) expik.a y(R)
Y(R a)
Periodicity of the environment Need specify
where the atoms are Unit cell a3 for cubic
systems sc, fcc, bcc, etc. OR Primitive cell for
sc, fcc, bcc, etc. OR Atomic basis
Think about need to specify
Most atoms
Fewer atoms
Least atoms
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A primitive Unit cell is the minimal volume that
contains an atomic arrangement that shows the
important symmetries of the crystal
Example What makes a face-centered cubic
arrangement of atoms unique? Hint Unique means
unique arrangement of atoms within an a3 cube.
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Answer Atoms on the faces
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Answer Atoms on the faces Also need two corner
atoms that give maximum dimension of volume of a3
cubic Unit cell
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Answer Atoms on the faces Also need two corner
atoms that give maximum dimension of volume of a3
cubic Unit cell This arrangement of 8 atoms does
represent the fcc primitive cell
But specifying the arrangement of 8 atoms is a
complicated description. There is a simpler way.
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• Switch to a simpler example
• How many atoms do you need to describe this
  simple cubic structure?
• Want to specify
• atomic arrangement
• minimal volume a3 for this structure

Start 8 atoms,1 on each corner. Do you need all
of them?
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Simpler Example Answer 4 atoms and 3 vectors
between them give the minimal volume l x w x
h. 4 red atoms Specify 3 vectors a l a x
0 y 0 z b w 0 x a y 0 z c h 0 x
0 y a z Minimal Vol a . b x c Specify the
atomic arrangement as one atom at every vertex
of the minimal volume.
h
w
l
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Return to fcc primitive cell example 8
atoms Simpler description 4 atoms and 3
vectors between them give the volume of a
non-orthogonal solid (parallelepiped) p.11
Volume a . b x c Specify the atomic
arrangement as one atom at every vertex.
rotate
b
c
a
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Better picture of the fcc parallelepiped
tilt
rotate
Ashcroft Mermin
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This is what Sze does in Chp.01, Pr. 03
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Picture and coordinate system for Pr. 03 For a
face centered cubic, the volume of a conventional
unit cell is a3. Find the volume of an fcc
primitive cell with three basis vectors (000) -gt
(a/2,0,a/2) a a/2 x 0 y a/2 z (000) -gt
(a/2,a/2,0) b a/2 x a/2 y 0 z (000) -gt
(0,a/2,a/2) c 0 x a/2 y a/2 z
z
c
a
y
b
(000)
x
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Volume of fcc primitive cell
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Sze, Chp.01, Pr. 03
For a face centered cubic, the volume of a
conventional unit cell is a3. Find the volume of
an fcc primitive cell with three basis
vectors (000) -gt (a/2,0,a/2) a a/2 x 0 y
a/2 z (000) -gt (a/2,a/2,0) b a/2 x a/2 y 0
z (000) -gt (0,a/2,a/2) c 0 x a/2 y a/2
z a, b and c are the primitive vectors of the
fcc Bravais lattice. P. 10 Three primitive
basis vectors a, b, and c of a primitive cell
describe a crystalline solid such that the
crystal structure remains invariant under
translation through any vector that is the sum of
integral multiples of these basis vectors. In
other words, the direct lattice sites can be
defined by the set R ma nb
pc. Translational invariance is great for
describing an e- wave function acknowledging the
symmetries of its crystal environment Block
function Y(R) expik.a y(R) Y(R a)
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Formal definition of a Primitive cell, Ashcroft
and Mermin A volume of space that when
translated through all the vectors of a Bravais
lattice just fills all the space without either
overlapping itself of leaving voids is called a
primitive cell or a primitive Unit cell of the
lattice.
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Steps for fcc were
1. four atoms
a a/2 x 0 y a/2 z b a/2 x a/2 y 0 z c
0 x a/2 y a/2 z
2. three vectors between them Anywhere R ma
nb pc
3. minimal Vol a3/4 (parallelepiped) atom at
each vertex of the minimal volume
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Typo, p. 08
No! Figure 1 shows conventional Unit cells!
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Conventional Unit Cells Vol. a3
Primitive Unit Cells Smaller Volumes
Vol a3/4
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Primitive cell for fcc is also the primitive cell
for diamond and zincblende
Conventional cubic Unit cell
Primitive cell for fcc, diamond and zinc-blende
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P. 08 the diamond (and zinc-blende) lattices
can be considered as two inter-penetrating fcc
lattices.
The two interpenetrating fcc lattices are
displaced (¼, ¼, ¼) x aNote also have pairs of
atoms displaced (¼, ¼, ¼) x a
a lattice constant
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P. 08 the diamond (and zinc-blende) lattices
can be considered as two inter-penetrating fcc
lattices.
The two interpenetrating fcc lattices are
displaced (¼, ¼, ¼) x aNote also have pairs of
atoms displaced (¼, ¼, ¼) x a
a lattice constant
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Example
What are the three primitive basis vectors for
the diamond primitive cell? (000) -gt (a/2,0,a/2)
a a/2 x 0 y a/2 z (000) -gt (a/2,a/2,0) b
a/2 x a/2 y 0 z (000) -gt (0,a/2,a/2) c 0 x
a/2 y a/2 z How to make it diamond two-atom
basis
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Picture and coordinate system for example problem
(000)
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Answer
Three basis vectors for the diamond primitive
cell (000) -gt (a/2,0,a/2) a a/2 x 0 y a/2
z (000) -gt (a/2,a/2,0) b a/2 x a/2 y 0
z (000) -gt (0,a/2,a/2) c 0 x a/2 y a/2 z
Same basis vectors as fcc Same primitive cell
volume a3/4
Make it diamond by specifying the atomic
arrangement as a two-atom basis at every vertex
of the primitive cell. Pair a 2nd atom at (¼ ,
¼, ¼) x a with every fcc atom in the primitive
cell
(000)
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Rock salt
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Rock salt can be also considered as two
inter-penetrating fcc lattices.Discussion Lec
03 13 Jan 14
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Direct space (lattice)
Direct space (lattice)
Conventional cubic Unit cell
Primitive cell for fcc, diamond, zinc-blende,
and rock salt
Rock salt
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Direct space (lattice)
Direct space (lattice)
Reciprocal space (lattice)
Conventional cubic Unit cell
Primitive cell for fcc, diamond, zinc-blende,
and rock salt
Reciprocal space first Brillouin zone for
fcc, diamond, zinc-blende, and rock salt
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HW01
Direct lattice
Reciprocal lattice
Reciprocal lattice
Reciprocal lattice
Needed for describing an e- wave function in
terms of the symmetries of its crystal
environment Block function Y(R) expik.a y(R)
Y(R a)