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### ECE 875:Electronic Devices

Prof. Virginia Ayres

Electrical & Computer Engineering

Michigan State University

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

VM Ayres, ECE875, S14

VM Ayres, ECE875, S14

Lecture 02, 10 Jan 14

VM Ayres, ECE875, S14

Crystal Structures: Motivation:

Electronics: Transport: e-’s moving in an environment

Correct e- wave function in a crystal environment: Block function:

Y(R) = expik.ay(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

VM Ayres, ECE875, S14

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

VM Ayres, ECE875, S14

Non-cubic

VM Ayres, ECE875, S14

VM Ayres, ECE875, S14

Crystal Structures: Motivation:Electronics: Transport: e-’s moving in an environment

Correct e- wave function in a crystal environment: Block function: Y(R) = expik.ay(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

VM Ayres, ECE875, S14

A primitive Unit cell is theminimal 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.

VM Ayres, ECE875, S14

Atoms on the faces

Also need two corner atoms that give maximum dimension of volume of a3 cubic Unit cell

VM Ayres, ECE875, S14

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.

VM Ayres, ECE875, S14

- 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?

VM Ayres, ECE875, S14

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 = ax + 0y + 0z

b = w = 0x + ay + 0z

c = h = 0x + 0y + az

Minimal Vol = a . b x c

Specify the atomic arrangement as: one atom at every vertex of the minimal volume.

h

w

l

VM Ayres, ECE875, S14

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

VM Ayres, ECE875, S14

This is what Sze does in Chp.01, Pr. 03:

VM Ayres, ECE875, S14

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) -> (a/2,0,a/2): a = a/2 x + 0 y + a/2 z

(000) -> (a/2,a/2,0): b = a/2 x + a/2 y + 0 z

(000) -> (0,a/2,a/2): c = 0 x + a/2 y + a/2 z

z

c

a

y

b

(000)

x

VM Ayres, ECE875, S14

= Volume of fcc primitive cell

VM Ayres, ECE875, S14

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) -> (a/2,0,a/2): a = a/2 x + 0 y + a/2 z

(000) -> (a/2,a/2,0): b = a/2 x + a/2 y + 0 z

(000) -> (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 fordescribing an e- wave function acknowledging the symmetries of its crystal environment: Block function: Y(R) = expik.ay(R) = Y(R + a)

VM Ayres, ECE875, S14

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.”

VM Ayres, ECE875, S14

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

VM Ayres, ECE875, S14

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

VM Ayres, ECE875, S14

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

VM Ayres, ECE875, S14

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 atomsdisplaced (¼, ¼, ¼) x a:

a = lattice constant

VM Ayres, ECE875, S14

Example:

What are the three primitive basis vectors for the diamond primitive cell?

(000) -> (a/2,0,a/2): a = a/2 x + 0 y + a/2 z

(000) -> (a/2,a/2,0): b = a/2 x + a/2 y + 0 z

(000) -> (0,a/2,a/2): c = 0 x + a/2 y + a/2 z

How to make it diamond: two-atom basis

VM Ayres, ECE875, S14

Answer:

Three basis vectors for the diamond primitive cell:

(000) -> (a/2,0,a/2): a = a/2 x + 0 y + a/2 z

(000) -> (a/2,a/2,0): b = a/2 x + a/2 y + 0 z

(000) -> (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)

VM Ayres, ECE875, S14

VM Ayres, ECE875, S14

Rock salt can be also considered as two inter-penetrating fcc lattices.Discussion: Lec 03 13 Jan 14

VM Ayres, ECE875, S14

Direct space (lattice)

Conventional cubic Unit cell

Primitive cell for:

fcc, diamond, zinc-blende, and rock salt

Rock salt

VM Ayres, ECE875, S14

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

VM Ayres, ECE875, S14

HW01:

Direct lattice

Reciprocal lattice

Reciprocal lattice

Reciprocal lattice

Needed fordescribing an e- wave function in terms of the symmetries of its crystal environment: Block function: Y(R) = expik.ay(R) = Y(R + a)

VM Ayres, ECE875, S14

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