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ECE 875: Electronic Devices. Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University [email protected] Course Content: Core: Part I: Semiconductor Physics Chapter 01: Physics and Properties of Semiconductors – a Review Part II: Device Building Blocks

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Ece 875 electronic devices

ECE 875:Electronic Devices

Prof. Virginia Ayres

Electrical & Computer Engineering

Michigan State University

[email protected]


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

VM Ayres, ECE875, S14


Course Content: Beyond core:

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Lecture 02 10 jan 14
Lecture 02, 10 Jan 14

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Crystal structures motivation
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
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


fcc lattice, to match Pr. 03

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14 atoms needed

VM Ayres, ECE875, S14


Crystal structures motivation electronics transport e s moving in an environment
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


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.

VM Ayres, ECE875, S14


  • 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?

VM Ayres, ECE875, S14


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 = 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

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Better picture of the fcc parallelepiped:

tilt

rotate

Ashcroft & Mermin

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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) -> (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

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= Volume of fcc primitive cell

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Sze chp 01 pr 03
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


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
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
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 lattices1
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
Example: considered as two inter-penetrating fcc lattices.

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


Picture and coordinate system for example problem

z considered as two inter-penetrating fcc lattices.

y

x

Picture and coordinate system for example problem:

(000)

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Answer
Answer: considered as two inter-penetrating fcc lattices.

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)

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Rock salt considered as two inter-penetrating fcc lattices.

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

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Direct space (lattice) considered as two inter-penetrating fcc lattices.

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) considered as two inter-penetrating fcc lattices.

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: considered as two inter-penetrating fcc lattices.

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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