ece 875: electronic devices

ECE 875:Electronic Devices

Prof. Virginia AyresElectrical & Computer EngineeringMichigan State [email protected]

VM Ayres, ECE875, S14

Course Content: Core:

Part I: Semiconductor PhysicsChapter 01: Physics and Properties of Semiconductors – a Review

Part II: Device Building BlocksChapter 02: p-n JunctionsChapter 03: Metal-Semiconductor ContactsChapter 04: Metal-Insulator-Semiconductor Capacitors

Part III: TransistorsChapter 06: MOSFETs


Course Content: Beyond core:


Lecture 02, 10 Jan 14


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

Correct e- wave function in a crystal environment: Block function:(R) = expik.a (R) = (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 carriersI-V measurementJ: Vext direction versus internal E-k: Egap direction

Fixed e-’s and holes:C-V measurement

Crystal Structures: Motivation:

x Probability f0 that energy level is occupied

q n, p velocity Area

(KE + PE) = E


Unit cells:

Non-cubic

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


fcc lattice, to match Pr. 03


14 atoms needed


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

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

Periodicity of the environment:Need specify where the atoms are

Unit cell a3 for cubic systems sc, fcc, bcc, etc.ORPrimitive cell for sc, fcc, bcc, etc.ORAtomic basis

Most atoms

Fewer atoms

Least atoms

Think about: need to specify:


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.


Answer:Atoms on the faces



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



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.


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?


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

lw


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

a

b

c


Better picture of the fcc parallelepiped:

Ashcroft & Mermin

rotate tilt


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


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

(000)

z

y

x

a

b

c


= Volume of fcc primitive cell


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 setR = ma + nb + pc.”

Translational invariance is great for describing an e- wave function acknowledging the symmetries of its crystal environment: Block function: (R) = expik.a (R) = (R + a)


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.”


a = a/2 x + 0 y + a/2 zb = a/2 x + a/2 y + 0 zc = 0 x + a/2 y + a/2 z

(000)

z

y

x

a

b

c

1. four atoms

2. three vectors between themAnywhere: R = ma + nb + pc

3. minimal Vol = a3/4 (parallelepiped)atom at each vertex of the minimal volume

Steps for fcc were:


Typo, p. 08:

No! Figure 1 shows conventional Unit cells!


Conventional Unit CellsVol. = a3

Primitive Unit Cells:Smaller Volumes

Vol = a3/4


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


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 a

Note: also have pairs of atoms displaced (¼, ¼, ¼) x a:

a =

lattice co

nsta

nt


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


Picture and coordinate system for example problem:

(000)

z

y

x


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:

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

(000)

Same basis vectors as fccSame primitive cell volume a3/4

z

y

x

z

y

x


Rock salt


Rock salt can be also considered as two inter-penetrating fcc lattices.

Discussion: Lec 03 13 Jan 14


Conventional cubic Unit cell Primitive cell for: fcc, diamond, zinc-blende, and rock salt

Direct space (lattice) Direct space (lattice)

Rock salt


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

Direct space (lattice) Direct space (lattice) Reciprocal space (lattice)


Direct lattice

HW01:

Reciprocal lattice

Reciprocal lattice

Reciprocal lattice

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

ece 875: electronic devices

Documents

corner atoms

atoms areunit cell a3

atoms neededcrystal

yr acorrect e

fcc primitive cell example

fcc parallelepiped

maximum dimension of

primitive unit cell