Key Questions ECE 340 Lecture 3 : Semiconductors Why is ...

1/28/14
 

ECE 340 Things you should know when you leave…
Lecture 3 : Semiconductors
Key Questions
and Crystal Structure
•  Why is crystal order important?
Class Outline: •  How is a crystal defined?
•  What are the most common
•S  emiconductor Crystal Lattices
•M  iller Indices types of crystal lattices used in
vS  emiconductor Crystal Growth semiconductor devices?

M.J. Gilbert ECE 340 – Lecture 3

Semiconductor Crystal Lattices Semiconductor Crystal Lattices

What is the crystal structure? What does it matter if it is crystalline or not?

amorphous • cCrCryryysssttatlalainllelsitpnroeupclotyulcryerscyrcsPyootlyamscrletylsatiiannllllinetienhreeaemabomarsipoAcmhrokrpopihnhoudusoss uscrycsrtya•s lltWianelleCcriyansntaelglienet Amorphous

a lot of information from the unit cell:

–  In the CRYSTALLINE state the atoms are ordered into a well-defined lattice that –  Density of atoms
extends over very long distances . –  Distance between nearest atoms

–  POLYCRYSTALLINE materials consist of small crystallites that are embedded in •  Calculate forces between atoms
AMORPHOUS regions of material.
–  Perform simple calculations

–  In the AMORPHOUS state there is little or no evidence for long-range crystalline •  Fraction of atoms filled in volume
order. •  Density of atoms

M.J. Gilbert ECE 340 – Lecture 3 M.J. Gilbert ECE 340 – Lecture 3

1
 

1/28/14
 

Semiconductor Crystal Lattices Semiconductor Crystal Lattices

Since we care about crystalline lattices, let’s examine the periodic lattice… In this section we consider some of the lattice types that will be important
for our discussion of semiconductors…
•  In the periodic
 
lattice:
–  Symmetric array •  Examine the simple
of points is the
lattice. cubic structure:

–  We add the atoms –  All primitive
to the lattice in an
arrangement called a vectors are equal in
a basis.
all three
–  We can define a z dimensions.
set of primitive
vectors which can y –  Here again, the
be used to trace
+= out the entire x a1 = axˆ (2.1) balls represent the
crystal structure. lattice points, but
a2 = ayˆ (2.2)
no basis has been

2D lattice where the grey Basis 2D Crystal a3 = azˆ (2.3) added.
balls denote lattice points

M.J. Gilbert ECE 340 – Lecture 3 M.J. Gilbert ECE 340 – Lecture 3

Semiconductor Crystal Lattices Semiconductor Crystal Lattices

A simple variant on the cubic lattice is the body-centered cubic lattice… A final variant is the face-centered cubic lattice…

•  Examine the body-centered cubic lattice (bcc): •  Examine the face-centered cubic lattice (fcc):
–  This is formed by adding an additional atom in the center
–  Same as simple cubic but with an additional atom at of each face of the simple cubic configuration.
the center of the cell. –  This is the most important configuration we will consider.
–  The primitive vectors have been written again by using
–  Primitive vectors are written in the more convenient symmetry considerations.
symmetric form but other representations exist.

a1 = a [xˆ + yˆ ] (2.7)
2
a
a1 = 2 [xˆ + yˆ − zˆ ] (2.4)
(2.5)
(2.6) a a2 = a [yˆ + zˆ ] (2.8)
2
z a
a a2 = 2 [− xˆ + yˆ + zˆ ] z

y y a
x 2
a3 = [zˆ + xˆ ] (2.9)
M.J. Gilbert
x a3 = a [xˆ − yˆ + zˆ ]
2

M.J. Gilbert ECE 340 – Lecture 3 ECE 340 – Lecture 3

2
 

1/28/14
 

Semiconductor Crystal Lattices Semiconductor Crystal Lattices

But be careful… Now let’s look at the silicon crystal…

•  There is a difference between unit cells and •  To discuss the crystal structure of different semiconductors we
primitive cells. will need to account for the basis unit that is added to each
lattice point.
–  The primitive cell is the volume associated with one –  Elemental semiconductors such as silicon and germanium both exhibit
lattice point. the diamond structure.
–  Named after one of the two crystalline forms of carbon.
 
–  Often it is more convenient to use a unit cell that is
 
larger than the primitive cell since such a cell
illustrates the crystal symmetry in a clearer way. •  Here we display the silicon
unit cell.
IN THIS FIGURE THE DOTTED LINES INDICATE THE
UNIT CELL OF THE FACE-CENTERED CUBIC LATTICE •  The balls each represent
one silicon atom.
a Vunit = a3 ⋅a1 ⋅a2 = a3 (2.10)
•  The solid lines represent
THE GRAY REGION DENOTES THE PRIMITIVE CELL FOR THIS LATTICE chemical bonds.

WHOSE VOLUME CAN BE DETERMINED FROM THE PRIMITIVE VECTORS •  Note how the bonds form a
tetrahedron.
INTRODUCED IN Eqns. 2.7 - 2.9 = a3 (2.11)
4 •  How many atoms per unit
Vprimitive = a3 ⋅ (a1 × a2 ) cell?

M.J. Gilbert ECE 340 – Lecture 3 M.J. Gilbert ECE 340 – Lecture 3

Semiconductor Crystal Lattices Semiconductor Crystal Lattices

Looks confusing, but it’s not so bad… What about compound semiconductors? Gallium
•  It is really just two inter-penetrating fcc lattices with a diatomic basis.
•  It looks more complex because we do not show atoms extending beyond •  Many compound Arsenide

the unit cell by convention. semiconductors such
•  In the figure below, each different color represents pairs of atoms
as Gallium Arsenide
from the same basis.
•  The black balls represent atoms with one atom in their basis outside the (GaAs) exhibit the

unit cell. zincblende crystal

structure.

a –  The atomic
4
(xˆ , yˆ , zˆ ) configuration is the

same as diamond.

–  The difference lies in Useful questions to ask:

(0, 0, 0) that each successive • How many atoms per unit cell?
atom is from a • Avogadro’s number: NA = # atoms / mole
LATTICE BASIS UNIT CELL different chemical • Atomic mass: A = grams / mole
M.J. Gilbert ECE 340 – Lecture 3 element. •A  tom counting in unit cell: atoms / cm3
• How do you calculate density?

 

M.J. Gilbert ECE 340 – Lecture 3

3
 

1/28/14
 

Miller Indices Miller Indices

How do we classify these periodic atomic arrangements? We can classify crystal planes through the use of Miller Indices.

•  The periodicity could possibly be used to 1.  Start with any arbitrary point in the crystal lattice.
define a series of planes. 2.  Take the three primitive lattice vectors as axes.
3.  Next we locate the intercepts of the desired plane with these
•  For any direction one could define many
different equivalent planes. coordinate axes.

•  For instance, consider the different 4.  Finally, we take the reciprocal of the intercepts and multiply
equivalent crystal planes in a cubic lattice. each of these by the smallest factor required to convert them
all to integers.
–  Indices which have no intercept have a Miller Index of zero.
 

 

LATTICE
POINT

a3 AXIS INTERCEPT RECIPROCAL
a1 a2 a1 3 1/3
a2 2 1/2
LATTICE a3 2 1/2
POINT

LATTICE
POINT

TO CONVERT THE RECIPROCALS TO INTEGERS WE NEED TO MULTIPLY BY
A FACTOR OF SIX THIS YIELDS MILLER INDICES (233)

M.J. Gilbert ECE 340 – Lecture 3 M.J. Gilbert ECE 340 – Lecture 3

Miller Indices Miller Indices

Applications of Miller Indices. More Miller Indices:

AXIS INTERCEPT RECIPROCAL MILLER INDEX •  Since we chose the origin arbitrarily when we began,
6 Miller indices define a family of parallel planes.
a1 2 1/2 3
a2 4 1/4 4 •  To denote crystal planes with the same symmetry, we
a3 3 1/3 use {hkl}.

•  “h” is the x-axis intercept inverse, “k” is the y-axis
intercept inverse, and “l” is the z-axis intercept
inverse .

z
  z
  z
 

(001)

AXIS INTERCEPT RECIPROCAL MILLER INDEX
1
4 (100) (010)
2
a1 4 1/4 y
  y
  y
 
a2 1 1/1
a3 -2 -1/2 x
  x
  x
 

Notice how we denoted the miller index for a plane with a negative intercept All three planes shown here are related by simple rotations, thus they
represent {100} family.

M.J. Gilbert ECE 340 – Lecture 3 M.J. Gilbert ECE 340 – Lecture 3

4
 

1/28/14
 

Miller Indices Crystal Growth - Silicon

We can also use Miller Indices to denote directions:

•  The direction Summary

[hkl] is used A crystal plane: (hkl)

to denote the Equivalent crystal planes: {hkl}

directions Crystal direction: [hkl]
perpendicular
to (hkl).
 
  (perpendicular to the plane (hkl) in a

cubic lattice)

[001]
  [001]
 

[hkl]

(hkl) [010]
 

[100]
  [010]
  M.J. Gilbert ECE 340 – Lecture 3

M.J. Gilbert [100]
 

ECE 340 – Lecture 3

Crystal Growth – Compound Semiconductors Crystal Growth – Compound Semiconductors
•  Careful control of the deposition rates and the substrate
•  Heterojunctions are typically produced by a
process known as MOLECULAR-BEAM EPITAXY temperature are required to realize heterojunctions with well-
–  This is performed in an ultra-high vacuum (UHV) defined interfaces.
evaporation chamber working at pressures of 10-11 Torr. –  In order to achieve high uniformity the substrate is heated to
–  The materials to be grown are provided from heated
KNUDSEN CELLS in which the individual elements are approximately 600 ºC and is slowly rotated in the vacuum chamber.
individually vaporized –  The growth rate of the epitaxial layer is of order several MICRONS

ELECTRON-­‐
  CHAMBER
  PER HOUR which allows for ATOMIC level resolution in the growth
DIFFRACTION
  WALLS
  process.
COOLED
 
  –  The growth is monitored in situ using electron diffraction and mass
SOURCE
  spectroscopy.

 WITH
 LIQUID
 
NITROGEN
 

600
 °C
 

KNUDSEN
 CELLS
 
CONTAINING
 
Ga,
 Al,
 As
 &
 Si
 

ELECTRON-­‐
 
DIFFRACTION
 

DETECTOR
 

UHV
 PUMP
 

A
 SCHEMATIC
 DIAGRAM
 SHOWING
 THE
 KEY
 COMPONENTS
  TEM
 IMAGES
 OF
 EPITAXIALLY
 GROWN
 GaAs/AlGaAs
 
OF
 A
 MOLECULAR-­‐BEAM
 EPITAXY
 SYSTEM
 
ECE 340 – Lecture 3
M.J. Gilbert ECE 340 – Lecture 3 M.J. Gilbert

5
 

1/28/14
 

Example Problem

Treating atoms as rigid spheres with radii equal to ½ the distance between
the nearest neighbors, show that the ratio of the volume occupied by atoms to
the total available volume in an FCC is 74%.

M.J. Gilbert ECE 340 – Lecture 3

6