Post on 06-Jul-2018
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•Miller index is used to describe directions and planes in acrystal.
•Directions - written as [u v w] where u, v, w.• Integers u, v, w represent coordinates of the vector in realspace.
•A family of directions which are equivalent due tosymmetry operations is written as
•Planes: Written as (h k l ).
•Integers h , k , and l represent the intercept of the plane withx -, y -, and z - axes, respectively.
• Equivalent planes represented by {h k l}.
Miller Index For Cubic Structures
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The intercepts of a crystal plane with the axis defined by a set of unit
vectors are at 2a, -3b and 4c. Find the corresponding Miller indices of this
and all other crystal planes parallel to this plane.
The Miller indices are obtained in the following three steps:
1. Identify the intersections with the axis, namely 2, -3 and 4.
2. Calculate the inverse of each of those intercepts, resulting in 1/2, -
1/3 and 1/4.
3. Find the smallest integers proportional to the inverse of theintercepts. Multiplying each fraction with the product of each of the
intercepts (24 = 2 x 3 x 4) does result in integers, but not always
the smallest integers.
4. These are obtained in this case by multiplying each fraction by 12.
5. Resulting Miller indices is6. Negative index indicated by a bar on top.
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z
y
x
z=
y=
x=a
x y z
[1] Determine intercept of plane with each axis a ∞ ∞
[2] Invert the intercept values 1/a 1/∞ 1/∞
[3] Convert to the smallest integers 1 0 0
[4] Enclose the number in round brackets (1 0 0)
Miller Indices of Planes
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• Planes (100), (010), (001), (100), (010), (001) areequivalent planes. Denoted by {1 0 0}.
• Atomic density and arrangement as well as electrical,optical, physical properties are also equivalent.
z
yx
(100)
plane
(010)plan
e
(001) plane
Equivalent Planes
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The (110) surface
Assignment
Intercepts : a , a ,
Fractional intercepts : 1 , 1 , Miller Indices : (110)
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z
y
x
x y z
[1] Determine intercept of plane with each axis 2a 2a 2a
[2] Invert the intercept values 1/2a 1/2a 1/2a
[3] Convert to the smallest integers 1 1 1
[4] Enclose the number in round brackets (1 1 1)
Miller Indices of Planes
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The (111) surface
Assignment Intercepts : a , a , a
Fractional intercepts : 1 , 1 , 1
Miller Indices : (111)
The (210) surface
Assignment
Intercepts : ½ a , a , Fractional intercepts : ½ , 1 ,
Miller Indices : (210)
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z
yx
Planes with Negative Indices
x y z
[1] Determine intercept of plane with each axis a -a a
[2] Invert the intercept values 1/a -1/a 1/a
[3] Convert to the smallest integers 1 -1 -1
[4] Enclose the number in round brackets 111
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x y z
[1] Draw a vector and take components 0 2a 2a
[2] Reduce to simplest integers 0 1 1
[3] Enclose the number in square brackets [0 1 1]
z
y
x
Miller Indices: Directions
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z
y
x
x y z[1] Draw a vector and take components 0 -a 2a
[2] Reduce to simplest integers 0 -1 2
[3] Enclose the number in square brackets 210
Negative Directions
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Miller Indices: Equivalent Directions
z
yx
12
3
1: [100]
2: [010]3: [001]
Equivalent directions due to crystal symmetry:
Notation used to denote all directions equivalent to [100]
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Directions
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In the cubic system the (hk l ) plane and thevector [hk l ] are normal to one another.
This characteristic is unique to the cubic
crystal system and does not apply to crystal
systems of lower symmetry.
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Importance of Miller Indices
• They are important because properties aredifferent along different directions
• The planes influence• Optical Properties• Reactivity• Surface Tension
• Dislocation
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Miller Indices of Wafers
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Angle between (100) and
(011) planes
cos(θ)=(1x0+0x1+0x1)/(√1x√2 ) = 0so θ=90 degrees(011) surface is normal to (100) surface
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
½ ½
u u v v w wcos
(u v w ) (u v w )
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24
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Sketch and Find Planar Density or Atomic
Density (atoms/unit area) For Simple Cubic (SC)
unit cell. If lattice constant is a and radius is R.
• (100) plane• (110) plane• (111) plane
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Simple Cubic(SC) Planer/Areal Density of
(100) plane
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Simple Cubic(SC) –Planer/Areal density (110)
plane
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STM image of Si (111)
STM image of Gold (111)
Miller Index For Cubic Structures
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Sketch and Find Planar Density or Atomic
Density (atoms/unit area) For Face Centered
Cubic (FCC) unit cell. If lattice constant is a and
radius is R.
• (100) plane• (110) plane• (111) plane
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http://www.google.com.pk/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&docid=jCnxNKfwBLjfVM&tbnid=sVB7IeBcwL_rkM:&ved=0CAUQjRw&url=http%3A%2F%2Fwww.chem.wisc.edu%2Fcourses%2F801%2FSpring00%2Fchemlecnotes%2Fchemln2.html&ei=SkDwU_vqOumd0QXM-oCYCg&bvm=bv.73231344,d.d2k&psig=AFQjCNGk0dVfMPxjmmz2SS4w4ArTLjSJOw&ust=1408340299658215
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1)(
24
1)(
4
12)(
2111
2110
22100
R FCC PD
R FCC PD
Ra
FCC PD
Planar Density or Atomic Density (atoms/unit
area) For Face Centered Cubic (FCC) unit cell
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Home Work:
Sketch the plane and Find Planar Density or Atomic Density
(atoms/unit area) for Body centered cubic Unit cell(BCC) and
Diamond Lattice for
• (100) plane• (110) plane• (111) plane
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Why MOS Devices are made on (100)
wafers and Bipolar Devices are made on
(111) wafers?Why we do not have (110) Wafers?
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1. (111) ingot is easy to pull therefore initially(111) ingots used to be made in olden
timings.
2. As BJT are old devices therefore they were
made on (111) wafers.
3. But now a days small scale ICs of BJT aremade on (100) wafers.
4. For (100) surfaces the interface charge
densities at Si-SiO2 interfaces is 1010
charges/cm2
for (111) wafer is is ten timeshigher therefore MOS devices and other
surface devices are made on (100) wafers.