Solid State Device Fundamentalscsivc.csi.cuny.edu/engsci/files/ens345/2017/2017... · 2017-02-10 · 9 Department of Engineering Science and Physics College of Staten Island / CUNY

Solid State Device Fundamentals

ENS 345

Lecture Course

by

Alexander M. Zaitsev

[email protected]

Tel: 718 982 2812

Office 4N101b

College of Staten Island / CUNY Department of Engineering Science and Physics

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2. Crystals

mailto:[email protected]



Solids

Crystal is a periodic atomic structure. This structure can be reproduced by translation of an elementary element which is known as unit cell. The least translation along one axis is known as lattice parameter.

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Three types of solids classified according to atomic arrangement:

Crystalline Polycrystalline Amorphous

2. Crystals

Department of Engineering Science and Physics


Crystal lattice

The unit cell of a simple cubic lattice (a) along

with an image of 2 repeats in each direction (b).

Example of a

complex cubic

lattice: Silicon

crystal lattice.

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College of Staten Island / CUNY

Different unit cells of cubic lattice:

primitive, body-centered, face-centered.

There are 7 types of crystal lattices (called Bravais lattices), shown below:

2. Crystals



Crystallographic positions

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Silicon crystal has so-called “diamond type lattice”.Each Si atom has 4 nearest neighbors.

Diamond lattice starts with a FCC lattice and then adds four additional INTERNAL atoms at locationsr = a/4+b/4+c/4 away from each of the atoms. In other words, diamond lattice is formed by two FCC lattices sifted by the vector r.

Crystallographic position is denoted by three numbers, which are coefficients of the position vector, e.g. ½ ½ ½ for the red atom.

2. Crystals

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3

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Crystallographic positions in Si crystal

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What are the positions of the blue atoms in silicon unit cell?

Find interatomic distance for Si lattice.

Tetrahedron

2. Crystals

6College of Staten Island / CUNY Department of Engineering Science and Physics


Homework 4

Crystallographic positions

Identify crystallographic position of all atoms in silicon unit cell.

2. Crystals



Crystallographic directions

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Crystallographic direction is a direction between any two atoms of crystal lattice

[221]

Cubic lattice

Hexagonallattice

2. Crystals

Family of directions:e.g. [123], [213], [312], [132], [231], [321] for cubic lattice.

In the cubic lattice directions having the same indices regardless of order or sign are equivalent.

Dot product of indices of two perpendicular directions is zero. Directions [100] and [010] are perpendicular: [100].[010]=0



Homework 5

Crystallographic directions in Si crystal

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Identify crystallographic directions from red atom towards all atoms in silicon unit cell.

Are directions of electron bonds (blue bonds) perpendicular?

2. Crystals



Crystallographic planes

Crystallographic planes are denoted by Miller indices.

3 5 31/3 1/5 1/35 3 5(535)

2. Crystals

In the cubic system, a plane and a direction with the same indices are orthogonal. E.g. [100] direction is perpendicular to (100) plane. Correspondingly, [123] direction is perpendicular to (123) plane.

Indices of crystallographic plane can be found from cross product of indices of any two non-parallel directions in this plane.



Homework 6

Crystallographic planes in Si crystal

Identify crystallographic planes comprising red atom and any two of the blue atoms in silicon unit cell.

Find directions perpendicular to these planes.

2. Crystals



Linear atomic density of crystallographic directions

Linear Atomic Density (LAD) of a crystallographic direction is measured by number of atoms per unit length along this direction.

There is a tendency: The higher direction indices the lower linear density.

a

LAD [100] = (0.5+0.5)/a = 1/a

LAD [110] = (0.5+0.5)/ 2a = 1/ 2a = 0.71/a

LAD [111] = (0.5+0.5)/ 3a = 1/ 3a = 0.58/a

LAD [111]BCC = (0.5+1+0.5)/ 3a = 2/ 3a = 1.15/a

2. Crystals



Calculate LAD in silicon along [120], [123] and [112] directions.

Homework 7

LAD of crystallographic directions in Si crystal

2. Crystals



Atomic density of crystallographic planes

Atomic Density (AD) of crystallographic planes is measured by number of atoms per unit area.

Tendency: The higher Miller indices the lower atomic density.

2. Crystals



Calculate AD in silicon for planes (120), (123) and (112).

Homework 8

Atomic density of crystallographic planes in Si crystal

2. Crystals



APF – Atomic Packing Factor

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APF is a proportion of space that would be filled

by spheres that are centered on the vertices of

the crystal structure and are as large as

possible without overlapping.

2. Crystals



Calculate APF of silicon lattice.

Homework 9

APF of Si crystal lattice

2. Crystals



Defects in crystals

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0D defects (point defects)

1D defects (linear defects)

2D defects (planar defects)

3D defects(bulk defects)

2. Crystals

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Solid State Device Fundamentalscsivc.csi.cuny.edu/engsci/files/ens345/2017/2017... · 2017-02-10 · 9 Department of Engineering Science and Physics College of Staten Island / CUNY