ece 474: principles of electronic devices · ece 474: principles of electronic devices prof....

ECE 474:Principles of Electronic Devices

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

V.M. Ayres, ECE474, Spring 2011

Example for VA Pr. 01 1.13: The {111} family of planes with the <111> family of directions. There are X = 8 of these.


Va Pr. 01 is: do likewise for the {110} family. You need only draw four out of the X possibilities.Hint: X is NOT equal to 8.


Lecture 08:

Lectures: Hexagonal nanosystems: graphene and carbon nanotubes

Introduction to graphene and CNTsThe Basis Vectors: a1 and a2Nearest neighbor distancesThe Chiral Vector: ChThe CNT diameter dtThe Translation Vector: TThe Unit Cell of a CNTNumber of hexagons NNumber of carbon atoms 2N (π electrons)Areal Density

Examples of each

Quantify physical structures of crystal systems that are important for devices:


The Unit Cell of a CNT (single wall)

Note that T and Ch are perpendicular.

Therefore T X Ch = the area of the CNT Unit Cell

= | T || Ch |sin(90o)

= | T || Ch |


Example: find the area of the Unit cell for a (10,10) CNT.


Example: find the area of the Unit cell for a (10,10) CNT.

From Lec 07:

|Ch| = 43.13 Ang

| T | = 2.49 Ang

Area of the Unit cell = | T || Ch | = 107.39 Ang2


Example: find the formula for the area of the Unit cell in terms of n, m and dR.


Example: find the formula for the area of the Unit cell in terms of n, m and dR.

Area of the Unit cell = | T || Ch |


Number of hexagons N in the Unit Cell:

Basic unit is the hexagon.

The number of hexagons in the Unit cell is called N

Therefore:the number of hexagons N per CNT Unit Cell is:

N = area of the Unit cellarea of one hexagon


Called: dividing by the Primitive Cell of a CNT:

a1 x a2 is the area of a single basic hexagon. This is also called the primitive cell.

Note: the area of the dotted rhombus is equal to the area of a single hexagon. The magnitude is:

Note the two inequivalent carbon atoms A and B inside the rhombus

23 2a

C C


The number of hexagons N per CNT Unit Cell is:

N = | T X Ch || a1 x a2 |

= 2(m2 + n2+nm)/dR

How many carbon atoms per hexagon?

Number of hexagons N versus the number of carbon atoms in the Unit Cell:


120o

120o

120o

Each C atom is 1/3 inside the hexagon.


120o

120o

120o

Each C atom is 1/3 inside the hexagon.

6 X (1/3) = 2 atoms per hexagon


The number of hexagons N per CNT Unit Cell is:

N = | T X Ch || a1 x a2 |

= 2(m2 + n2+nm)/dR

How many carbon atoms per hexagon?

Answer: 2



How many carbon atoms in the Unit cell?

Answer:First solve for the number of hexagons N.

The number of carbon atoms in the Unit cell = 2N.



Each primitive cell contains two carbon atoms.

There is one pz-orbital per each carbon atom.

Therefore there are 2N pzorbitals available per CNT (or graphene) Unit Cell = cloud of electrons = greatconductivity.

The shared electrons from the pz orbitals are called πelectrons.

π electrons and electrical properties:


a) Example:

Find the number of hexagons N in the unit cell of a (10, 10) CNT.Find the number of carbon atoms in the unit cell of a (10, 10) CNT.


Example: find the areal density of the unit cell of a (10, 10) CNT.


Example: find the general areal density formula and evaluate it.

This is the conclusion of the discussion we started on the board. The answer is obvious when you think about it!


= the same number as for the (10,10) CNT. Of course! The density of a material doesn’t change when you cut it.The areal density on different faces within the cubic system changes but graphene has only one face.


Interesting fact: there are Endcap restrictions on CNT diameter dt:

CNTs have closed endcapswhich are ½ a buckyball.

Armchair

Zigzag

Chiral


C60

XZ Ke, et al, Physics Letters A 255 (1999) 294-300

C36

C60 mean ball diameter 6.83 Å

C60 ball outer diameter 10.18 Å

C60 ball inner diameter 3.48 Å

http://www.sesres.com/PhysicalProperties.asp

Buckyballs are ‘round CNTs’. They have some pentagons amongst their hexagons for curvature. C60 forms quite easily, C36 is a more strained structure.


The smallest possible buckyball endcap is half of the dodecahedral C20, a shape consisting of 12 pentagonal faces and no hexagonal faces. So the smallest CNT diameter dt is about 4 Angstroms.

C20

~4 angstroms

ece 474: principles of electronic devices · ece 474: principles of electronic devices prof....

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