ECE 802-604: Nanoelectronics

1
ECE 802-604Nanoelectronics

• Prof. Virginia Ayres
• Electrical Computer Engineering
• Michigan State University
• ayresv_at_msu.edu

2
Lecture 18, 29 Oct 13
Carbon Nanotubes and Graphene Carbon
nanotube/Graphene physical structure Carbon
bond hybridization is versatile sp1, sp2, and
sp3 sp2 origin of mechanical and electronic
structures Carbon nanotube/Graphene
electronic structure
R. Saito, G. Dresselhaus and M.S.
Dresselhaus Physical Properties of Carbon
Nanotubes Imperial College Press, London, 1998.
3
CNT Structure

• The Basis Vectors a1 and a2
• The Chiral Vector Ch
• The Chiral Angle cos(q)
• The Translation Vector T
• The Unit Cell of a CNT
• Headcount of available p electrons

4
(No Transcript)
5
Lec 17 The Basis Vectors
a1 v3 a x 1 a y 2
2 a2 v3 a x - 1 a y 2
2
where magnitude a a1 a2
6
Lec 17 The Basis Vectors
1.44 Angstroms is the carbon-to-carbon distance
in individual ring
7
Lec 17 The Basis Vectors
a1 v3 a x 1 a y 2
2 a2 v3 a x - 1 a y 2
2
1.44 A
1.44 A
120o
a
Magnitude a 2 (1.44 Angstroms)cos(30)
2.49 Angstroms
8
The Basis Vectors
a1 v3 a x 1 a y 2
2 a2 v3 a x - 1 a y 2
2
where magnitude a 2.49 Ang
Example label the vectors shown in red
9
The Basis Vectors
a1 v3 a x 1 a y 2
2 a2 v3 a x - 1 a y 2
2
a1
where magnitude a 2.49 Ang
a2
Example label the vectors shown in red Answer
as shown
10
The Basis Vectors
a1 v3 a x 1 a y 2
2 a2 v3 a x - 1 a y 2
2
a1
where magnitude a 2.49 Ang
a2
Example what is a1 a2 ?
11
The Basis Vectors
a1 v3 a x 1 a y 2
2 a2 v3 a x - 1 a y 2
2
a1
where magnitude a 2.49 Ang
Example what is a1 a2 ? Answer
non-orthogonal in ai system
a2
12
The Chiral Vector Ch
13
The Chiral Vector ChBasic definition
14
Lec 17 Introduction
Buckyball endcaps
Many different types of wrapping result in a
seamless cylinder. But The particular cylinder
wrapping dictates the electronic and mechanical
properties.
15
The Chiral Vector ChBasic definition
Example Prove that the magnitude of Ch is
avn2 m2 mn
16
The Chiral Vector ChBasic definition
Answer
17
The Chiral Vector ChBasic definition
Example Evaluate Ch for a (10,10) SWCNT
18
The Chiral Vector ChBasic definition
Example Evaluate Ch for a (10,10)
SWCNT Answer
19
The Chiral Vector ChA number associated with the
Chiral Vector the Greatest Common Divisor d(or
gcd)
20
The Chiral Vector ChThe Greatest Common Divisor
(d, or gcd)
Example Find d for the following
SWCNTs (10,10), (9,9), (9,0), (7,4), and (8, 6)
21
The Chiral Vector ChThe Greatest Common Divisor
(d, or gcd)
Example Find d for the following
SWCNTs (10,10), (9,9), (9,0), (7,4), and (8,
6) Answer (10,10) d10 (9,9) d 9 (9,0)
d9 (7,4) d1 and (8, 6) d2
22
The Chiral Vector ChDefine the CNT tube diameter
dt(Note that diameter dt is different than
greatest common divisor d!)
23
The Chiral Vector ChCNT diameter dt
Example Find the nanotube diameter for a (10,
10) CNT.
24
The Chiral Vector ChCNT diameter dt
Example Find the nanotube diameter for a (10,
10) CNT. Answer dt 136.38 Ang/p 43.41 Ang
25
The Chiral Angle q
26
The Chiral Angle q
Defines the tilt of Ch with respect to a1
Direction cosine of a1 Ch a1 Ch a1 Ch
cos q cos q a1 Ch a1
Ch ( n m/2) vn2 m2
mn
27
The Chiral Angle q
Example Prove that cosq ( n m/2)
vn2 m2 mn
28
The Chiral Angle q
Answer
29
The Translation Vector T
30
The Translation Vector T
Note that T and Ch are perpendicular. Therefore
TCh 0 Let T t1 a1 t2 a2 Take TCh 0
Solve for t1 and t2
31
The Translation Vector T
Vector T t1 a1 t2 a2 t1 2m n/ dR t2
- (2n m) /dR Magnitude T v3 Ch /dR
What is dR?
32
The Translation Vector TdR is a NEW greatest
common divisor
dR the greatest common divisor of 2m n and
2n m dR d if n-m is not a multiple of 3d
33
The Unit Cell of a CNT (single wall)
34
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
35
The Primitive Cell of a CNT (single wall)
Also a1 x a2 is the area of a single basic or
primitive cell. Therefore the number of
hexagons N per CNT Unit Cell is N T X Ch
a1 x a2 2(m2 n2nm)/dR
36
The Unit Cell of a CNT (single wall)
Each primitive cell contains two C atoms. There
is one pz-orbital per each C atom. Therefore
there are 2N pz orbitals available per CNT Unit
Cell
37
Example Problems CNT Structure

1. Find Ch, Ch , cosq, q, T, T , and N for a
   (10,10) SWCNT
2. What does N count?
3. What type of CNT is this?
4. Find Ch, Ch , cosq, q, T, T , and N for a
   (9,0) SWCNT
5. What type of SWCNT is this?
6. Find Ch, Ch , cosq, q, T, T , and N for a
   (7,4) SWCNT
7. What type of SWCNT is this?

38
For use in Example Problems
Ch n a1 m a2 Ch avn2 m2 mn dt
Ch/p cos q a1 Ch
a1 Ch T t1 a1 t2 a2 t1 (2m
n)/ dR t2 - (2n m) /dR dR the
greatest common divisor of 2m n and 2n m T
v 3 Ch / dR N T X Ch a1 x a2
2(m2 n2nm)/dR
39

1. Find Ch, Ch , cosq, q, T, T , and N for a
   (10,10) SWCNT
2. What does N count?
3. What type of CNT is this?
4. Find Ch, Ch , cosq, q, T, T , and N for a
   (9,0) SWCNT
5. What type of SWCNT is this?
6. Find Ch, Ch , cosq, q, T, T , and N for a
   (7,4) SWCNT
7. What type of SWCNT is this?

40
Answer 1
41
Answer 1 continued
42
Answers 2 and 3
2.
3.
T v3 Ch /dR 0.25 nm
43

1. Find Ch, Ch , cosq, q, T, T , and N for a
   (10,10) SWCNT
2. What does N count?
3. What type of CNT is this?
4. Find Ch, Ch , cosq, q, T, T , and N for a
   (9,0) SWCNT
5. What type of SWCNT is this?
6. Find Ch, Ch , cosq, q, T, T , and N for a
   (7,4) SWCNT
7. What type of SWCNT is this?

44
Answer 4
45
Answers 4 and 5
4.
5.
46

1. Find Ch, Ch , cosq, q, T, T , and N for a
   (10,10) SWCNT
2. What does N count?
3. What type of CNT is this?
4. Find Ch, Ch , cosq, q, T, T , and N for a
   (9,0) SWCNT
5. What type of SWCNT is this?
6. Find Ch, Ch , cosq, q, T, T , and N for a
   (7,4) SWCNT
7. What type of SWCNT is this?

47
Answer 6
48
Answers 6 and 7
6.
7.