ECE 802-604: Nanoelectronics

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ECE 802-604Nanoelectronics

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

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Lecture 25, 26 Nov 13
Carbon Nanotubes and Graphene CNT/Graphene
electronic properties sp2 electronic
structure 2DEG E-k relationship/graph for
graphene and transport 1DEG E-k
relationship/graph for CNTs and transport
R. Saito, G. Dresselhaus and M.S.
Dresselhaus Physical Properties of Carbon
Nanotubes
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CNT Unit cell in green
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(m2 n2nm)/dR v 3Ch/dR N T X Ch
a1 x a2 2(m2 n2nm)/dR
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K1 is in same direction as Ch Specify direction
of Ch using choral angle
K2 is in same direction as T
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Transport
Real space Ch Reciprocal space K1
Real space T Reciprocal space K2
Transport along CNT Along a Unit vector in the
K2 direction Can have any magnitude (hbar)k
(10,10)
(9,0)
(7,4)
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For an e- described as a wave
Quantization of Energy E is here
Standing wave Quantization by m in Ch / K1
direction
Travelling wave with an unquantized wave vector
k in T/ K2 direction
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Transport ECNT is proportional to Egraphene2D ?
conduction energy levelECNT is proportional to
the value of the transfer integral t

Conduction and valence energy levels
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k ? hbark is in the transport direction. Where k
is relative to kx and ky depends on the nanotube
(n,m)
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ZIGZAG
a1
Zigzag Ch in a1 direction
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ZIGZAG
kx
ky
Example which is the Ch direction, kx or ky?
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ZIGZAG
kx
ky
Answer ky
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Lec 24 Consider an (n, 0) zigzag CNT. This is
where the periodic boundary condition on ky comes
from in
That leaves just kx as open, MD calls it just k.
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ZIGZAG
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kx
ARMCHAIR
a1
ky
Example Which components cancel? Which
components add?
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kx
ARMCHAIR
a1
ky
Answer Which components cancel? kx Which
components add? ky
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Lec 24 Consider an (n, n) armchair CNT. This is
where the periodic boundary condition on kX comes
from in
That leaves just kY as open, MD calls it just k.
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ARMCHAIR
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ARMCHAIR
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(4,2) CHIRAL where Ch and T are
a1
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For chiral from Lec 23
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Therefore
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(4,2) CHIRAL where Ch and T are
a1
a2
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Real space Ch Reciprocal space K1
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(4,2) CHIRAL where Ch and T are
a1
a2
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(4,2) CHIRAL where Ch and T are
a1
a2
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Real space Ch Reciprocal space K1
Transport direction Real space T Reciprocal
space K2
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Transport in a 1-D
Real space Ch Reciprocal space K1
Real space T Reciprocal space K2
A Unit vector in the K2 direction
(10,10)
(9,0)
(7,4)
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Lec 05
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Lec 05
6. Current I ? q x n x vgroup
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Lec 06
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Lec 24 What you can do with an E-k diagram
Answer
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Lec 07
2-DEG
1-DEG
1-DEG
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2DEG Graphene
Conduction energy level for p
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2DEG Graphene
N(E)
E
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1DEG CNT
Conduction energy levels
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Specify example (n, 0) zigzag CNT. You can
write a periodic boundary condition on ky and
substitute into eqn 2.29. That leaves just kx
as open, MD calls it just k.
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Lec 07
2-DEG
1-DEG
1-DEG
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Same as Datta Pr. 1.3
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Same as Datta Chp. 02
Four terminal
Two terminal
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Same as Datta Chp. 02
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Coherent
Same as Datta Chp. 03
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Incoherent
Same as Datta Chp. 03