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 17, 24 Oct 13
Datta 3.1 scattering/S matrix Combining two
2x2 scattering matrices combine the scatterers
coherently combine the scatterers
incoherently combine the scatterers with
partial coherence Goal find the transmission
probability T through a complete structure that
contains the scatterers
Dresselhaus Graphene and Carbon Nanotubes
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Lec 15 What is a scattering S matrix?
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Lec 15
1.
2.
3.
Game plan for each Si j element determine
is it a reflection or a transmission?
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Datta, Sec. 3.1
Two propagating modes into one propagating
mode expect scattering due to occupied states
Coherent Conductor
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Lec 15 Example find the S-matrix for the
Buttiker/Enquist diagrams shown.
Incoherent Conductor
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Example find the S-matrix for both of the
diagrams shown.
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Example find the scattering matrix S for the
diagram shown within the red box
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Basic form
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Two points Point 01 Reflections are not really
the same. One incorporates the influence of Leads
2 and 4 and the other doesnt. Same is true for
transmissions. Therefore Let r -gt r and r
with the influence of Leads 2 and 4 Let t -gt t
and t with the influence of Leads 2 and 4
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Two points Point 01 Reflections are not really
the same. One incorporates the influence of Leads
2 and 4 and the other doesnt. Same is true for
transmissions. Therefore Let r -gt r, and r
with the influence of Leads 2 and 4 Let t -gt t,
and t with the influence of Leads 2 and 4
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Two points Point 02 the influence of Lead 3
must be included. Lead 3 directly influences a1
and b1.
Not usual a13 a1 3
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Answer With these two points included
Not usual a13 a1 3
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Example find the scattering matrix S for the
diagram shown within the red box
Answer With these two points included
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a13
b24
The individual S matrices little s(1) and
little s(2) are
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HW03 VA Pr. 01 Combine the two 2x2 scattering
matrices given on p. 126 by eliminating a5 and b5
to obtain the S-matrix for the composite
structure with the matrix elements given in eqn
3.2.1.
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Could analyze any of the 4 terms. Looking at t
Re-write
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Why t
a13 into b24 via combined S-matrix element t
a13
b24
Because it represent the goal find the overall
transmission.
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Goal find the transmission probability T through
a complete structure that contains the
scatterers So far found element t in a
combined S-matrix. What is its relationship to T?
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Lec 15
1.
2.
3.
Game plan for each Si j element determine
is it a reflection or a transmission?
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More accurately, transmission probabilities T ? t
an t elements. You could also solve for
individual reflection probabilities.
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HW03 VA Pr. 02
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Lecture 17, 24 Oct 13
Datta 3.1 scattering/S matrix Combining two
2x2 scattering matrices combine the scatterers
coherently combine the scatterers
incoherently combine the scatterers with
partial coherence Goal find the transmission
probability T through a complete structure that
contains the scatterers
Dresselhaus Graphene and Carbon
Nanotubes Carbon nanotube structure Carbon
bond hybridization is versatile sp1, sp2, and
sp3 Graphene
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CNT Structure

• Introduction
• 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

R. Saito, G. Dresselhaus and M.S.
Dresselhaus, Physical Properties of Carbon
Nanotubes, Imperial College Press, London, 1998.
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Introduction
A single wall Carbon Nanotube is a single
graphene sheet wrapped into a cylinder.
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Introduction
Buckyball endcaps
Many different types of wrapping result in a
seamless cylinder. But The particular cylinder
wrapping dictates the electronic and mechanical
properties.
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Introduction
Example of mechanical properties Raman
Spectroscopy phonons
Light in
Different wavelength light out
(10, 10) SWCNT
Phonons
Breathing mode
Tangential mode
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Introduction
Example of mechanical properties Raman
Spectroscopy phonons
Tangential Mode Semiconducting CNT
Light in
Different wavelength light out
Tangential Mode Metallic CNT
Phonons
Semiconducting Metallic CNT Mix
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Introduction
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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
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The Basis Vectors
1.44 Angstroms
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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
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The Basis Vectors
Note that the 1.44 Angstrom value is slightly
different for a buckyball (0-D), a CNT (1-D) and
in a graphite sheet (2-D). This is due to
curvature effects.