ECE : Nanoelectronics Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University

VM Ayres, ECE , F13 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

VM Ayres, ECE , F13 Lec 15: What is a scattering S matrix?:

VM Ayres, ECE , F13 Lec 15: Game plan: for each S i j element: determine : is it a reflection or a transmission? 3.

VM Ayres, ECE , F13 Datta, Sec. 3.1: Two propagating modes into one propagating mode: expect scattering due to occupied states Coherent Conductor

VM Ayres, ECE , F13 Lec 15: Example: find the S-matrix for the Buttiker/Enquist diagrams shown. Incoherent Conductor

VM Ayres, ECE , F13 Example: find the S-matrix for both of the diagrams shown.

VM Ayres, ECE , F13 Example: find the scattering matrix S for the diagram shown within the red box:

VM Ayres, ECE , F13

Basic form:

VM Ayres, ECE , F13 Two points: Point 01: Reflections are not really the same. One incorporates the influence of Leads 2 and 4 and the other doesn’t. Same is true for transmissions. Therefore: Let r -> r and r’ with the influence of Leads 2 and 4 Let t -> t and t’ with the influence of Leads 2 and 4

VM Ayres, ECE , F13

Two points: Point 01: Reflections are not really the same. One incorporates the influence of Leads 2 and 4 and the other doesn’t. Same is true for transmissions. Therefore: Let r -> r, and r’ with the influence of Leads 2 and 4 Let t -> t, and t’ with the influence of Leads 2 and 4

VM Ayres, ECE , F13

Two points: Point 02: the influence of Lead 3 must be included. Lead 3 directly influences a 1 and b 1. Not usual a 13 = a 1 3

VM Ayres, ECE , F13 Answer: With these two points included: Not usual a 13 = a 1 3

VM Ayres, ECE , F13 Example: find the scattering matrix S for the diagram shown within the red box: Answer: With these two points included:

VM Ayres, ECE , F13 The individual S matrices = little s (1) and little s (2) are: a 13 == b 24

VM Ayres, ECE , F13 HW03: VA Pr. 01: Combine the two 2x2 scattering matrices given on p. 126 by eliminating a 5 and b 5 to obtain the S-matrix for the composite structure with the matrix elements given in eq’n

VM Ayres, ECE , F13 Could analyze any of the 4 terms. Looking at “t”: Re-write

VM Ayres, ECE , F13 Why “t”: a 13 into b 24 via combined S-matrix element “t” a 13 == b 24 Because it represent the goal: find the overall transmission.

VM Ayres, ECE , F13 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?

VM Ayres, ECE , F13 Lec 15: Game plan: for each S i j element: determine : is it a reflection or a transmission? 3.

VM Ayres, ECE , F13 More accurately, transmission probabilities T  t an t’ elements. You could also solve for individual reflection probabilities.

VM Ayres, ECE , F13

HW03: VA Pr. 02:

VM Ayres, ECE , F13 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 : sp 1, sp 2, and sp 3 Graphene

VM Ayres, ECE , F13 CNT Structure Introduction The Basis Vectors: a 1 and a 2 The Chiral Vector: C h The Chiral Angle: cos(  The Translation Vector: T The Unit Cell of a CNT – Headcount of available  electrons R. Saito, G. Dresselhaus and M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, 1998.

VM Ayres, ECE , F13 Introduction A single wall Carbon Nanotube is a single graphene sheet wrapped into a cylinder.

VM Ayres, ECE , F13 Introduction Many different types of wrapping result in a seamless cylinder. But The particular cylinder wrapping dictates the electronic and mechanical properties. Buckyball endcaps

VM Ayres, ECE , F13 Introduction Example of mechanical properties: Raman Spectroscopy & phonons Breathing mode Tangential mode Light in Phonons Different wavelength light out (10, 10) SWCNT

VM Ayres, ECE , F13 Introduction Example of mechanical properties: Raman Spectroscopy & phonons Light in Phonons Different wavelength light out Tangential Mode Semiconducting CNT Semiconducting & Metallic CNT Mix Tangential Mode Metallic CNT

VM Ayres, ECE , F13 Introduction

VM Ayres, ECE , F13 The Basis Vectors a 1 = √3 a x + 1 a y 2 2 a 2 = √3 a x - 1 a y 2 2 where magnitude a = |a 1 | = |a 2 |

VM Ayres, ECE , F13 The Basis Vectors 1.44 Angstroms

VM Ayres, ECE , F13 The Basis Vectors a 1 = √3 a x + 1 a y 2 2 a 2 = √3 a x - 1 a y 2 2 Magnitude a= 2 [ (1.44 Angstroms)cos(30) ] = 2.49 Angstroms 1.44 A 120 o a

VM Ayres, ECE , F13 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. The Basis Vectors