ECE 802 604 Nanoelectronics Prof Virginia Ayres Electrical

ECE 802 -604: Nanoelectronics Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu. edu

Lecture 25, 26 Nov 13 Carbon Nanotubes and Graphene CNT/Graphene electronic properties sp 2: electronic structure 2 DEG: E-k relationship/graph for graphene and transport 1 DEG: E-k relationship/graph for CNTs and transport R. Saito, G. Dresselhaus and M. S. Dresselhaus Physical Properties of Carbon Nanotubes VM Ayres, ECE 802 -604, F 13

CNT Unit cell in green: Ch = n a 1 + m a 2 |Ch| = a√n 2 + mn dt = |Ch|/p cos q = a 1 • Ch |a 1| |Ch| T = t 1 a 1 + t 2 a 2 t 1 = (2 m + n)/ d. R t 2 = - (2 n + m) /d. R = the greatest common divisor of 2 m + n and 2 n+ m |T| = √ 3(m 2 + n 2+nm)/d. R = √ 3|Ch|/d. R N = | T X Ch | | a 1 x a 2 | = 2(m 2 + n 2+nm)/d. R VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

K 1 is in same direction as Ch Specify direction of Ch using choral angle K 2 is in same direction as T VM Ayres, ECE 802 -604, F 13

Transport: Real space: Ch Reciprocal space: K 1 Real space: T Reciprocal space: K 2 Transport: along CNT: Along a Unit vector in the K 2 direction Can have any magnitude (hbar)k (10, 10) (9, 0) (7, 4) VM Ayres, ECE 802 -604, F 13

For an e- described as a wave: Quantization of Energy E is here Standing wave: Quantization by m in Ch / K 1 direction Travelling wave: with an unquantized wave vector k in T/ K 2 direction VM Ayres, ECE 802 -604, F 13

+ Transport ECNT is proportional to Egraphene 2 D conduction energy level ECNT is proportional to the value of the transfer integral t Conduction and valence energy levels VM Ayres, ECE 802 -604, F 13

k hbark is in the transport direction. Where k is relative to kx and ky depends on the nanotube (n, m) VM Ayres, ECE 802 -604, F 13

a 1 ZIGZAG: Zigzag: Ch in a 1 direction VM Ayres, ECE 802 -604, F 13

ZIGZAG: kx ky Example: which is the Ch direction, kx or ky? VM Ayres, ECE 802 -604, F 13

ZIGZAG: kx ky Answer: ky VM Ayres, ECE 802 -604, F 13

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. VM Ayres, ECE 802 -604, F 13

ZIGZAG: VM Ayres, ECE 802 -604, F 13

a 1 ARMCHAIR: kx ky Example: Which components cancel? Which components add? VM Ayres, ECE 802 -604, F 13

a 1 ARMCHAIR: kx ky Answer: Which components cancel? kx Which components add? ky VM Ayres, ECE 802 -604, F 13

Lec 24: Consider an (n, n) armchair CNT. This is where the periodic boundary condition on k. X comes from in: That leaves just k. Y as open, MD calls it just k. VM Ayres, ECE 802 -604, F 13

ARMCHAIR: VM Ayres, ECE 802 -604, F 13

ARMCHAIR: VM Ayres, ECE 802 -604, F 13

a 1 (4, 2) CHIRAL: where Ch and T are: VM Ayres, ECE 802 -604, F 13

For chiral: from Lec 23: VM Ayres, ECE 802 -604, F 13

Therefore: VM Ayres, ECE 802 -604, F 13

a 1 (4, 2) CHIRAL: where Ch and T are: a 2 VM Ayres, ECE 802 -604, F 13

Real space: Ch Reciprocal space: K 1 VM Ayres, ECE 802 -604, F 13

a 1 (4, 2) CHIRAL: where Ch and T are: a 2 VM Ayres, ECE 802 -604, F 13

a 1 (4, 2) CHIRAL: where Ch and T are: a 2 VM Ayres, ECE 802 -604, F 13

Real space: Ch Reciprocal space: K 1 Transport direction: Real space: T Reciprocal space: K 2 VM Ayres, ECE 802 -604, F 13

Transport in a 1 -D Real space: Ch Reciprocal space: K 1 Real space: T Reciprocal space: K 2 A Unit vector in the K 2 direction: (10, 10) (9, 0) (7, 4) VM Ayres, ECE 802 -604, F 13

Lec 05 VM Ayres, ECE 802 -604, F 13

Lec 05: 6. Current I q x n x vgroup VM Ayres, ECE 802 -604, F 13

Lec 06: VM Ayres, ECE 802 -604, F 13

Lec 24: What you can do with an E-k diagram: Answer: VM Ayres, ECE 802 -604, F 13

Lec 07: 2 -DEG: 1 -DEG: VM Ayres, ECE 802 -604, F 13

2 DEG Graphene: Conduction energy level for p* VM Ayres, ECE 802 -604, F 13

2 DEG Graphene: N(E) E VM Ayres, ECE 802 -604, F 13

1 DEG CNT: Conduction energy levels VM Ayres, ECE 802 -604, F 13

Specify: example: (n, 0) zigzag CNT. You can write a periodic boundary condition on ky and substitute into eq’n 2. 29. That leaves just kx as open, MD calls it just k. VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

Lec 07: 2 -DEG: 1 -DEG: VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

Same as Datta Pr. 1. 3 VM Ayres, ECE 802 -604, F 13

Same as Datta Chp. 02 Four terminal Two terminal VM Ayres, ECE 802 -604, F 13

Same as Datta Chp. 02 VM Ayres, ECE 802 -604, F 13

Coherent: Same as Datta Chp. 03 VM Ayres, ECE 802 -604, F 13

Incoherent: Same as Datta Chp. 03 VM Ayres, ECE 802 -604, F 13