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 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 VM Ayres, ECE802-604, F13

dR = the greatest common divisor of 2m + n and 2n+ m CNT Unit cell in green: Ch = n a1 + m a2 |Ch| = a√n2 + 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| = √ 3(m2 + n2+nm)/dR = √ 3|Ch|/dR N = | T X Ch | | a1 x a2 | = 2(m2 + n2+nm)/dR VM Ayres, ECE802-604, F13

VM Ayres, ECE802-604, F13

K1 is in same direction as Ch Specify direction of Ch using choral angle K2 is in same direction as T VM Ayres, ECE802-604, F13

Transport: Real space: Ch Reciprocal space: K1 Real space: T Transport: along CNT: Along a Unit vector in the K2 direction Can have any magnitude (hbar)k (10,10) (9,0) (7,4) VM Ayres, ECE802-604, F13

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 VM Ayres, ECE802-604, F13

+ Transport ECNT is proportional to Egraphene2D  conduction energy level ECNT is proportional to the value of the transfer integral t Conduction and valence energy levels VM Ayres, ECE802-604, F13

k  hbark is in the transport direction k  hbark is in the transport direction. Where k is relative to kx and ky depends on the nanotube (n,m) VM Ayres, ECE802-604, F13

ZIGZAG: a1 Zigzag: Ch in a1 direction VM Ayres, ECE802-604, F13

ZIGZAG: kx ky Example: which is the Ch direction, kx or ky? VM Ayres, ECE802-604, F13

ZIGZAG: kx ky Answer: ky VM Ayres, ECE802-604, F13

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, ECE802-604, F13

ZIGZAG: VM Ayres, ECE802-604, F13

ARMCHAIR: kx a1 ky Example: Which components cancel? Which components add? VM Ayres, ECE802-604, F13

ARMCHAIR: kx a1 ky Answer: Which components cancel? kx Which components add? ky VM Ayres, ECE802-604, F13

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. VM Ayres, ECE802-604, F13

ARMCHAIR: VM Ayres, ECE802-604, F13

ARMCHAIR: VM Ayres, ECE802-604, F13

(4,2) CHIRAL: where Ch and T are: VM Ayres, ECE802-604, F13

For chiral: from Lec 23: VM Ayres, ECE802-604, F13

Therefore: VM Ayres, ECE802-604, F13

(4,2) CHIRAL: where Ch and T are: VM Ayres, ECE802-604, F13

Real space: Ch Reciprocal space: K1 VM Ayres, ECE802-604, F13

(4,2) CHIRAL: where Ch and T are: VM Ayres, ECE802-604, F13

(4,2) CHIRAL: where Ch and T are: VM Ayres, ECE802-604, F13

Real space: Ch Reciprocal space: K1 Transport direction: Real space: T VM Ayres, ECE802-604, F13

Transport in a 1-D Real space: Ch Reciprocal space: K1 Real space: T A Unit vector in the K2 direction: (10,10) (9,0) (7,4) VM Ayres, ECE802-604, F13

Lec 05 VM Ayres, ECE802-604, F13

Lec 05: 6. Current I  q x n x vgroup VM Ayres, ECE802-604, F13

Lec 06: VM Ayres, ECE802-604, F13

Lec 24: What you can do with an E-k diagram: Answer: VM Ayres, ECE802-604, F13

Lec 07: 2-DEG: 1-DEG: 1-DEG: VM Ayres, ECE802-604, F13

2DEG Graphene: Conduction energy level for p* VM Ayres, ECE802-604, F13

2DEG Graphene: N(E) E VM Ayres, ECE802-604, F13

1DEG CNT: VM Ayres, ECE802-604, F13 Conduction energy levels

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, ECE802-604, F13

VM Ayres, ECE802-604, F13

Lec 07: 2-DEG: 1-DEG: 1-DEG: VM Ayres, ECE802-604, F13

VM Ayres, ECE802-604, F13

Same as Datta Pr. 1.3 VM Ayres, ECE802-604, F13

Same as Datta Chp. 02 Four terminal Two terminal VM Ayres, ECE802-604, F13

Same as Datta Chp. 02 VM Ayres, ECE802-604, F13

Coherent: Same as Datta Chp. 03 VM Ayres, ECE802-604, F13

Incoherent: Same as Datta Chp. 03 VM Ayres, ECE802-604, F13