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 23, 19 Nov 13 Carbon Nanotubes and Graphene CNT/Graphene electronic properties sp 2: electronic structure E-k relationship/graph for polyacetylene E-k relationship/graph for graphene E-k relationship/graph for CNTs R. Saito, G. Dresselhaus and M. S. Dresselhaus Physical Properties of Carbon Nanotubes VM Ayres, ECE 802 -604, F 13

Polyacetylene E-k: E -kx kx VM Ayres, ECE 802 -604, F 13

To really finish: Need to model the wavefunctions: Could let f = |2 px> Could let f = |p> (f = |sp 2> is the s-bond) ECE 802 -604: Use this result, p. 24: t = -1. 0 s = +0. 2 e 2 p = 0. 0 VM Ayres, ECE 802 -604, F 13

Real space VM Ayres, ECE 802 -604, F 13

Two different spring constants: tighter k 1 (double bond) and looser k 2 single bond k 1 k 2 VM Ayres, ECE 802 -604, F 13

Graphene E-k: VM Ayres, ECE 802 -604, F 13

e 2 p A To really finish: Need to model the wavefunctions: Could let f = |2 px> Could let f = |p> (f = |sp 2> is the s-bond) ECE 802 -604: Use this result, p. 27: t = -3. 033 Units s = 0. 129 Units e 2 p = 0. 0 Units Example: what are the Units? VM Ayres, ECE 802 -604, F 13

e 2 p A To really finish: Need to model the wavefunctions: Could let f = |2 px> Could let f = |p> Answer: (f = |sp 2> is the s-bond) ECE 802 -604: Use this result, p. 27: t = -3. 033 e. V s = 0. 129 pure number e 2 p = 0. 0 e. V VM Ayres, ECE 802 -604, F 13

Graphene E-k: VM Ayres, ECE 802 -604, F 13

Bottom of the conduction band: the 6 equivalent K-points E ky kx VM Ayres, ECE 802 -604, F 13

What you can do with an E-k diagram: Example: What is “k” in 2 D? In 1 D? VM Ayres, ECE 802 -604, F 13

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

Lecture 23 & 24, 19 Nov 13 Carbon Nanotubes and Graphene CNT/Graphene electronic properties sp 2: electronic structure E-k relationship/graph for polyacetylene E-k relationship/graph for graphene E-k relationship/graph for CNTs R. Saito, G. Dresselhaus and M. S. Dresselhaus Physical Properties of Carbon Nanotubes VM Ayres, ECE 802 -604, F 13

Graphene: the 6 equivalent K-points Bottom of the conduction band the 6 equivalent K-points metallic E ky kx Therefore: CNTs are metallic at the conditions for the K-points of graphene VM Ayres, ECE 802 -604, F 13

Rules for finding the electronic structure (p. 21): 1 Use to find k 2 Find k 3 4 Find HAA, HAB, SAA, SAB Find det|H – ES| = 0 => E = E(k) Plot E versus k VM Ayres, ECE 802 -604, F 13

l sp 2 electronic structure: CNTs – Real space (Unit cell) – Reciprocal space – Use Real and Reciprocal space to find E 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 N = | T X Ch | | a 1 x a 2 | = 2(m 2 + n 2+nm)/d. R VM Ayres, ECE 802 -604, F 13

Example: Evaluate K 1 for a (4, 2) CNT: VM Ayres, ECE 802 -604, F 13

In class: VM Ayres, ECE 802 -604, F 13

In class: VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

Example: Evaluate K 2 for a (4, 2) CNT: VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

Example: add a set of axes VM Ayres, ECE 802 -604, F 13

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

VM Ayres, ECE 802 -604, F 13

0 through 27 28 of these: VM Ayres, ECE 802 -604, F 13

ECNT is quantized VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

Example: For a (4, 2) CNT evaluate: Ch, |Ch|, T, |T|, K 1, K 2, |K 1|, |K 2| VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

Example: Compare |b 1|, |b 2|, with |K 1|, |K 2| for a (4, 2) CNT VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

CNT E-k; Energy dispersion relations (E vs k curves): Quantization of Energy E is here K 1 is quantized by m in Ch direction K 2 = k is continuous in T direction VM Ayres, ECE 802 -604, F 13