Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Solid State Nanoelectronics - Homework 8 Solved | ELEG 667, Assignments of Electrical and Electronics Engineering

University of Delaware (UD)Electrical and Electronics Engineering

Material Type: Assignment; Class: ELECTRIC POWER II; Subject: Electrical Engineering; University: University of Delaware; Term: Fall 2005;

Typology: Assignments

Pre 2010

Uploaded on 09/02/2009

koofers-user-x35
koofers-user-x35 🇺🇸

8 documents

1 / 1

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
ELEG 667–016; MSEG-667-016 - Solid State Nanoelectronics – Fall 2005
Homework #8 - due Tuesday, 22 November 2005, in class
1. Carbon Nanotube Structure: Derive the expressions for the unit vectors of graphene (unrolled
nanotube) a1, a2, in terms of the Cartesian unit vectors x^, y^. (a) Using trigonometry, show your
calculations for the numerical factors a and b (e.g. ½ or whatever) in terms of the length of the
carbon-carbon bond length, ao. (b) sketch and indicate the unit cell in real space, which is the
parallelogram spanned by a1, and a2. Hint, consider the figures below.
2. Carbon Nanotube Dispersion: Consider the dispersion relation for graphene:
W(kx, ky) = ± γo[1+4cos(√3kxa/2) cos(kya/2) + 4cos2(kya/2)]½ ,
following the notation in Waser, where a = √3ao is the length of the unit vector ai, and ao is the
length of the carbon-carbon bond (0.142 nm). Note that this “a” differs from the convention used
above in question 1. (a) Find the six Fermi level conduction points in k-space (which are the
corners of the hexagonal Brillouin zone below) by solving for the k values where W(kx, ky) = 0. (b)
On the hexagonal Brillouin zone, sketch and label the coordinates of these 6 points in terms of a, or
ao .
Hint: in the dispersion relation first let kx = 0 and solve for the corner points along ky; and then let
√3kxa/2 = π, and get the corners with kx ≠ 0. This approach makes it easier to factor the dispersion
terms under the root as a perfect square. Then take the square root and solve for kx,y.
ky
kx
3. Carbon Nanotube Metallic condition: Show that the condition for metallic conductivity of
chiral nanotubes: 2n1 + n2 = 3q, where q is an integer, can be obtained by substituting the k vector
of one of the corner points of the Brillouin zone into the periodic boundary condition: Ch •·k =
2πm, where Ch = n1 a1 + n2 a2 is the chiral vector, and m is an integer. (Hint: use vector coordinates
with respect to x and y, and a zone boundary point that has both x and y components).
Homework assignments will appear on the web at:
http://www.ece.udel.edu/~kolodzey/courses/eleg667_016f05.html
Note: On each submission, give your name, due date, assignment number and course number.

Partial preview of the text

Download Solid State Nanoelectronics - Homework 8 Solved | ELEG 667 and more Assignments Electrical and Electronics Engineering in PDF only on Docsity!

ELEG 667–016; MSEG-667-016 - Solid State Nanoelectronics – Fall 2005

Homework #8 - due Tuesday, 22 November 2005, in class

  1. Carbon Nanotube Structure: Derive the expressions for the unit vectors of graphene (unrolled nanotube) a 1 , a2 , in terms of the Cartesian unit vectors x^ , y^. (a) Using trigonometry, show your calculations for the numerical factors a and b (e.g. ½ or whatever) in terms of the length of the carbon-carbon bond length, ao. (b) sketch and indicate the unit cell in real space, which is the parallelogram spanned by a 1 , and a2. Hint, consider the figures below.
  2. Carbon Nanotube Dispersion: Consider the dispersion relation for graphene:

W(kx, ky) = ± γo[1+4cos(√3kx a /2) cos(k (^) y a /2) + 4cos 2 (ky a /2)] ½^ ,

following the notation in Waser, where a = √ 3 ao is the length of the unit vector ai , and ao is the length of the carbon-carbon bond (0.142 nm). Note that this “ a ” differs from the convention used above in question 1. (a) Find the six Fermi level conduction points in k-space (which are the corners of the hexagonal Brillouin zone below) by solving for the k values where W(kx, k (^) y) = 0. (b) On the hexagonal Brillouin zone, sketch and label the coordinates of these 6 points in terms of a , or ao. Hint: in the dispersion relation first let kx = 0 and solve for the corner points along k (^) y; and then let √3k (^) xa/2 = π, and get the corners with kx ≠ 0. This approach makes it easier to factor the dispersion terms under the root as a perfect square. Then take the square root and solve for k (^) x,y. k (^) y

kx

  1. Carbon Nanotube Metallic condition: Show that the condition for metallic conductivity of chiral nanotubes: 2n 1 + n 2 = 3q, where q is an integer, can be obtained by substituting the k vector of one of the corner points of the Brillouin zone into the periodic boundary condition: Ch • · k = 2 πm, where Ch = n 1 a1 + n 2 a2 is the chiral vector, and m is an integer. (Hint: use vector coordinates with respect to x and y, and a zone boundary point that has both x and y components).

Homework assignments will appear on the web at: http://www.ece.udel.edu/~kolodzey/courses/eleg667_016f05.html Note: On each submission, give your name, due date, assignment number and course number.