### Homework Assignments QIC 890/891, Module 7: Nanowires in QIP

```Homework Assignments, QIC 890/891, Module 7: Nanowires in QIP
Daryoush Shiri, July 2014
PRJ1: Band structure of Graphene and CNT
Write the on-site (H11) and nearest neighbour (H1n) Tight
Binding Hamiltonians of Graphene (a periodic 2D solid) by
assuming an appropriate unit cell and finding out its
neighbours.
Use Bloch method and write the total Hamiltonian.
Diagonalize (or simply use eig function in MATLAB) the
Hamiltonian to find E(kx,ky) and plot it.
Show that by wrapping a portion of Graphene to make a zig zag
CNT of (5,0) type, the band structure looks like as shown in the
figure.
HINT: Consult with the following book to know the procedure.
Quantum Transport, Supriyo Datta,
Cambridge University press, (2005).
ECE 730, Winter 2008,
M. P. Anantram, University of Washington
1
PRJ2: Electron- polar LO phonon scattering in a nanowire:
Using the Hamiltonian given for Fröhlich interaction and Effective Mass Approximation(EMA):
(a) Write the correctly normalized wave function for the nanowire as shown in the figure.
(b) Write the Energy band structure and plot it for (n,m)=(0,0),(0,1),(1,0),(1,1).
(c) Plot scattering rate for electrons starting from the bottom of the first conduction sub-band up to 100meV.
Hints: Start from Fermi golden rule to obtain the scattering rate and simplify it as much as possible.
To compute the structure factors consult with the following references:
Phonons in Nanostructures, Michael A. Stroscio, Mitra Dutta, Cambridge
University Press (2005).
K. W. Kim, et al, J. Appl. Phys. 70 (1), 319, 1 July 1991.
Assumptions: nanowire is made of InAs and it is confined along x and y direction.
It has square cross section (Lx=Ly=50nm).
Find other numerical parameters from the following reference:
http://www.ioffe.ru/SVA/NSM/Semicond/InAs/index.html
2
1- Show that if a magnetic field (B) is applied along Y direction, then with A=(Bz, 0, 0),
the Rashba SOI Hamiltonian looks like as follows:
𝑎46
𝑝 + 𝑒𝐵𝑧
𝐻𝑅 =
𝜀𝑦 𝑥
−𝑝𝑧
ℏ
−𝑝𝑧
−𝑝𝑥 − 𝑒𝐵𝑧
2
2- Show that [𝜎. (𝑝 + 𝑒𝐴)] = (𝑝 + 𝑒𝐴)2 +2𝑚0 𝜇𝐵 𝐵. 𝜎
2- In the following nanowire Hamiltonian determine:
(a) which SOI effect (Rashba or Dresselhaus) causes a magnetic field in parallel with the external magnetic field?
(b) which one generates an effective magnetic field perpendicular to the external magnetic field?
and
propose an experiment with which you cam measure Rashba and Dresselhaus SOI strengths i.e. factors in a nanowire.
See page.20 of Spin-Orbit-Interaction slides
3
```