Homework QM Nanodevices assignments Fall 2013
1. Quiz I on September 18 (UG and G students) at the beginning of class: HW set 1;
To practice the material covered in chapter 1; not from the book, 1a. Plot the first
3 allowed 1D wave shapes versus x and derive an expression for their energies for
a wave constrained to a space of length L for periodic and stationary boundary
conditions. How does the energy between mode n and (n+1) depend on the size of
the space if we associate this wave with an electron for modelling purposes? 1b.
Show that if we associate a fictitious wave packet with a particle the group
velocity equals the classical particle velocity. What do you find if you use a plane
wave instead? From the book 2.1, 2.3, 2.5, 2.6, 2.16, 2.19. Note that on pg 31
below eq. 2.3 the units of max.T are meter.Kelvin not millikelvin.
2. Quiz on October 9 (UG and G students) at the beginning of class on the material
of HW set 2:
2.1 The channel of a modern n-channel MOSFET could have a length of 90 nm
(y-axis, Source-Drain spacing), width of 1 m (x-axis), with an electron
confinement under the SiO2/Si interface of 100 Angstrom (z-axis).Using
stationary boundary conditions in the x and z-direction and periodic bc in the
y- direction, calculate the 3 lowest electron energy levels E using EC for
potental energy. Express your answers in eV and use mn*= 0.3 x mo with mo
= 9.11x10-31 kg. Also draw a picture of the 3D k-space associated with this
device.
2.2 Calculate the required thickness [in x-direction] of a silicon wafer in nm to
create an 0.2 eV energy difference between the two lowest Enx energy states,
use mn*= 0.3 x mo with mo = 9.11x10-31 kg. Assume the y and z-direction
infinitely long.
2.3 What is the wavelength of the photon produced by an electron dropping from
the higher to the lower state calculated in 2.2?
2.4 Derive an expression for the electron density of a 2D system in terms of the
DoS and Fermi level EF. Simplify it for low temperature if EF lies between E1
and E2, and below E1.
2.5 Repeat problem 2.4 for a 1D system.
3. Quiz on October 23 (UG and G students) at the beginning of class on the
material of HW set 3: book hw problems 4.13 [answer E1=1.3eV, E2=2.9eV],
4.14 [answer mn*= 8 x m0], and 4.15 [answer i. mn*=8.1 x mo, ii. mn*=-7.2 x
mo, iii.mn*=+/- infinity]. Note to solve this homework some numerical matlab
or smart calculator work is needed so start early.
4. Quiz 4 on November 13 on the following assignments: HW problem 5.1 and the
following Problem Statement: Design the width and the minimum mol fraction x
of an AlxGa1-xAs/GaAs/ AlxGa1-xAs Quantum Well Infrared Photodetector
(QWIP) device such that it is able to detect infrared radiation of 10 m
wavelength (night vision, skin thermography application. Note that 10 m is the
peak blackbody radiation wavelength at about T=300K). Design the QWIP such
that it has 2 confined energy levels E1 and E2 (p.153 Singh, fig. 5.3) in the
conduction band with the difference between E2 and E1 corresponding to the
incoming photon energy and E2 aligning with the barrier height. Use m*=0.067
mo, assume that 60% of the EG shows up in the conduction band, and employ
the theory of graphical solution explained on page 151 of Singh’s book.
Numerical answers: V0=0.191 eV, W=5.4 nm.
5. Quiz 5 will take place on Dec 4. and is based on book problems 5.2, 6.1, 6.2, 6.6,
and 6.11. This is the last quiz for the class.