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Real space
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5/17/2007
Electron Velocity and the Dispersion Relation
Until now, we have not considered the velocity of electrons because we have not
considered the time dependence of solutions to the Schrdinger equation. In Part 1, we
broke the full Schrdinger Equatio

5/17/2007
Problems
Q1. Find the effective mass for an electron in a conductor with the dispersion relation:
E ( k ) = 5 2V cos ( ka ) , k <
a
where V and a are positive constants.
Q2.(a) A 1nm x 1nm molecule is 30 away from a metal contact. Calculate the

5/17/2007
Propagation of a Gaussian Wavepacket
Next we will examine the propagation of a Gaussian wavepacket in free space. Again, we
will expand the wavefunction in terms of its eigenfunctions.
Consider an electron in free space. Let the initial wavepack

5/17/2007
Periodic Boundary Conditions in 2-d
Applying periodic boundary conditions to 2d materials follows the same principles as in
1d.
Lets assume that the long axes of the quantum well are aligned with the x and y axes,
and that the dimensions of the

5/17/2007
Momentum and Energy
Two key experiments revolutionized science at the turn of the 20th century. Both
experiments involve the interaction of light and electrons. We have already seen that
electrons are best described by wavepackets. Similarly, li

5/17/2007
Completeness
We will not prove completeness in the class. Instead we merely state that the
completeness property of eigenfunctions allows us to express any well-behaved function
in terms of a linear combination of eigenfunctions. i.e. if n is an

5/17/2007
6
Energy/EL
5
E=4.68EL
4
3
E=2.32EL
2
1
E=0.60EL
0
-L/2
0
L/2
x
Fig. 1.22. The three bound states for electrons in a well with confining potential
V0 = 5.EL. Note that the higher the energy, the lower the effective confining potential,
and the g

5/17/2007
Problems
1. i) Using MATLAB, Generate Figure 2.13.
3
which mode has a lower energy cfw_nx=3, ny=1 or cfw_nx=2, ny=2
5
2. The transistor illustrated below has an oxide ( = 4 0 ) thickness d = 1.2nm and
channel depth Lz = 2.5nm. Considering the ch

5/17/2007
In nanoelectronics, however, the uncertainty principle can play a role.
For example, consider a very thin wire through which electrons pass one at a time. The
current in the wire is related to the transit time of each electron by
q
(1.83)
I= ,
T

5/17/2007
The 0-d DOS: single molecules and quantum dots confined in 3-d
The 0-d DOS is a special case. Now the particle is confined in all directions. Periodic
boundary conditions are inappropriate.
Like a particle in a well with discrete energy levels,

5/17/2007
Problems
1. Suppose we fire electrons through a single slit with width d. At the viewing screen
behind the aperture, the electrons will form a pattern. Derive and sketch the expression
for the intensity at the viewing screen. State all necessary

5/17/2007
The Density of States
To calculate the number of electrons that conduct charge, and hence the magnitude of the
current, we need the density of states (DOS), which as you recall is a measure of the
number of states in a conductor per unit energy.

5/17/2007
The Ideal Contact Limit
Interfaces between molecules and contacts vary widely in quality. Much depends on how
close we can bring the molecule to the contact surface. Here, we have modeled the source
and drain interfaces with the parameters S and

5/17/2007
Part I. The Quantum Particle
This class is concerned with the propagation of electrons in conductors.
To begin, we will need a way to describe electrons. It is often convenient to imagine
electrons as little projectiles pushed around by an elect

5/17/2007
Summary
The net change in potential at the molecule, U, is the sum of electrostatic and charging
effects:
U = U ES + U C
(3.17)
Note that by fixing the Fermi energy of the molecule at equilibrium, EF = 0, and
assuming that = 0.5, we have forced

5/17/2007
Current Flow in Two Terminal Quantum Dot/Single Molecule Devices
In this section we present a simplified model for conduction through a molecule. It is
based on the toy model of Datta, et al.1 which despite its relative simplicity describes
many

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6.085/6.975
Introduction to Nanoelectronics
Introduction
Modern technology is characterized by its emphasis on miniaturization. Perhaps the most
striking example is electronics, where remarkable technological progress has come from
reductions in

5/17/2007
Part 3. Two Terminal Quantum Dot Devices
In this part of the class we are going to study electronic devices. We will examine devices
consisting of a quantum dot or a quantum wire conductor between two contacts. We will
calculate the current in t