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University of Notre Dame
Quantum Transport in Nanoscience
EE47065- 67065 Fall Term 2025
PROBLEM SET
Due: THURSDAY, OCTOBER 16
th
, at 9:30 a.m. (preferably via email or else) in Room 100
Stinson-Remick
1. (20 points) Polarizability of an Ethane Molecule. Consider a particle of mass m
constrained to move in the xy - plane on a circular ring of radius “ a”. The only variable of
the system is the azimuthal angle called The state of the system is described by a
wavefunction , which must have the property that:
(a) The kinetic energy of the particle can be written as follows:
Calculate the eigenvalues and eigenfunctions of H
0
. Which of the energy levels are
degenerate?
(b) Now assume that the particle has a charge and that it is placed in a uniform electric field
E in the x - direction. We must therefore add to the Hamiltonian the perturbation:
Calculate the new wave function of the ground state to first order in E. Use this wave
function to evaluate the induced electric dipole moment in the x - direction: i.e.,
Determine the proportionality constant between the dipole moment and the applied field.
This proportionality constant is called the “polarizability” of the molecule.
(c) Now consider the ethane molecule CH
3
—CH
3
, and attempt to analyze the rotation of
one CH 3
group relative to the other about the straight line joining the two carbon atoms. To
lowest order, the rotation is free and the Hamiltonian resembles H
0
above. However, the H
is different. To account for the electrostatic interaction energy between the two CH
3
groups
we assume a perturbation:
where “ b” is a real constant. By appealing to elementary chemistry, justify the
dependence of the perturbation. Calculate the energy and wave function for the new
ground state (to first order in “ b” for the wave function and second order for the energy.)
Give a physical interpretation to the result.
2
0
2
d
where
2
2
2
2
0
2 d
d
ma
H
H b cos 3
qx
cos
H q E a
2. (15 points) Scattering from a spherical delta-function shell. Consider the case of low-
energy scattering from a spherical delta-function shell defined by:
where and “a” are constant. Calculate the scattering amplitude, f (), the differential
cross-section D () and the total cross-section , assuming that ka << 1 so that only the l =
0 term contributes significantly. Express your answer in terms of the dimensionless
quantity
2
2 ma /
.
3. (25 points) Finite Size Hydrogen Nucleus. When you studied the hydrogen atom, you
assumed that the Coulomb potential extended all the way to the origin. In reality, the proton
charge is smeared out over a sphere of radius of 1 fm (or 1 Fermi = 10
m ). This has a
slight effect on the energy levels of the hydrogen atom. Model the electric charge distribution
of the proton as a uniformly charged sphere of radius R.
(a) Find the electrostatic potential energy of the electron for all r. [ Hint: Use Gauss’s law to
find the electric field everywhere and then integrate to find the potential energy .]
(b) Use lowest order perturbation theory to calculate the shift in the energy of the ground
state of hydrogen due to this modification of the potential. Evaluate the answer numerically
using R = 100 fm and express your answer as a fraction of the binding energy of the ground
state (i.e., 1 Rydberg= 13.6 eV ) [ Hint: To simplify the integrals notice that the unperturbed
wavefunction varies slowly over the range 0 < r < R. ]
4. (20 points) Scattering Matrix. The theory of scattering generalizes in an obvious way to
arbitrary localized potentials. Consider the figure below.
To the left (Region I), V ( x ) = 0 , so:
To the right in Region III, V ( x ) is again zero so the solution takes the form:
2
2
( )
mE
x Ae Be wherek
ikx ikx
forr R
R
R r
R
q
forr R
r
q
V r
1
( )
2
1
( )
2 2
3
2
2
V ( r )( r a )
2
2
( )
mE
x Ae Be wherek
ikx ikx
5. (40 points) Multi-slice Simulation of Electron Beam Propagation through SrTiO3 [001]****.
We aim to demonstrate the basic concepts and functions of a
transmission electron microscope (TEM) image simulations
with Dr. Probe simulation tools step-by-step, such that it may
serve as a template for your own TEM image simulations
someday. The applied simulation approach realistically captures
the dynamics of elastic electron diffraction and partially
coherent imaging by a modern aberration-corrected TEM.
Specifically, our main aim is to illustrate the steps in a
multi-slice simulation of an electron beam propagating through
a crystal of cubic perovskite SrTiO 3
in [001] zone-axis
orientation. The structural model of SrTiO 3
was taken form Abramov et al. [Yu.A. Abramov,
V.G. Tsirel'son, V.E. Zavodnik, S.A. Ivanov, and I.D. Brown, Acta Cryst. B 51 (1995) p.
942]. The space group describing the symmetry of the crystal is Pm-3m (221) with a lattice
constant of a = 0.3901 nm, Sr in the corners, Ti in the center and three oxygen atoms at the
face centers as illustrated by the figure. For full credit you will have to work through an
example illustrated at: http://www.er-c.org/barthel/drprobe/
To get started, you will have to first download Dr. Probe software/GUI. (To run the GUI,
you will need a 64-bit Windows Operating system with some memory—8 GB is desired but
not absolutely necessary.) The Dr. Probe software is a tool package for multi-slice image
simulations in high-resolution scanning and imaging transmission electron microscopy. It
comprises a graphical user interface version for direct visualization of STEM/TEM image
calculations, as well as a bundle of command-line modules for more comprehensive
calculation tasks. While the graphical user-interface version is primarily designed to make
quick simulation setups with intuitive parameter input and to check meaningful setups for
experiment or intensive calculations, the command-line version modules allow you to script
calculations with parameter variations for time-consuming image calculation. (The Dr. Probe
software package is distributed freely via download in form of an installer or a ZIP archive.
The installation procedure is simple and can be performed by following the download and
installation instructions available at the website. Using the Dr. Probe STEM simulation
graphical user interface requires a registration of the installation. You should also be able to
find Dr. Probe in the stacks (virtual workstations). The stack can be accessed through the
link:https://esc-stack-graphics-design-xlarge-cad-apps.escvcl.nd.edu/)
(a) To appreciate the issues, let’s dispense with a few preliminaries first. In a TEM, the
image is not usually determined by the resolving power of the electron, but by the available
contrast. Let’s assume that the electrons in a FEI Titan 80- 300 kV TEM are accelerated
through a voltage V 0
= 300 kV , so that the wavelength is more than suitable for imaging.
What is the electron wavelength, , corresponding to this accelerating voltage?
(b) On the other hand, the actual point resolution of this microscope, s
, is determined by:
where C
s
is the spherical aberration coefficient of the objective lens in the microscope.
Estimate the point resolution assuming that the spherical aberration coefficient is C
s
mm.
31 / 4
0. 64 ( )
s s
C
(c) To obtain a STEM simulation of the cubic perovskite SrTiO
3
in [001] orientation we
use the Dr. Probe GUI version 1.7. The main purpose of this example is to demonstrate the
basic concepts and functions of STEM image simulations with Dr. Probe. The example
documents the simulation procedure step by step, such that it may serve as a template for
your own STEM image simulations (in the future). The corresponding documents that will
guide you through the example in detail can be found at:
http://www.er-c.org/barthel/drprobe/drprobegui-example-sim1-sto001.html. Initially, let’s
setup the microscope and calculation parameters.
First, open the Dr. Probe GUI, and build the Basic Illumination Parameters, complete the
Detector Setup, and input the Microscope Parameters.
Second, set up the parameters for the multi-slice calculation.
Load the input atomic structure data of SrTiO 3
[001], which
can be downloaded from: http://www.er-
c.org/barthel/data/Example01.SrTiO3-input.zip.
Next, show the atomic structure of SrTiO 3
cubic perovskite
unit cell image in the orientation of projection vector [0.4 0.
1] generated by the free-download software of VESTA
available at: http://jp-minerals.org/vesta/en/download.html.
Then load this input structure in .cif format to the Setup
Multi-slice Parameters in the GUI and modify the structure
model to a larger super-cell and create the phase gratings.
Finally, finish the setup of slice data, calculation, and scan
frame.
Now, it’s time to run the simulations. Activate the object data view and select the
maximum object thickness of 25 nm for the calculation. Further, select the item Phase of
object functions from the object data list and scan image from the calculation type list. Start
the image calculation by clicking on the Start Calculation… button, and you will see the
progress bar and result preview. Typically, the progress may take up to 35 minutes to
complete the simulations. Once the full STEM image is calculated, use the controls of the
calculation results section to navigate through the calculated images using Bright Field (BF),
Annular Bright Field (ABF) and High-angle Annular Dark Field detectors. Finally, convolute
the STEM images by clicking on the Apply Source Profile button, and the resulting images
will appear with a significantly smoothed and lower contrast. Save all the simulation results,
and show an array consisting of RAW data and with PSC applied for BF (0-5 mrad), ABF
(12-24 mrad), HAADF (80-250 mrad) (six images with the selected Z = 65 ).
(d) In a transmission electron microscope (TEM), the absolute square of a wave function,
which has been magnified by passing through a system of lenses, the so-called image-plane
wave function. The image-plane wave function is additionally modulated due to aberrations,
and the imaging process itself is affected by partial losses of coherence and further
incoherent phenomena, which all essentially lead to a reduction of the image contrast.
TEM image simulations with Dr. Probe are composed of three major steps that separate the
creation of important intermediate data. This separation allows a few short-cuts when doing
parameter variations, as not all steps need to be repeated depending on the level where the
parameter variation takes place. Each of the three step is performed by calling a particular
command-line tool:
CELSLC - creating phase gratings of object slices,
MSA – multi-slice algorithm calculating the electron diffraction
WAVIMG - image calculated from an exit-plane wave function
necessary for focal series calculation. Run WAVIMG with the new parameter file, and a
series of 41 raw binary data files is created and stored in the subfolder img with the file
names STO_110_300_001.dat, ... , STO_110_300_041.dat. The files contain images of
the 2.8 nm thick SrTiO 3
crystal in [110] projection with image defocus changing between
- 10 nm and +10 nm in steps of 0.5 nm. Show three calculated HRTEM images at
defocuses of - 10, 0 and +10 nm. To generate these HRTEM images, import the
STO_110_300_xxx.data file to DigitalMicrograph with Real 4 Byte data type and
128 135 1 size with Binary selected.
6. (2 0 points) Time-dependent Perturbation Theory. Suppose that a perturbation takes the
form of a delta-function (in time): i.e.,
If c
a
( t = ∞ ) = 1 and c
b
( t = ∞ ) = 0 , and U
aa
= U
bb
= 0, and U
ab
= U
ba
c
a
( t ) and c
b
( t ) and check that |c
a
( t ) |
2
+|c
b
( t ) |
2
= 1. What is the net probability P
a b
( t
= ∞ ) that a transition occurs? Hint: try representing the delta-function as a sequence of
rectangles of the form :
7. (20 points) The Effect of an Electric Field on Transmission through a Double Well.
(a) Write a program in MATLAB that uses the
propagation matrix method to find the
transmission resonances of a particle of mass m =
0.07 m 0
(where m 0
is the bare electron mass) in the
one-dimensional well shown on the right. Your
results should include:
i. a MATLAB computer program listing;
ii. a computer generated plot of the particle
transmission as a function of energy from
0 1.25 eV. (Plot the negative of the natural
log of the transmission coefficient versus
energy); and
iii. a list of the energy level values and the resonant line widths.
(b) Repeat the above calculation, but now apply a uniform electric
field that falls across the double barrier and single well structure only,
as shown in the figure below. The right-hand edge of the 3 nm thick
barrier as at a potential of V = – 0.63 eV below the left-hand edge
of the 4 nm thick barrier. Notice the changes in the transmission
8. (2 0 points) Scattering of single electrons. The scattering of a single
electron from a potential V ( x ) can be formulated in terms of the
scattering eigenstates
( t )
S
that satisfy the Schrödinger equation as:
H ( t ) U ( t ).
otherwise
t
0
H V t E t
S S
To prescribe the scattering boundary conditions, which we take to be the outgoing wave
function,
( t )
S
is re-expressed as the solution to the scattering integral equation:
where p represents the incident plane wave. To ensure that t p
S
() represents an
outgoing scattered wave, must be chosen real and positive with
0. The operator
E H i
0
is diagonal in the p - (momentum) representation and its matrix elements form
the outgoing wave Green’s function of the free particle Schrödinger equation:
In the x - representation for the wave function in Equation (2), it was shown in LECTURE 11
that the Fourier transform can be used to find:
which shows the required asymptotic behavior for the scattered wave. Now consider a set of
N electrons, all scattering from the same potential as above. In the absence of interactions,
the Hamiltonian is given by:
The N - electron wave functions for the unperturbed problem ( V ( x ) = 0) can be written as a
Slater determinant of plane wave solutions: i.e.,
You will recall from LECTURE 8 that the many-particle states can be expressed in terms of
creation and annihilation operators as follows:
where 0 represents the unoccupied vacuum state. The Hamiltonian represented in Equation
(5) is a sum of one-body operators, which can be written as:
0
t
E H i
t p
S S
0
0
pp p p
p
G E
E E i
p
E H i
p
p
3
|
2
V xd x
m e
x x e
S
i
i
S S
x
x-x|
px-x| /
px/
2
i
i
i
V x
m
p
H
px/
x
i
p
e
L
u
1 2
1 2
r
r p p p
p p p c c c
( 3
3
2 2
,
2
p
p q p
q
p
p
p p
p pp
i
i
i
i
i
i
i p p
pp
i
H E c c V q c c
where V p p V e d x
V x e d x E V p p
m
p
V x p e
m
p
where p
V x p c c
m
p
H p
p-p) x/
px/ px/
x
TEAM A: NetID
Ortiz Ortega, Eibar [email protected]
Chin, Darren [email protected]
Carlin, Riley [email protected]
Sinclair, Will [email protected]
TEAM B: NetID
Burnard, Travis [email protected]
Keppinger, Brandon [email protected]
Niu, Xuezhong [email protected]
