1. No category

FYS-1350 Nanofysiikka
TTY / Syksy 2014
Laskuharjoitus 6
Tehtävä 1 (Exercise 6.14).
We can look at the band structure of an element to get an idea of how many electrons
per atom participate in conduction. We can also determine this number based on the
free electron density, the atomic mass, and the mass density of the material.
a) Calculate the free electron density of lithium. The Fermi energy is 4.72 eV.
226 ◾ Nanotechnology
b) Derive a formula for the number of atoms per unit volume.
c) Calculate the number of atoms per unit volume of lithium (535 kg/m3, 6.9
6.22 Determine
the temperature
at which
a cube
of gold
3 nm
g/mol)
using the formula
that you
derived
in part
(b). on a side becomes
d) Determine
of free electrons per
lithium atom.
quantum
confined the
(vianumber
the metal-to-insulator
transition).
Gold has a density
e)
What
is
the
electron
configuration
for
Li?
Explain
whether
your answer to (d)
3
of 19,300 kg/m and its atomic mass is 197 g/mol.
sense according to this configuration. The free electron(s) of Li is (are) in
6.23 Why is a makes
physical
model different from the one used for conductors needed to
which sublevel?
6.24
6.25
6.26
6.27
determine the conditions of quantum confinement for semiconductors?
What name is given to the distance separating an electron–hole pair in a
Tehtävä 2 (Exercise 6.22).
semiconductor?
Determine
temperature
a cubedot
of gold
3 nm on
a side becomes
True
or false?the
The
band gap atofwhich
a quantum
is directly
proportional
to itsquantum
confined (via the metal-to-insulator transition). Gold has a density of 19,300 kg/m3 and
size.
its atomic mass is 197 g/mol.
How must the boundary conditions of a potential energy well change to allow
forTehtävä
tunneling?
3 (Exercise 6.27).
wave
function,ψ(x),
ψ(x),ofofananelectron
electronin
in aa potential
potential well
width, L,
TheThe
wave
function,
well of
of finite
finitedepth
depthand
and
is
given
width, L, is given by
ψ(x) = AeCx
for x ≤ 0
ψ(x) = F sin(kx) + G cos(kx)
for 0 ≤ x ≤ L
ψ(x) = Be−Cx
for x ≥ L
a. Given
F = G F= =5,Gk == 5,
1, kand
= 2,Cdetermine
A. A.
a) Given
= 1,Cand
= 2, determine
b) What
is ψ (0)?
b. What
is ψ (0)?
c) Given
= ψdetermine
(0), determine
values
andBBwhether
whether this
c. Given
ψ(L) =ψ(L)
ψ (0),
the the
values
of of
L Land
thiswave
wavefunction
corresponds to the ground energy state. See the attached Figure (below).
function corresponds to the ground energy state. See Figure 6.22.
d) Given ψ(L) = ψ (0), determine the values of L and B if this wave function corresponds
d. Given toψ(L)
= ψ lowest
(0), determine
the values of L and B if this wave function
the third
energy state.
corresponds
to the third
lowest
state.
e) Use a spreadsheet
program
to energy
graph the
wave function from part (d) over the range −2
≤
x
≤
(L
+
2).
Indicate
the
location
and
value
L along the
x-axis.
e. Use a spreadsheet program to graph the waveoffunction
from
part (d) over
the range −2 ≤ x ≤ (L + 2). Indicate the location and value of L along the
x-axis.
6.28 About what size are the smallest features we can see with a light microscope?
6.29 What variable does the STM attempt to keep constant as it scans?
ψ(x) 8
7
the range −2 ≤ x ≤ (L + 2). Indicate the location and value of L along the
x-axis.
6.28 About what size are the smallest features we can see with a light microscope?
6.29 What variable does the STM attempt to keep constant as it scans?
ψ(x) 8
7
6
5
4
3
2
1
0
–1
FIGURE 6.22
L
x
Homework
6.27c.
Tehtävä Exercise
4.
In the book and in the lecture notes, the density of states was given for a 3D system.
Reconsider the case but now for a 2D system (quantum well): what is the density of states for
a 2D system of this kind?
This exercise goes slightly beyond the scope of the material discussed in the book, but
please encourage yourself to consider it anyway. It is called research, which often means that
one should step out from the comfort zone.