Transcribed Text
The KronigPenney model is a 1D squarewell approximation for a periodic array of
charged
nuclei, as shown below.
d(x)
d,

E
X
 C
+b +a
Period =a
In terms of the parameters a2 = 2mE/ h2 and B2 =  the Schrödinger
equation can be written as
dty
+ d2 = 0
05x5b
dx²
dty B2=0 =
Cdx2
where E is the electron wavefunction energy and we assume E<©0. From Bloch's theorem,
4(x) = u(x)etter where u(x) is a periodic function with period a: u(+++ a) = u(x) .
In terms of the parameters a2 = 2mE/n² and
B2 =  the Schrödinger
equation can be written as
dty
+ ² 4 = 0
05x5b
dx²
d
4 B24 = 0
Cdx2
where
E
is
the electron wavefunction energy and we assume E<©0. From Bloch's theorem,
4 (x) = u(x)eth where u(x) is a periodic function with period a: u(x+a)=1(x). =
a) Both 4 and dy dx must be continuous in order for the Schrödinger equation to have finite
energies.
Find general forms for 4 in the barrier and the well and also find four equations
from boundary conditions that solve for the undetermined coefficients.
b)
Solve
this
system of four equations (possibly by setting a determinant to zero) to find an
equation of the form f(b,c,a,B) = cos(ak).
c)
Plot
f(E) versus E for representative values a = 1nm, b = 0.9nm, and Do =0.5, 5, 15, and
50eV (1 eV = 1.602x1019 J). Label the allowed ranges of energy and the forbidden ranges
(gaps).
d)
Plot
the dispersion relation, i.e. the electron energy values as a function of k, for the
representative values of part (c). On each of these plots, also draw the freeelectron E VS. k.
e)
Redo
part
(b)
if
Do
is
very
small.
This
is
the
case
for
most
metals,
such
as
alkali
metals
where
electrons are very weakly bound. What are the allowed energies for a given k? What
does
this mean physically for the electron motion? Does this help explain the plot in (d) for
small DO?
f)
Redo part (b) if Do is very large. This is the case for solids in which electrons are tightly
bound
to
ion cores. What are the allowed energies for a given k? What does this mean
physically for the electron motion? Does this help explain the plot in (d) for large Do?
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