Hi, everyone!
When the potential is constant, no force is
applied to an electron. In this case, as discussed
in Part 4, the electron is called “free.”
In the crystal structure, instead, we have
position-dependent potential.
That introduces discontinuity in the E-k relation.
In the left figure, we can see the crystal
structure of silicon.
Since atomic cores are positively charged,
electrons are attracted by them.
Therefore, the potential energy is periodically
modulated.
Along a certain direction, it can be schematically
drawn like the right figure.
Now we understand that, due to the periodic
atomic arrangement, we have the periodic potential.
Once the potential is periodic, it can be
expanded with periodic functions.
With a period of a, it can be expressed as
this form, the Fourier expansion.
Maybe the simplest example would be just a
cosine function.
We know that the cosine function, cos (arg),
can be decomposed into exp( +i arg ) and exp(
-i arg ).
Therefore, the potential multiplied by the
wavefunction yields exp ( i(k + 2 pi/a)x)
and exp ( i(k - 2 pi/a)x).
It
means that the free electron of exp (ikx)
is now coupled with other free electrons with
different wavenumbers.
Especially, when k = pi/a, we have the coupling
between exp(i pi/a x) and exp(-i pi/a x).
And they have the same energy!
They have the same energy. And they are coupled.
What happens?
In this case, the problem can be reduced to
solve a simple eigenvalue problem, shown here.
The diagonal component, 1, represents the
free electron, while the off diagonal component,
V, provides the coupling between two modes.
Of course, when V = 0, two eigenvalues are
1. When V is not zero, two eigenvalues are
1 + V and 1 – V.
As shown in the figure, when we have larger
coupling energy, V, the energy difference
becomes larger.
The
Kronig-Penny model is a model with a square
potential, as shown in the right upper side.
In the left figure, the E-k relation for V
= 0 is shown. It is just a free electron.
Nothing special.
However, when we introduce a non-zero V, the
energy discontinuity in the E-k relation is
clearly shown in the right figure.
The discontinuity occurs at points, which
are integer multiples of pi/a.
In the next video, the band structure of silicon
will be introduced. Thank you!
