Solid State Theory
Exercise 1
WS 2015/16
Prof. M. Eckstein
1) Kronig Penney model
A simplified model for a periodic crystal was studied by Kronig and Penney in 1931 [R. de Kronig and W. G. Penney, Proc. R. Soc. London A 130, 499 (1931)]. The model describes a one-dimensional chain of “atoms”, whose
atomic potentials are approximated by rectangular quantum wells. In this exercise, we discuss a simplified version
of this model, in which the rectangular wells are replaced by delta functions, such that the potential energy of the
electrons is given by (see Fig. 1)
∞
X
V (x) = V0
δ(x − an).
(1)
n=−∞
V(x)
...
(n-1)a
na
...
(n+1)a
x
Figure 1: Schematic picture of the Kronig Penney model.
2
2
~
a) Determine the energy bands of the model, i.e., the spectrum of − 2m
dx2 Ψ(x)+V (x)Ψ(x) = EΨ(x). In general,
the equation can only be solved numerically, but the structure of the solution can be understood by showing that
the energy E for a given quasi-momentum k satisfies the equation
cos(ka) =
p
v sin(βa)
+ cos(βa), β = 2mE/~2 , v = 2mV0 a/~2 .
2 βa
(2)
To proof this equation, determine the solution of the free Schrödinger equation in each of the intervals [na, (n +
1)a], and connect the solutions in neighboring intervals by matching their boundary values and derivatives, taking
into account the δ-function potential. Show that there are energy gaps. Discuss the opening of the gap for small V0
(what is the relevant scale to compare to?). What determines the gap for large V0 (v → ∞)?
b) A crystal with an open surface can be modeled with the potential
U0 P
x<0
V (x) =
.
∞
V0 n=1 δ(x − an) x > 0
(3)
Following Kronig and Penney, the model can be used to study the reflection of an incoming electron wave, i.e., for
x < 0 the wave function of incoming and reflected electron with energy Ei = p2 /2m + U0 is given by
ψ(x) = e
Discuss the behavior of the reflectivity R = |A|2 .
ipx
~
+ Ae
−ipx
~
.
(4)