PHYS 7440
Assignment 8
Due Fri. March 21, 2003
Reading:
1. Sections on semiclassical electron dynamics, Marder Chapter 16 and on phonons,
Marder Chapter 13.
Problems:
In this problem set, we consider semiclassical electron dynamics in somewhat more
detail. Let's consider the case of Bloch oscillations for electrons in a dc electric field:
1. Marder 16.1. Skip Marder's part (e) and instead, consider the following:
e) In Marder's (d), you consider how to use a semiconductor superlattice (see
attached notes) to make an unusually large crystalline unit cell, so that Bloch
oscillations are easier to achieve. This proposal has been considered as a possible
practical way to cause electronic oscillations in semiconductors for making
microwave sources and lasers that operate in the far infrared. However, there is a
problem: Think about the high field restriction Marder equation 16.86. Use this
restriction along with equation 16.13 as rough guess for the band structure in the
long-a direction (based for example on the NFE or tight binding picture) to
explain why this idea might not work in practice. Use your restriction on the
magnitude of E to produce a new requirement on the length of a for observing
Bloch oscillations.
2. Marder 13.1.
Physics 7440
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Spring 2003
Notes on Semiconductor Superlattices
A semiconductor superlattice is created by alternately growing layers of two different
semiconductors. The most commonly studied materials are the GaAs - AlxGa1-xAs
system. Here, the replacement of Ga with Al results in a material of higher bandgap
than GaAs while maintaining a nearly unchanged set of lattice constants. Thus, it is
possible to grow alternating layers of GaAs and AlGaAs and achieve nearly perfectly
ordered crystals. However, because of the new periodicity introduced by the GaAs AlGaAs layer-repeat period (which is controlled by the crystal grower) the GaAs band
structure in the growth direction is folded back into a much smaller Brillouin zone
than would be the case in bulk GaAs. By controlling the layer thicknesses, rather
interesting transport properties can be achieved.
Because the AlGaAs conduction band is higher than in GaAs, it is typical to think of
the GaAs layers as potential wells separated by AlGaAs barriers. Within these wells,
the band structure is well represented by a single parabolic band with an effective
mass, m*, of roughly 0.07me. The superlattice periodicity causes this band structure to
be folded back into a reduced zone just as we saw for the NFE picture. Mixing of
degenerate states at the zone boundary also opens small gaps (as for the NFE case) so
that in the direction of the superlattice periodicity, the original band is broken up into
many small bands, generally referred to as mini-bands. Structures of this type are the
first real success stories in band structure engineering.
Physics 7440
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Spring 2003
2. Let's continue with our discussion of semiconductor superlattices: Suppose that we
have a GaAs-AlGaAs superlattice composed of repeating layers of GaAs and
AlGaAs. In the growth direction, the potential is very similar to the simple KronigPenney model (See the discussion in Marder Chapter 7, especially on page 182. Also,
the AlGaAs layers present regularly spaced square barriers to transport from the GaAs
conduction band). In this problem, we consider how this additional periodicity affects
the cyclotron resonance effective mass for magnetic field aimed in different
directions. For actual data on this problem and further discussion, see the attached
paper (Electron mass tunneling along the growth direction of (Al,Ga)As/GaAs
semiconductor superlattices, T. Duffield et al. Phys. Rev. Lett. 56, 2724 (1986),
hereafter referred to as Duffield.).
Consider a superlattice like that discussed by Duffield composed of 8nm GaAs
followed by 2nm of AlGaAs and then repeated a huge number of times. We will
approximate this structure with a Kronig-Penney model of square barriers of height
Vx (which varies with Al concentration), thickness, d=2nm, and (center-to-center)
spaced a=10nm, in the direction of growth (in this case, the z-direction). Since we are
dealing with a semiconductor, we have the flexibility of doping it to control number
of carriers. For low densities of electrons, the energy bands of interest can be
represented by a single parabolic band centered on the middle of the first Brillouin
zone. Thus,
2 2
k
EGaAs k  E0 
2m0
 
where
m0  0.07me
From this very small effective mass, it should be clear that we are dealing with a
rather high band number (think about why). From Marder Prob. 7.5, we know that the
Kronig-Penney energy bands, Ekp, in the growth direction can be calculated from
Marder equation 7.113 for any assumed barrier shape. What we consider here is how
cyclotron resonance data can be used to test different models for real structures.
a) From the basic equation for the cyclotron effective mass,
mc* 
 A  kH , Ei 
2
E
2
(where kH is the component of k in the field direction and Ei is the initial energy of
the electron of interest) and the assumption:
 
Esl k  E0 
Physics 7440
2
2m0
k
2
x
 k y2   Ekp  k z 
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Spring 2003
Here, Esl is the band structure of the superlattice and Ekp is the (not yet known, but
presumably calculable) shape of the Kronig-Penney band due to the period GaAsAlGaAs superlattice. Show for H applied in the z-direction that mc*  m0
b) Similarly, for H in the x-y plane, show that the cyclotron effective mass is related to
Ekp by:
kmax
dk z
1/ 2
*
mc   2m0 
1/ 2


0  Ei  Ekp  k z 
where Ekp  kmax   Ei .
c) Assume that the barriers are rather strong so that we may calculate Ekp in the tight
binding nearest neighbor limit. Using results from class, we would find a KronigPenney cosine band:
Ekp  kZ   2 cos  2kZ a 
Then, show that for small Ei,
mc*   m0 mtb 
1/ 2
where
mtb 
2
2 a 2
and  is an appropriately defined overlap integral.
Thus, the cyclotron effective mass is just the geometric mean between the GaAs band
mass and the tight binding effective mass.
d) Argue on the basis of A&M 8.79 and a simple WKB estimate for the barrier
transmission that in the limit of high aluminum concentration barriers (i.e., low
transmission) that the cyclotron effective mass gives a measure of the single barrier
tunneling probability.
e) In this limit of large mtb, use the semiclassical equations of motion to sketch a picture
of the k-space orbit for an electron. Also sketch the real space orbit paying particular
attention to the orientation of the orbit to the direction of the superlattice. Give a
simple physical interpretation of the shape of this orbit in terms of band velocities and
tunneling rates through the AlGaAs barriers (Note: Do not believe Duffield, FIG. 1
for the orbit shape).
f) Now relax the requirement of large mtb. Use A&M 8.76, the well-known result for
the square barrier that:
t  eikd
2k
2k cosh  d  i  k 2   2  sinh  d
Physics 7440
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Spring 2003
where    2m0 Vx  Ei  2  and the empirical relation that Vx  x  0.5eV (with
x the atomic percent of Al vs. Ga and 0  x  1) to calculate the value of mtb for a
few Al concentrations. Compare your results to the data quoted in Duffield FIG 3.
1/ 2
Physics 7440
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Spring 2003