10.2. ZENER-BLOCH OSCILLATIONS 491
10.2 ZENER-BLOCHOSCILLATIONS ........................
In a perfect crystal electrons see a periodic potential and according to Bloch theorem an elec-
tron wavefunction is described by a plane wave with a central cell periodic part. Of course the
crystal has to be rigid since lattice vibrations even in a defect-free structure will cause scattering.
There are many interesting effects that occur when electrons move without scattering in crystals.
One such effect is Zener-Bloch oscillations. The equation of motion of electrons in an electric
field is simply
dk
dt=eE (10.2.1)In the absence of any collisions the electron will simply start from the bottom of the band (fig-
ure 10.2) and go along theEvskcurve until it reaches the Brillouin zone edge.
It must be noted that just as the electron sees a periodic potential in real space in a crystal the
bandstructure E vskis also periodic ink-space. The electron at the zone edge is thus “reflected”
as shown in figure 10.2 and now starts to lose energy in its motion in the field. Thek-direction of
the electron changes sign as the electron passes through the zone edge representing oscillations
ink-space and consequently in the real space. These oscillations are called the Zener-Bloch
oscillations.
If we have a spatial periodicity defined by distanceathe bandstructure is periodic in the
reciprocal vectorΓ=2π/a. As a result the frequency of Bloch-Zener oscillation is
ωb=eEa
(10.2.2)
The oscillation frequency is quite high and can easily be in the several terrahertz regime. Note
that the oscillations depend upon field direction since the edges of the Brillouin zone (see chap-
ter 3) are at different points along different directions. From a practical device point of view it
has not been possible to exploit Bloch oscillations since the scattering mechanisms are usually
strong enough to cause a electron to scatter before it can go through a complete oscillation, it
has not been possible to observe these oscillations.
IfτSCis the scattering time oscillations can occur if we have the condition
ωbτSC≥ 1 (10.2.3)From the oscillation condition given above we see that if the periodic distance in real space
is increased, it will take less time to reach the zone edge and one can expect Bloch oscillations
to survive. The periodicity can be increased by using superlattices. In figure 10.3 we show a
schematic of the effect of enlarging the periodic distance (by making superlattices) on an energy
band. On the top we show the energy band schematic of a crystal with a unit cell periodicity
represented by the distancea. The zone edge ink-space is at 2 π/a. Now if a superlattice with
aperiodnais made as shown in the lower panel the zone edge occurs at 2 π/na. Assuming that
the scattering time is not changed much due to superlattice formation, it can be expected that
an electron will be able to reach the superlattice zone edge without scattering, thus Bloch-Zener
oscillations could occur. Although these considerations seem promising, real devices have not
been created.