In a simple
model of the coupled electron-phonon system the
lattice energy is assumed to be 
, where s is the amplitude of a static lattice
distortion of wavevector q, and 
is an elastic
constant of the crystal. The static distortion of the
lattice acts on the electrons as a perturbation,
which is described by a weak periodic potential,
, where 
and 
is a coupling
constant describing how sensitive the electrons are
to a lattice distortion.
The periodic potential will induce a gap in the electronic
spectrum.
If the wavenumber of the modulation is 
, then the
energy gap will open at 
, and the Fermi energy
will be in the middle of the band gap. Assuming that the perturbation
is much less than the bandwidth, 
,
the density of states around 
can be approximated as
where 
is the density of states for the nonperturbed
system and 
.
(a) Calculate the electronic contribution to the free energy
and for 
, and discuss if the
free energy increases or decreases when the gap opens.
(b) Investigate the stability of the metallic state ( 
)
of the coupled electron-phonon system at low temperatures.
Determine the magnitude of the energy gap.
(c) Calculate the temperature of the Peierls transition, when
a spontaneous lattice distortion develops in the system.
Laszlo Mihaly
Thu Oct 31 13:23:11 EST 1996
Cover Page
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