Ph.D. thesis (pdf) - dirac

Chapter 8
Boson Peak
The low energy vibrations in crystalline solids are often quite well accounted for
by the Debye model. In the Debye model the modes are assumed to be plane
waves and the number of possible modes are counted in the Q-space. This leads
to a density of vibrational states which is proportional to the square of the mode
frequency g D (ω) ∝ ω 2 . It is this dependence of the vibrational density of states
which leads to the well known T 3 dependence of the heat capacity in solids at low
temperatures. The Debye model is described in textbooks on solid states physics
and thermodynamics e.g. [Kittel, 1996; Bairlein, 1999].
Disordered solids, on the other hand, have an excess in the vibrational density of
states as compared to the ω 2 Debye behavior followed by crystals in the corresponding
frequency range (approx 2 meV - 10 meV depending on system). The excess
gives rise to a peak in the reduced density of vibrational states g(ω)/ω 2 (rDOS) and
the peak is seen directly in incoherent inelastic neutron spectra, which essentially
probes g(ω)/ω 2 as well as in Raman scattering spectra. The excess also gives rise
to a peak in the reduced low temperature heat capacity c P (T)/T 3 .
The boson peak has been studied intensively experimentally as well as theoretically
over the last decade, but its origin is still controversial. The proposed explanations
for the boson peak fall in three categories. (i) Localized (harmonic) modes in soft
potentials (SPM), the concept of soft localized soft modes being expressed differently
in different models [Gurevich et al., 2003, 2005; Klinger, 1999, 2001]. (ii) Quasilocalized
sound like modes in structural regions of the nanometer size in the glass
[Duval et al., 1990; Schroeder et al., 2004; Quitmann and Soltwisch, 1998], and (iii)
Fluctuating elastic constants (FEC) giving rise to a peak in g(ω)/ω 2 [Maurer and
Schirmacher, 2004]. The total density of vibrational states is in the two first cases
described by Debye modes plus extra modes, which are ascribed to the disorder.
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