Back to top
SUMMARY OF RESULTS OF THE LEVERHULME TRUST FUNDED RESEARCH F/00144/AS
Vibrational and Thermal Properties of Phononic Crystals
Aims and objectives of the project:
This project aimed to study the vibrational and thermal properties of
phononic crystals at a microscopic level. To accomplish our aims, we
set out the following four objectives:
 accurate theoretical determination of the number and energy
locations of band gaps,
 numerical results for negative group velocity and explanation of
'negative refraction',
 calculations of anharmonic phonon lifetime, and
 calculations of thermal properties, in particular specific heat
and thermal conductivity.
Research activity:
Phononic crystals are the acoustic analogues of photonic crystals and
offer the possibility of novel applications for phonon engineering
including phonon focusing and sound filters. A phononic crystal can be
created by growing periodically two or more different materials which
have contrasting vibrational properties. Most realizable phononic
structures rely on solid/fluid composites on the scale of mmmm, but
recent technological advances have led to the fabrication of
nanophononic solid/solid materials. In particular, semiconducting
superlattices with nanometer scale periodicity have recently been
fabricated, and there are possibilities of twodimensional as well as
threedimensional array systems with phononic properties. The creation
of 'new periodicity' in a phononic crystal leads to band gaps in its
vibrational spectrum. Creation of phononic band gaps leads to the
concept of 'negative refraction' of sound waves, relating directly to
'negative group velocity' of the crystal vibrational quasiparticles
called phonons. Nanophononic semiconductor superlattices, embedded
wires or embedded dots are potential candidates for application as
highefficiency thermoelectric materials over a large temperature
range. For possible phonon engineering applications of such materials
it is vitally important to gain insight into (i) lattice dynamics
(phonon dispersion relations), (ii) phonon interaction mechanisms, and
(iii) magnitude and temperature variation of phonon conductivity. This
project has made some progress in this direction. We summarise the
theoretical methods used (including new developments) and the results
obtained.
 Theoretical developments:
 Lattice dynamics: A simple springandball model was developed and
used to obtain trends in phonon dispersion relations. An Enhanced
Bond Charge Model was developed (as an extension of the original
model developed in 1977 by Weber) and used for accurate
determination of phonon dispersion relations in nanophononic
semiconductors in the form of superlattices, embedded nanowires,
and embedded nanodots.
 Phonon scattering mechanisms: One of the most important features
of phononic crystals is very low lattice thermal conductivity
compared to either of the bulk constituents used for fabricating
these. In order to explain this behaviour considerations of new
phonon scattering mechanisms which arise due to the formation of a
phononic composite, and not present in individual bulk
constituent, are required. Three most important considerations
are: scattering by mass mixing across interfaces, scattering by
straindriven interface dislocations, and modification of
anharmonic interactions due to the presence of more than one type
of material and the generation of 'miniUmklapp' processes
corresponding to the longer periodicity introduced by composite
crystal formation. We developed atomicscale theories (the first
effort to the best of our knowledge) for phonon scattering rates
arising from these three types of scattering in superlattice
phononic structures. The expression for the strength of the
anharmonic crystal potential is derived from the application of an
isotropic anharmonic continuum theory, developed by the PI in the
1970s and detailed in his book The Physics of Phonons. The
presence of two material regions gives rise to a 'Dual Mass' term
in a superlattice structure. This term has been expressed by
applying the theory of the diatomic linear chain. The relaxation
rate expressions have been numerically evaluated by employing the
concept of 'special qpoints' for Brillouin zone integration
(often used in modern abinito electronic band structure
calculations) and by using the phonon dispersion relations
obtained from our lattice dynamical calculations.
 Phonon conductivity: An expression for phonon conductivity
tensor was derived within the single mode relaxation time
approximation (detailed description in the book The Physics
of Phonons authored by the PI). Numerical evaluation of the
conductivity tensor for Si/Ge and GaAs/AlAs superlattices was
performed by employing the 'special qpoints' scheme for
Brillouin zone integration, and using the phonon dispersion
relations and phonon group velocity components obtained from our
lattice dynamical calculations.
Main results and predictions:

Phononic behaviour and gaps:
From our theoretical modelling we have concluded that phononic band
gaps, and their sizes, in a phononic composite A/B may be controlled
by a combination of five factors: (i) periodicity (i.e. size of unit
cell, L), (ii) length fraction Lf (i.e. size of one material in
relation to the other), (iii) mass ratio between materials A and B,
(iv) force constant differences between materials A and B, and (v)
dimensionality of system (i.e. 1D, 2D, or 3D). In particular, the
following conclusions were reached:

Provided that a phononic (polarization, or true) band gap exists,
both its location as well as its width decrease as 1/L.

For a given A/B composite material, there is an optimum length
fraction Lf for a true phononic gap to appear. Within the range 0.9 >
Lf > 0.1, a true phononic gap will appear for 2D systems with mass
ratio larger than 10, and for 3D systems with mass ratio larger than
15. For the tetrahedrally bonded Si/Ge and Si/Sn composite systems,
the optimum length fraction appears to be close to 1/3.

Our investigations confirm the presence of the lowest three
longitudinal acoustic gaps in the GHz range for the
Si(4nm)/Si0.4Ge0.6(8nm)[100] superlattice fabricated by Ezzahri et
al. In addition, we predict several transverse acoustic gaps. More
importantly, we predict that this system is characterized by a true
phononic gap between the hypersonic range of 507 GHz and 515 GHz. We
have also concluded that the opening of the gaps at the zone centre
and at the zone edge can lead to negative phonon group velocities of
the order of 4.0 km/s to 5.9 km/s.

The effect of interface mixing rarely leads to the closure of any
existing phononic gaps.

We have developed a methodology for predicting phononic gaps in
thicker superlattices once these are found in thin structures.
 Phonon relaxation times:

The strengths of phonon scattering rates due to interface
massmixing and interfacedislocations are greatest for thin
superlattice phononic structures and decrease as the structures become
thicker.

The anharmonic phonon lifetime in a superlattice phononic
structure is shorter than the average of results for constituent bulk
materials. This is due to additional decay routes (including
'miniUmklapp' processes) and the 'Dual Mass factor', both of which
become available upon superlattice structure formation.

Thermal conductivity:

The phonon conductivity of phononic superlattice structures is
usually two orders of magnitude lower than that its constituent bulk
materials. The conductivity also shows a strong anisotropic behaviour,
with the growthdirection result being up to three times lower than
the inplane value. (b) The measured thermal conductivity results for
Si(19)/Ge(5)[001] and Si(72)/Ge(30)[001] superlattices in the low
temperature range 80150 K can be successfully explained by
considering phonon scattering from sample size, isotopic defects,
interface massmixing and interface dislocations, with practically no
contribution from anharmonic scatterings. The conductivity is more
heavily controlled by interface massmixing for thinner periods and by
interface dislocations for thicker periods. The interplay between the
interface massmixing and dislocation scatterings in controlling
phonon lifetime explains the apparent dip in the thermal conductivity
measured by Lee et al in 1997 as a function of superlattice period.

It is important to consider the role of phonon anharmonic
interactions in order to explain the thermal conductivity results,
measured by Capinski et al (1999), for ultrathin GaAs(n)/AlAs(n)[001]
superlattices in the intermediate temperature range 100400 K. In such
structures there is no evidence of interface dislocations, but
interface massmixing produces a significant reduction in the
conductivity values. Our work predicts accurate results both at low
and high temperatures. An experimental group in France has been
contacted for making further measurements of conductivity to validate
our predictions.
Conclusions and achievements:
To the best of our knowledge, this project has produced the first
accurate and detailed theoretical and computational research effort at
atomic level to study vibrational and thermal properties of
nanophononic crystals. Our work presents useful phononic gap results
for thin semiconducting composites, and provides guidelines for
predicting phononic gaps for thick superlattices. We have developed
theories of phonon scattering by interface massmixing, interface
dislocations, and anharmonic inrecations in solid/solid phononic
composites. We have identified the relative importance of these three
types of phonon interactions in determining the thermal conductivity
of Si/Ge and GaAs/AlAs superlattices. Our work also identifies the key
parameters for controlling the magnitude of the lattice thermal
conductivity of nanostructured semiconductors. We hope that this
research will form a basis for further theoretical and computational
works, and in collaboration with experimental groups will make useful
contribution in developing nanophononic materials for useful
vibrational and thermal applications of technological importance.
Publications and dissemination:
Publications:

Hypersonic Modes in Nanophononic Semiconductors, S. P. Hepplestone
and G. P. Srivastava, Phys. Rev. Lett. 101, 105502 (2008)
[Editors' suggestion for wider reading]

Atomic Theory of Phononic Gaps in Nanopatterned Semiconductors,
S. P. Hepplestone and G. P. Srivastava, Transport and Optical
Props. of Nanomaterials (AIP Conf. Proc. 1147, 135 (2009).

Anharmonic Lifetime of Phonons in Nanophononic Semiconductors,
S. P. Hepplestone and G. P. Srivastava,
Mater. Res. Soc. Symp. Proc. 1172, 1172T0309 (2009).

Theory of thermal conductivity of micro and nanostructured
materials, G. P. Srivastava, Mater. Res. Soc. Symp. Proc. 1172,
1172T0807 (2009).

Phononic Gaps in Thin Semiconductor Superlattices,
S. P. Hepplestone and G. P. Srivastava, J. Appl. Phys. 107, 043504
(2010).

Theory of Interface Scattering of Phonons in Superlattices,
S. P. Hepplestone and G. P. Srivastava, Phys. Rev. B 82, 144303
(2010).
Seminars:

PDRA: 2008: School of Physics, University of Exeter.

PI: 2008: Physics Department, Banaras Hindu University, Varanasi,
India.

PI: 2009: School of Physics, University of Exeter.
Conferences:

Phonons2007, Paris, July 2007 (attended by PI and PDRA).

MRS Fall2007, Boston, USA (Symposium EE 'Phonon Engineering:
Theory and Applications', coorganised by PI).

International Conference on Transport and Optical Properties of
Nanomaterials, Jan 2009, Allahabad, India (invited talk by PI entitled
'Atomic theory of phononic gaps in nanopatterned semiconductors').

MRS Spring2009, SanFrancisco, USA (oral talk by PDRA, entitled
'Anharmonic lifetime of phonons in nanophononic semiconductors').

MRS Spring2009, SanFrancisco, USA (invited talk by PI, entitled
'Theory of thermal conductivity of micro and nanostructured
materials').

MRS Fall2010, Boston, USA (oral talk by PDRA, entitled 'Defect and
interface scattering in nanophononic semiconductors').

MRS Fall2010, Boston, USA (Symposium CC 'Phonon Engineering for
Enhanced materials Solutions: Theory and Applications',
coorganised by PI).

ICREA workshop phonon engineering, Girona, Spain, May 2010
(invited talk by PI, entitled 'Phonon transport in nanophononic
semiconductors').
