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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:
  1. accurate theoretical determination of the number and energy locations of band gaps,
  2. numerical results for negative group velocity and explanation of 'negative refraction',
  3. calculations of anharmonic phonon lifetime, and
  4. 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 mm-mm, 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 two-dimensional as well as three-dimensional 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 quasi-particles called phonons. Nanophononic semiconductor superlattices, embedded wires or embedded dots are potential candidates for application as high-efficiency 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.

  1. Theoretical developments:
    1. Lattice dynamics: A simple spring-and-ball 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.

    2. 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 strain-driven interface dislocations, and modification of anharmonic interactions due to the presence of more than one type of material and the generation of 'mini-Umklapp' processes corresponding to the longer periodicity introduced by composite crystal formation. We developed atomic-scale 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 q-points' for Brillouin zone integration (often used in modern ab-inito electronic band structure calculations) and by using the phonon dispersion relations obtained from our lattice dynamical calculations.

    3. 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 q-points' 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:

  1. 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:

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

    2. 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.

    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.

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

    5. We have developed a methodology for predicting phononic gaps in thicker superlattices once these are found in thin structures.

  2. Phonon relaxation times:
    1. The strengths of phonon scattering rates due to interface mass-mixing and interface-dislocations are greatest for thin superlattice phononic structures and decrease as the structures become thicker.

    2. 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 'mini-Umklapp' processes) and the 'Dual Mass factor', both of which become available upon superlattice structure formation.

  3. Thermal conductivity:
    1. 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 growth-direction result being up to three times lower than the in-plane 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 80-150 K can be successfully explained by considering phonon scattering from sample size, isotopic defects, interface mass-mixing and interface dislocations, with practically no contribution from anharmonic scatterings. The conductivity is more heavily controlled by interface mass-mixing for thinner periods and by interface dislocations for thicker periods. The interplay between the interface mass-mixing 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.

    2. 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 100-400 K. In such structures there is no evidence of interface dislocations, but interface mass-mixing 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 mass-mixing, 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:


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

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

  3. Anharmonic Lifetime of Phonons in Nanophononic Semiconductors, S. P. Hepplestone and G. P. Srivastava, Mater. Res. Soc. Symp. Proc. 1172, 1172-T03-09 (2009).

  4. Theory of thermal conductivity of micro- and nano-structured materials, G. P. Srivastava, Mater. Res. Soc. Symp. Proc. 1172, 1172-T08-07 (2009).

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

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


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

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

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


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

  2. MRS Fall2007, Boston, USA (Symposium EE 'Phonon Engineering: Theory and Applications', co-organised by PI).

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

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

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

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

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

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

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