Physics and Astronomy

## Quantum Systems and Nanomaterials Group

Home Research Quantum Systems and Nanomaterials Electrons & phonons EPSRC-EP-H046690-1.html

# SUMMARY OF RESULTS OF THE EPSRC FUNDED RESEARCH EP-H046690-1

Phonon Engineering of Nanocomposite Thermoelectric Materials

## Aims and objectives of the project:

The overall aim of this project is to identify the key parameters for developing the phonon engineering of Si-based nanocomposite thermoelectric materials by undertaking a systematic state-of-the-art theoretical study of their thermoelectric figure of merit. The systems of study will be: (i) thin Si/Ge and Si/SiGe superlattices, (ii) Si nanowires embedded in Ge and SiGe matrices, and (iii) Si nanodots embedded in a Ge or SiGe matrix. The specific objectives are: (1) Calculation of electrical conductivity and thermoelectric power, (2) Calculation of equilibrium atomic geometry at interfaces, (3) Calculations of phonon dispersion relations, acoustic phonon velocities, and phonon density of states, (4) Theoretical development of phonon interactions in nanocomposite thermoelectric systems, (5) Phonon conductivity tensor using a more complete theoretical model than is currently available, (6) Identification of key paramters for improved thermoelectric efficiency parameter ZT for a system with a given dimensionality, (7) Establishing the role of dimensionality (2D vs 1D vs 0D) in the enhancement of thermoelectric efficiency parameter ZT.

## Research activity:

Under construction xxxx

1. Theoretical developments:
1. Lattice dynamics: The lattice dynamics (phonon dispersion relations and phonon eigenvectors) of bulk Si, bulk Ge, ultra-thin Si/Ge[001] superlattices, ultra-thin Si nanowires embedded in Ge matrix, and ultra-thin Si nanodots embedded in Ge matrix was studied. Two different theoretical methods were employed: adiabatic bond charge model and home-built code; density-functional perturbation theory using the Quantum Espresso package. These calculations were made with fully relaxaed atomic cordinates, determined from the plane wave pseudopotential method within the local density functional approximation.

2. Phonon scattering mechanisms: For a realistic and full-scale investigation of phonon transport in nanocomposite thermoelectric materials it is important to formulate two specific phonon scattering mechanisms that do not occur in bulk materials: interface scattering and composition-dependent anharmonic scattering. These have not been adequately discussed by any other research group. In this project we developed [1-5] a revised version of Hamiltonian for interface mass smudging and anharmonicity for acoustic as well as optical modes, and derived the phonon scattering rates from interface mass smudging and anharmonicity in ultra-thin Si/Ge[001] superlattices.

3. Thermoelectric transport coefficients: The figure of merit $ZT$ of a TE material is expressed as % \be ZT=\frac{\sigma S^2T}{\kappa_{\rm tot}}, \ee % \noindent where $\sigma$ is the electrical conductivity, $S$ the Seebeck coefficient (or thermopower) and $\kappa_{\rm tot}$ the total thermal conductivity. The latter can be expressed as $\kappa_{\rm tot}=\kappa_{\rm el}+\kappa_{\rm ph}$, where the electronic part $\kappa_{\rm el}$ is the sum of the charge carrier (electron or hole) and bipolar (electron and hole) contributions, and $\kappa_{\rm ph}$ is the lattice (phonon) thermal conductivity. We made use of the schema of Lin-Chung and Reineke [6] (based on the more general work of Bergman et al [7]), to describe the in-plane and cross-plane thermoelectric quantities in terms of the values of their bulk or quantum-well constituents in order to obtain our final results.

4. Phonon conductivity: The phonon conductivity tensor was computed for Si$_x$Ge$_{1-x}$ alloys, Si(n)Ge(n)[001] superlattices, Si nanowires embedded in Ge matrix, and Si nanodots embedded in Ge matrix. Some of the results are published in Refs. [3-5] and other results are being analysed and prepared for future publications.

5. Fermi level: The temperature dependent Fermi energy (i.e. Fermi level, or chemical potential) was computed by solving the charge conservation condition within the effective mass scheme. This was done first using an approximate formulation [8], and later numerically using the Van-Wjingaarden-Decker-Brent method [9] at a given temperature $T$.

6. Electronic components of thermoelectric coefficients The nearly-free-electron model D-dimensional systems (D=3 for bulk, D=2 for superlattice structure). Using the computed values of the Fermi level, the Seebeck coefficient $S$ and the electrical conductivity $\sigma$. The scateering of electrons from acoustic phonons was considered for calculating $\sigma$. The electronic component of the thermal conductivity, $\kappa_{\rm el}$, was calculated by employing the Wiedeman-Franz relations, and taking into account the monopolar and bipolar contributions.

## Main results and predictions:

• Thermoelectric figure of merit in an n-doped SiGe alloy We have made use of density functional methods in order to obtain the required phonon eigensolutions, and a detailed calculation of the anharmonic contribution to phonon scattering based on a semi-empirical model for anharmonic crystal potential. The subsequent calculation of $\kappa_{\rm ph}+\kappa_{\rm e}$ shows good agreement with measurements in the entire temperature range $300-1100$ K. From this information we have calculated the dimensionless figure of merit $ZT$ and compared it with the (incomplete) calculation in Ref. \cite{MP}. While in general the qualitative behaviour of the $ZT$ vs. $T$ curve in our work is quite similar to that in Ref. [10] with the more complete theoretical treatment we have predicted values of greater than 0.5 in the temperature range 750--1250 K with a maximum of approximately 0.7 at around 1000 K. Publication: see Ref. [3].
• Effects of atomic relaxation on phonon dispersion relations and thermal properties of ultrathin (Si)$_n$(Ge)$_n$[001] superlattices Using {\it ab initio} density-functional perturbation theory we have examined the effects of atomic relaxation on the phonon dispersion relations and thermal properties of ultrathin (Si)$_n$(Ge)$_n$[001] ($1 \le n \le 5$) superlattices. It is found that atomic relaxation effects governed by the minimum energy requirement lead to significant changes in the location of phonon frequencies above 200 cm$^{-1}$ as well as in the location and width of phononic gaps. These changes result in 8\% decrease in the zone-average phonon relaxation time and up to 5\% decrease in the thermal conductivity tensor components $\kappa_{ZZ}$ and $\kappa_{XX}$ of the (Si)$_1$(Ge)$_1$[001] superlattice. Publication: see Ref. [4].
• On the thinning down of the thermal conductivity in ultra-short period superlattices We have performed a systematic theoretical investigation of the reduction of the lattice thermal conductivity in ultrathin Si($n$)Ge($n$)[001] superlattices, with $1 \le n\le 8$, where $n$ represents the number of atomic bilayers of a species within a repeat period. The calculations are performed using a model anharmonic Hamiltonian describing three-phonon interactions involving acoustic as well as optical phonons in a two-material superlattice structure, and an improved scheme for phonon scattering due to mass smudging at interfaces. The variation of the components with the superlattice period and sample temperature have been examined and trends extracted. The theoretical results for Si($8$)Ge($8$) have been successfully compared with experimental measurements for Si(22 {\AA})/Ge(22 {\AA}) reported in Ref. [11]. We have found that the cross-planar thermal conductivity is roughly 4.1-4.8 times smaller than the in-plane thermal conductivities for the (8,8) case for temperatures between 100 and 300 K. We have examined in some detail the effects of various parameterisations of the interface mass smudging (IMS) scattering as described by the parameter $\mathcal{P}$. Keeping $\mathcal{P}$ constant with $n$ results in a decrease in $\kappa_{zz}$ (between 3-14\% for a sample size of 4.4mm); however, this involves making problematic assumptions regarding interface quality, which is best described by a $\mathcal{P}$ that decreases with $n$. Examining various models of the decay of $\mathcal{P}$ we have found that while in general the behaviour of $\kappa_{zz}$ at high $T$ shows a monotonic decrease with $n$, at $T$ close to 100 K we see that $\kappa_{zz}$ reaches a minimum at $n=4-5$ before beginning to increase. It is likely that this is because IMS scattering outcompetes the effect of phonon velocity that is described in Refs. [12,13] at larger values of $n$, so that the decay in $\mathcal{P}$ causes $\kappa_{zz}$ to increase. Note that it cannot be said that this minimum is a global one; that would depend on the effects of dislocation scattering when the lattice size exceeds $n=10$ and is a matter beyond the scope of this study. We also observe that for the $T=10-100$ K region boundary scattering is the dominant effect, consistent with the findings of [14]. Since the dimensionless thermoelectric figure of merit parameter $ZT$ is inversely proportional to the sum of the electronic and phonon contributions to the thermal conductivity, we can see that the above results have implications with respect to the thermoelectric efficiency of ultra-thin superlattices, in that we should not expect the very thinnest SLs to be as thermoelectrically efficient in the cross-planar direction as slightly thicker ones in the limit where dislocation effects are unimportant; moreover, within that limit an optimal SL length where $\kappa_{zz}$ is at a minimum may exist. However, the precise extent of any enhancement will be dependent upon how the electronic components of $ZT$ change upon superlattice formation. Publication: see Ref. [5].
• Phonon Engineering for Enhanced Thermoelectric Efficiency in Ultra-Thin Superlattices From a systematic evaluation of relevant transport coefficients we show that in the mid-to-high-temperature range (500-1200 K) an enhancement in $ZT$ to values of up to about 6 may be achieved along the growth direction of a ultra-thin Si/Ge[001] superlattice of periodicity 2.2 nm with reasonably dirty interfaces, an improvement of almost an order of magnitude from that for bulk Si, bulk Ge and SiGe alloys and about 2-4 times that for some current state-of-the art thermoelectric materials. Publication: see Ref. [15]; and paper submitted for publication.

## References:

1. S. P. Hepplestone and G. P. Srivastava, Phys. Rev. B {\bf 82}, 144303 (2010).
2. S. P. Hepplestone and G. P. Srivastava, Phys. Rev. B {\bf 84}, 115326 (2011).
3. I. O. Thomas and G. P. Srivastava, Phys. Rev. B {\bf 86}, 045205 (2012).
4. I. O. Thomas and G. P. Srivastava, Phys. Rev. B {\bf 87}, 085410 (2013).
5. I. O. Thomas and G. P. Srivastava, Phys. Rev. B {\bf 88}, 115207 (2013).
6. P. J. Lin-Chung and T. L. Reinecke, Phys. Rev. B {\bf 51}, 13244 (1995).
7. D. J. Bergman and O. Levy, J. Appl. Phys. {\bf 70}, 6821-6833 (1991).
8. J. P. McKelvey, {\it Solid State and Semiconductor Physics (International Edition)} (Harper \& Row,New York-Evanston-London, and John Weatherhill Inc., Tokyo, 1966).
9. W. H. Press, B. P. Flanney, S. A. Teukolsky, and W. T. Vetterling, {\em Numerical Recipies: The Art of Scientific Computing} (Cambridge University Press, Cambridge, 1986) pp. 352-355.
10. H. R. Meddins and J. E. Parrott, J. Phys. C: Solid State Phys. {\bf 9}, 1263 (1976).
11. T. Borca-Tascuic, W. Liu, T. Zeng, D. W. Song, C. D. Moore, G. Chen, K. L. Wang, M. S. Goorsky, T. Radetic, R. Gronsky, T. Koga, and M. S. Dresselhaus, Superlattices and Microstructures {\bf 28}, 199 (2000).
12. P. Hyldgaard and G. D. Mahan, Phys. Rev. {\bf 56}, 10754 (1997).
13. S. Tamura, Y. Tanaka, and H. J. Maris, Phys. Rev. {\bf 60}, 2627 (1999).
14. M. N. Luckyanova, J. Garg, K. Esfarjani, A. Jandl, M. T> Bulsara, A. J. Schmidt, A. J. Minnich, S. Chen, M. S. Dresselhaus, Z. Ren, E. A. Fitzgerald, G. Chen, Science {\bf 338}, 936 (2012).
15. I. O. Thomas and G. P. Srivastava, AIP Conf Proc {\bf 1590}, 95 (2014).

## Publications and Presentations:

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