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SUMMARY OF RESULTS OF THE EPSRC FUNDED RESEARCH EPH0466901
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 Sibased nanocomposite
thermoelectric materials by undertaking a systematic stateoftheart
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
 Theoretical developments:
 Lattice dynamics:
The lattice dynamics (phonon dispersion relations and phonon
eigenvectors) of bulk Si, bulk Ge, ultrathin Si/Ge[001]
superlattices, ultrathin Si nanowires embedded in Ge matrix, and
ultrathin Si nanodots embedded in Ge matrix was studied. Two
different theoretical methods were employed: adiabatic bond charge
model and homebuilt code; densityfunctional 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.
 Phonon scattering mechanisms:
For a realistic and fullscale 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 compositiondependent anharmonic
scattering. These have not been adequately discussed by any other
research group. In this project we developed [15] 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 ultrathin
Si/Ge[001] superlattices.
 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 LinChung and Reineke [6] (based on the
more general work of Bergman et al [7]), to describe the inplane and
crossplane thermoelectric quantities in terms of the values of their
bulk or quantumwell constituents in order to obtain our final
results.
 Phonon conductivity:
The phonon conductivity tensor was computed for Si$_x$Ge$_{1x}$
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. [35] and other results are being analysed and
prepared for future publications.
 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
VanWjingaardenDeckerBrent method [9] at a given temperature $T$.
 Electronic components of thermoelectric coefficients
The nearlyfreeelectron model Ddimensional 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 WiedemanFranz relations, and taking into
account the monopolar and bipolar contributions.
Main results and predictions:
Thermoelectric figure of merit in an ndoped 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 semiempirical
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 $3001100$ 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 7501250 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} densityfunctional 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 zoneaverage 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 ultrashort
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 threephonon interactions involving acoustic as
well as optical phonons in a twomaterial 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 crossplanar thermal
conductivity is roughly 4.14.8 times smaller than the inplane
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
314\% 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 $n=45$ 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
$T=10100$ 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 ultrathin superlattices, in that we should not expect
the very thinnest SLs to be as thermoelectrically efficient in the
crossplanar 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
UltraThin Superlattices
From a systematic evaluation of relevant transport coefficients we
show that in the midtohightemperature range (5001200 K) an
enhancement in $ZT$ to values of up to about 6 may be achieved along
the growth direction of a ultrathin 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 24 times that for some current stateofthe art
thermoelectric materials.
Publication: see Ref. [15]; and paper submitted for publication.
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Publications and Presentations:
