Tunable thermal conductivity of Si 1 − x Ge x nanowires
Jie Chen, Gang Zhang, and Baowen Li
Citation: Applied Physics Letters 95, 073117 (2009); doi: 10.1063/1.3212737
View online: http://dx.doi.org/10.1063/1.3212737
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APPLIED PHYSICS LETTERS 95, 073117 共2009兲
Tunable thermal conductivity of Si1−xGex nanowires
Jie Chen,1 Gang Zhang,2,a兲 and Baowen Li1,3
1
Department of Physics and Centre for Computational Science and Engineering,
National University of Singapore, Singapore 117542, Singapore
2
Institute of Microelectronics, 11 Science Park Road, Singapore Science Park II, 117685 Singapore
3
NUS Graduate School for Integrative Sciences and Engineering, 117456 Singapore
共Received 27 May 2009; accepted 3 August 2009; published online 21 August 2009兲
By using molecular dynamics simulation, we demonstrate that the thermal conductivity of
silicon-germanium nanowires 共Si1−xGex NWs兲 depends on the composition remarkably. The thermal
conductivity reaches the minimum, which is about 18% of that of pure Si NW, when Ge content is
50%. More interesting, with only 5% Ge atoms 共Si0.95Ge0.05 NW兲, SiNW’s thermal conductivity is
reduced to 50%. The reduction of thermal conductivity mainly comes from the localization of
phonon modes due to random scattering. Our results demonstrate that Si1−xGex NW might have
promising application in thermoelectrics. © 2009 American Institute of Physics.
关DOI: 10.1063/1.3212737兴
Silicon and germanium can form a continuous series
of substitutional solid, Si1−xGex. Single crystalline Si1−xGex
nanowires 共NWs兲 have been grown and the electronic band
gap modulation with composition has been reported.1 Recently, the experimental synthesis of core-shell structures2
provides intriguing opportunities for the development of NW
based devices. One of the promising applications for NWs is
as thermoelectric cooler.3,4 In thermoelectric application, low
thermal conductivity is preferred to increase the figure of
merit.5–7 Compared with bulk Si, SiNWs exhibits 100-fold
reduction in thermal conductivity because of the strong
boundary inelastic scattering of phonons.3,4,8 However, it is
still indispensable to reduce the thermal conductivity of NW
further in order to achieve higher thermoelectric performance. One possible way to achieve this is compound. In
this case, Si1−xGex NW seems to be a promising candidate
because both Si and Ge belong to the same group in the
periodic table, have the same crystal structure, and display
total solubility. In spite of an increasing number of works
devoted to the electronic and optical properties,9–11 very little
has been done for thermal conductivity of Si1−xGex NW.
In this letter, we investigate the thermal conductivity of
Si1−xGex NWs with x 共Ge content兲 changing from 0 to 1. The
thermal conductivity calculated in this paper is exclusively
from the lattice vibration 共phonon thermal conductivity兲, because phonons dominate the heat transport in SiNWs.12
In our simulations, nonequilibrium molecular dynamics
共NEMD兲 method is adapted to calculate the temperature distribution and thermal conductivity. To derive the force term,
Stillinger–Weber 共SW兲 potential13,14 is used for Si and Ge.
SW potential is widely used in the study of the thermal properties of silicon and germanium.15–18 It is a Keating type19
potential and consists of a two-body term and a three-body
term that can stabilize the diamond structure of silicon and
germanium. The details of the potential can be found in Ref.
15, and the parameters of Si–Ge interactions are taken to be
the arithmetic average of Si and Ge parameters for ␴Si–Ge,
and the geometric average for ␭Si–Ge and ␧Si–Ge.20
a兲
Electronic mail: [email protected].
We study the thermal conductivity of NWs along 共100兲
direction with cross section of 3 ⫻ 3 unit cells 共lattice constant is 0.543 nm兲, which corresponds to a cross section area
of 2.65 nm2. The atomic structure is initially constructed
from diamond structured bulk silicon. Then Si atoms are
randomly substituted by Ge atoms in the NW and the geometry is relaxed to its closest minimum total energy. The two
ends of NWs are put into heat bathes with temperature TL
and TR for the left and right end, respectively. Both
Langevin,21 and Nosé–Hoover22,23 heat bathes are used to
ensure our results are independent of heat bath. All results
given in this letter are obtained by averaging about 1 ⫻ 108
time steps, a time step is set as 0.8 fs. Free boundary condition is used to atoms on the outer surface of the NWs. The
thermal conductivity is calculated from the Fourier law,
␬ = −JL / ⵜT, where JL is the local heat current along the longitudinal direction, and ⵜT is the temperature gradient. The
MD calculated temperature TMD is corrected by taking into
account the quantum effects of phonon occupation, using the
relation: 3NkBTMD = 兰0␻DD共␻兲n共␻ , T兲ប␻d␻, where T is the
real temperature and TMD is the MD temperature, ␻ is the
phonon frequency, D共␻兲 is the density of states, n共␻ , T兲 is
the phonon occupation number given by the Bose–Einstein
distribution, and ␻D is the Debye frequency, Debye temperature TD = 645 K for Si.16,17 Correspondingly, according to the
Fourier law, the final effective thermal conductivity is rescaled by k = kMD共兩ⵜTMD兩 / 兩ⵜT兩兲 = kMD关⳵TMD / ⳵T兴. Using this
approach, we perform a quantum correction to temperature
and calculate the rescale rate ␣ = ⳵TMD / ⳵T for SiNWs. When
TMD is at room temperature, 300 K, the rescale rate ␣
= 0.91 for silicon, which gives a quite small quantum correction effect on thermal conductivity. Moreover, in our following study, we fix the simulation temperature at room temperature, and mainly focus on the compositional dependence
of thermal conductivity. Therefore, in the following part, we
do not do the quantum correction to MD temperature and
thermal conductivity.
We study the thermal conductivity of Si1−xGex NWs with
Ge atoms randomly distributed, 0 ⱕ x ⱕ 1. The NW we studied has a length of 20 unit cells, which corresponds to 10.86
nm. For each Ge content x, in order to reduce the fluctuation,
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073117-2
Appl. Phys. Lett. 95, 073117 共2009兲
Chen, Zhang, and Li
FIG. 2. The phonon participation ratio vs Ge content x for Si1−xGex NWs
with 0 ⱕ x ⱕ 1.
FIG. 1. 共Color online兲 The thermal conductivity ␬ vs x at T = 300 K. The
inset results are from Refs. 24 and 25.
the localization with O共1兲 for delocalized states and O共1 / N兲
for localized states. P is given by, P−1 = N兺i共兺␣␧iⴱ␣␧i␣兲2,
where ␧i␣ is the vibrational eigenvector component. We show
the results are averaged over 20 realizations. In Fig. 1 we
the participation ratio versus Ge content x in Fig. 2. For both
plot ␬ / ␬0 versus Ge content x at room temperature. Here ␬ is
pure Si NW and pure Ge NW, a high participation ratio
the thermal conductivity of Si1−xGex NWs, and ␬0 is the
appears, indicating the delocalized characteristics of
corresponding thermal conductivity of pure Si NW. The therphonon modes and corresponds to high thermal conductivity.
mal conductivity of pure SiNW calculated with Nosé–
However, in the Si1−xGex NWs 共0 ⬍ x ⬍ 1兲, with the impurity
Hoover heat bath is 2.43 W/m K, and it is 3.18 W/m K with
concentration increasing, the participation ratio decreases
Langevin heat bath. The lowest ␬ is only 18% 共Langevin兲
significantly, indicates strong localization due to impurity
and 15% 共Nosé–Hoover兲 of that of pure SiNW calculated
scattering and corresponds to low thermal conductivity. The
with the same heat bath. We also did the calculation with
Ge content dependent participation ratio is consistent with
NW of ten unit cells in the longitudinal direction. The lowest
the changes in thermal conductivity.
thermal conductivity is 17% of that pure Si NW, which demWe now turn to the reduction of thermal conductivity by
onstrates that the composition dependence of thermal conusing superlattice 共SL兲 structure. Some experimental and
ductivity is a general characteristic for Si1−xGex NWs. It is
theoretical works30–32 have been carried out to study the efquite remarkable that with only 5% Ge atoms 共Si0.95Ge0.05
fects of interface and SL period on thermal conductivity of
NW兲, its thermal conductivity can be reduced 50%. The best
various kinds of SL structures. Here we study the thermal
fitting gives rise to ␬ = A1e−x/B1 + A2e−共1−x兲/B2 + C, where A1,
conductivity of SL structured Si/Ge NWs. In our simulations,
B1, A2, B2, and C are fitting parameters. The decaying rates
the SL NWs consist of alternating Si and Ge layers with
B1 and B2 by Nosé–Hoover heat bath coincide with those by
changeable period length in the longitudinal direction. It has
Langevin methods 共as shown in Table I兲 indicating that the
a fixed cross section of 3 ⫻ 3 unit cells and a fixed length of
low thermal conductivity observed in Si1−xGex NWs is indeten unit cells in the longitudinal direction. In the Si/Ge SL
pendent of the heat path used.
structured NWs, the NWs and heat bathes may have three
We also show the simulation24 and experimental25 results
different contacts: both heat bathes are chosen to be Si; both
for bulk Si1−xGex alloy in Fig. 1. Although the thermal conare Ge; and one is Si, the other one is Ge, as the same
ductivity of NW is about two orders of magnitude smaller
material adjacent to the heat bath, respectively. Figure 3共a兲
than that of bulk material, the dependence of ␬ on the Ge
shows the thermal conductivity ␬ of the SL NWs versus the
atom content is similar.
period length for these three contacts. Langevin heat bath is
There is significant difference between nano and bulk
used here. The thermal conductivity calculated from different
material in the thermal property. In nanoscale system such as
NW-heat bath contacts has a good agreement with each
carbon nanotube, anomalous thermal conductivity has been
other. Therefore, the thermal conductivity of SL structured
observed both numerically26,27 and experimentailly.28 In orSi/Ge NWs has weak dependence on the detailed heat bath
der to study the physical mechanism of the reduction of thercontact. As shown in Fig. 3共a兲, ␬ decreases monotonically
mal conductivity, we have studied the participation ratio P,29
with the period length decreasing from 4.43 nm 共32 layers兲,
which is an important measure for the fraction of phonons
until period length reaches a critical value of 1.11 nm 共eight
participating in thermal transport, and effectively indicates
layers兲. At this critical period length, the thermal conductivity is only one sixth of that of pure SiNWs. This reduction in
TABLE I. The fitting parameters for the best fitting thermal conductivity
thermal conductivity is due to the fact that when decreasing
formula:␬ = A1e−x/B1 + A2e−共1−x兲/B2 + C, for both Langevin and Nosé–Hoover
heat baths.
the period length of SL structured NWs with a fixed total
length, the increasing number of interface will lead to an
B1
A2
B2
C
A1
enhanced interface scattering, which is responsible for the
reduction in ␬. As period length decreases further from the
Langevin heat bath
2.50
0.056
1.47
0.065
0.65
critical value, there exists a rapid increase in ␬. At room
Nosé–Hoover heat bath
2.01
0.063
1.23
0.066
0.40
temperature,
dominant phonon wavelength inDownloaded
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at: http://scitation.aip.org/termsconditions.
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073117-3
Appl. Phys. Lett. 95, 073117 共2009兲
Chen, Zhang, and Li
have also investigated the thermal conductivity of SL structured Si/Ge NWs. The dependence of thermal conductivity
on the period length is explained by the overlap of phonon
power spectrum of different layers. The low thermal conductivity of Si1−xGex NW significantly enhances its figure of
merit ZT and has raised the exciting prospect for application
in on-chip thermoelectric cooler.
J.C. would like to thank Nuo Yang, Donglai Yao, and
Lifa Zhang for fruitful discussions. This work is supported
in part by an ARF Grant No. R-144-000-203-112 from the
Ministry of Education of the Republic of Singapore and
Grant No. R-144-000-222-646 from National University of
Singapore.
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2
FIG. 3. 共Color online兲共a兲 The thermal conductivity ␬ of Si/Ge SL structured
NWs vs period length at T = 300 K. Here the atoms in heat baths are: both
heat baths are Si atoms 共black squares兲; both are Ge 共red circles兲; and atom
in one heat bath is Si, in the other one is Ge, as the same material adjacent
to the heat bath 共green triangles兲. 共b兲 The normalized power spectrum of
phonons of different atoms 共Si or Ge兲 along the longitudinal direction with
period length is 1.11 nm 共eight layers兲. 共c兲 The normalized power spectrum
with period length is 0.28 nm 共two layers兲. 共d兲 Overlap ratio S vs period
length.
about 1–2 nm,33 which is quite close to the critical value of
1.11 nm in the present paper. When period length is smaller
than the dominant phonon wavelength, ballistic transport
will replace diffusive mode and dominates the transport characteristic, which gives rise to a rapid increase in thermal
conductivity.15,34
In order to get a better understanding of the underlying
mechanism of the period length dependence of thermal conductivity, we calculate the power spectrum of both Si and Ge
layers of SL structured NWs with different period length.
Figure 3共b兲 and 3共c兲 show the normalized power spectrum in
two typical cases: SL structured NWs with period length of
1.11 nm 共eight layers兲; and SL structured NWs with period
length of 0.28 nm 共two layers兲. It is clear that there exists a
larger overlap of power spectrum in the case which has a
larger thermal conductivity. It is well understood that in low
dimensional systems, a large overlap of power spectrum
means that heat flow can easily go through the system and,
therefore, results in a high thermal conductivity.35,36 In order
to quantify the above power spectrum analysis, the overlap
共S兲 of the power spectra are calculated as36
S=
兰⬁0 PSi共␻兲PGe共␻兲d␻
.
⬁
兰0 PSi共␻兲d␻兰⬁0 PGe共␻兲d␻
共1兲
In Fig. 3共d兲 we plot the overlap ratio S versus period length.
A comparison between Figs. 3共a兲 and 3共d兲 demonstrates that
a larger overlap S corresponds to a higher thermal conductivity.
To summarize, we have investigated the composition dependence of thermal conductivity of Si1−xGex NWs with x
changing from 0 to 1. A remarkable composition effect on
thermal conductivity is observed. With only 5% Ge atoms
共Si0.95Ge0.05 NW兲, its thermal conductivity can be reduced to
50%. This composition dependence of thermal conductivity
is explained by phonon participation ratio. In addition, we
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