The composition dependent mechanical properties of ge si core–shell nanowires

Physica E 40 (2008) 3042–3048
The composition-dependent mechanical properties
of Ge/Si core–shell nanowires
X.W. Liu, J. Hu, B.C. Pan
Ã
Hefei National Laboratory for Physical Sciences at Microscale, Department of Physics,
University of Science and Technology of China, Hefei, Anhui 230026, PR China
Received 7 November 2007; received in revised form 14 March 2008; accepted 25 March 2008
Available online 12 April 2008
Abstract
The Stillinger–Weber potential is used to study the composition-dependent Young’s modulus for Ge-core/Si-shell and Si-core/Ge-shell
nanowires. Here, the composition is defined as a ratio of the number of atoms of the core to the number of atoms of a core–shell
nanowire. For each concerned Ge-core/Si-shell nanowire with a specified diameter, we find that its Young’s modulus increases to a
maximal value and then decreases as the composition increases. Whereas Young’s modulus of the Si-core/Ge-shell nanowires increase
nonlinearly in a wide compositional range. Our calculations reveal that these observed trends of Young’s modulus of core–shell
nanowires are essentially attributed to the different components of the cores and the shells, as well as the different strains in the interfaces
between the cores and the shells.
r 2008 Elsevier B.V. All rights reserved.
PACS: 61.46.Àw; 11.15.Kc; 46.80.þj; 74.62.Dh
Keywords: Nanowires; Mechanical properties; Calculation
1. Introduction
There has been fast-growing interest in semiconductor
nanowires due to their unique size- dependent electronic,
optical and transport properties [1–4]. Among various
kinds of nanowires, composite nanowires, where their sizes
and the composition can be modulated, provide potential
applications in thermoelectronics, nanoelectronics and
optoelectronics [5–7]. Therefore, much effort has focused
on the synthesis of core–shell nanowires with different
compositions in the past few years [8–10]. Typically, core–
shell nanowires consisting of germanium and silicon have

been synthesized using chemical vapor deposition method
successfully [5]. Later on, Musin and Wang [11, 12] studied
the composition- and size-dependent band gaps of Ge-
core/Si-shell and Si-core/Ge-shell nanowires at the level of
density functional theory, where the composition is defined
as the ratio of the number of atoms of the core to the
number of the atoms of the whole nanowire. They found
that the band gaps of both kinds of core–shell nanowires
decrease when composition o0:3 and increase after that.
However for a given composition, the band gap decreases
noticeably as the diameter of the nanowire increases.
Therefore, the Ge-core/Si-shell and Si-core/Ge-shell nano-
wires exhibit promising perspective for band gap engineer-
ing and optoelectronic applications [11–13]. Basically, for
most of novel materials, their mechanical properties are of
great importance for their potential application. Since so,
assessing the mechanical properties of such core–shell
nanowires is necessary.
Usually, the mechanical property of a nanowire can be
described by Young’s modulus. Previous publications
showed that Young’s modulus of nanoscale structure relies
upon the size of the structure and the orientation of lateral
facets. For example, the Young’s modulus of single-crystal
GaN nanotube increases as the ratio of the surface area to
its volume grows; Young’s modulus of single-crystal GaN
nanotube oriented along ½110 is higher than that of ½001
oriented single-crystal GaN nanotube [14]. The similar
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doi:10.1016/j.physe.2008.03.011
Ã
Corresponding author.
E-mail address: (B.C. Pan).
results were also observed for the case of ZnO nanowires in
experiment [15]. Although there have been an increasing
work devoted to Young’s modulus of nanowires [16–18],
there are few studies on Young’s modulus of core–shell
nanowires up to now.
In this work, we perform our calculations on Ge-core/Si-
shell and Si-core/Ge-shell nanowires to evaluate their
Young’s modulus. we find that Young’s modulus are
dependent on the composition, which is explained using the
strains at the interface between the cores and the shells.
2. Computational details
Experiments showed that the cross sections of the
produced Ge-core/Si-shell or Si-core/Ge-shell nanowires
are all hexagonal [5]. Because of this, we initially generate
core–shell nanowires with hexagonal cross section on the
basis of diamond-structured crystals [19]. For example, a
Si-core (that is a nanowire) orientated along ½111
direction is isolated from Si crystal. For convenience, the
distance between the axis and the vertex of hexagon on the
cross section is defined as the radius R
core
of a core as
shown in Fig. 1. For a Ge-shell with inner radius R
in
shell
and

outer radius R
out
shell
, it is generated using the scheme
proposed in previous work [14]. By filling a Ge-shell with
a suitable Si-core, we ach ieve a Si-core/Ge-shell nanowire.
With using this scheme, Ge-core/Si-shell nanowires are also
generated. Clearly, through adjusting the diameter of the
core and the thickness of the shell, we can obtain various
Si-core/Ge-shell and Ge-co re/Si-shell nanowires. For a
given core–shell nanowire, the composition, N
core
=ðN
core
þ
N
shell
Þ, where the N
core
and N
shell
are the number of core
atoms and shell atoms, respectively, does actually reflect
the structural feature described by ð R
core
/R
core2shell
Þ
2
.In

order to reveal how the composition of a core–shell
nanowire affects its Young’s modulus, we select a core–
shell nanowire with a specified radius R
core2shell
of about
31 a
˚
, where the radius of the core (R
core
) and the thickness
of the shell ðT
shell
¼ R
out
shell
À R
in
shell
Þ are adjustable with a
limitation of ðR
core
þ T
shell
¼ R
core2shell
Þ. Tables 1 and 2 list
the structural features of the core–shell nanowires we
considered.
As listed in Tables 1 and 2, 20 core–shell nanowires are
taken into account, each of which contains 2524 atoms. To

fully optimize these large systems and study their mechan-
ical properties, the proposed Stillinger–Weber (SW)
potentials for Si, Ge and Ge–Si [20] are employed for our
calculations, where the total energy of a system contains
one-body, two-body and three-body contributions. The
potential functions for the two-body and three-body are
parameterized. By fitting to some bulk properties, the
parameters for Si [20] and Ge [21] were, respectively,
achieved. According to these parameters, the parameters
for the Ge–Si were taken to be the geometric means of
the both sets of parameters [22]. Previously, this empirical
potential has been used to handle silicon–germanium
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Fig. 1. (Color online) Top view of a core–shell nanowire. The larger
circles stand for the core atoms, and the smaller ones for the shell-atoms.
Table 2
The optimal structural parameters of Si-core/Ge-shell nanowires
Radius of
core
Inner
radius of
shell
Outer
radius of
shell
Number of
atoms in
core
Composition
6.68 9.02 32.22 148 0.0586

8.91 11.25 32.18 244 0.0967
11.15 13.48 32.12 364 0.1442
13.37 15.71 32.05 508 0.2013
15.60 17.93 31.97 676 0.2678
17.83 20.15 31.87 868 0.3439
20.05 22.36 31.75 1084 0.4295
22.27 24.57 31.64 1324 0.5246
24.47 26.76 31.48 1588 0.6292
26.65 28.92 31.32 1876 0.7433
The number of atoms of core–shell nanowires are fixed to be 2524. The
unit of the length is in a
˚
.
Table 1
The optimal structural parameters of Ge-core/Si-shell nanowires
Radius of
core
Inner
radius of
shell
Outer
radius of
shell
Number of
atoms in
core
Composition
6.87 9.01 31.04 148 0.0586
9.16 11.29 31.08 244 0.0967
11.45 13.58 31.14 364 0.1442

13.74 15.87 31.20 508 0.2013
16.04 18.16 31.27 676 0.2678
18.33 20.46 31.36 868 0.3439
20.63 22.76 31.47 1084 0.4295
22.93 25.07 31.60 1324 0.5246
25.25 27.41 31.76 1588 0.6292
27.58 29.77 31.95 1876 0.7433
The number of atoms of core–shell nanowires are fixed to be 2524. The
unit of the length is in a
˚
.
X.W. Liu et al. / Physica E 40 (2008) 3042–3048 3043
alloys [22]. More recen tly, the Young’s modulus of Si
nanowires have been studied based on this classical
potential, which is in good agreement with the density
functional theory calculations [16,17]. Such agreement
shows that it is reliable to handle the Si/Ge core–shell
nanowires using this potential.
Commonly, Young’s modulus of a nanowire can be
calculated according to the following expression:
Y ¼
1
V
0
q
2
E
qe
2






e¼0
, (1)
where E is the total energy, V
0
is the equilibrium volume,
which is defined as the product of axial equilibrium lengt h
ð‘
0
Þ and the cross-section area S
0
. e is longitudinal strain.
In our calculations, the periodic condition along the axis of
each wire is imposed. Initially, the lattice constant
corresponding to the ideal bulk Ge is employed for
a concerned Si–Ge core–shell nanowire. Clearly, this
lattice constant is not optimal. Then, we adjust the lattice
constant of the nanowire. For each specified lattice
constant, the nanowire is fully relaxed. We thus obtain
the energies of the nanowire with different lattice constant.
From these energies, the optimal lattice constant of the
nanowire is achieved. Furthermore, the nanowire is
elongated and compressed axially from À1.8% to 1.8%
by increment of 0.3% around its equilibrium, to obtain an
energy curve (the total energy of a system vs the loaded
strain). This curve is fitted by using a cubic polynomial
function [14]. Inserting the cubic polynomial function into

Eq. (1), we obtain Young’s modulus of the nanowire.
3. Results and discussion
Fig. 2(a) plots the calculated Young’s modulus of the
core–shell nanowires with different compositions. It is
found that in the case of Si-core/Ge-shell, the Young’s
modulus increases with increasing the component of
Si-core, whereas in the case of Ge-core/Si-shell, the
Young’s modulus goes up and then decreases. Such an
increasing-and-decre asing trend in Young’s modulus curve
of Ge-core/Si-shell and monotonous increment in Young’s
modulus of Si-core/Ge-shell strongly indicate that the
mechanical property of a core–shell nanowire is dependent
on its composition.
Structurally, for a given core–shell nanowire, as the
diameter of the core increases, the thickness of the shell
decreases correspondingly. In a sense, the core and the shell
may be analogous to the isolated wire and the isolated
tube, respectively. Our calculations show that Young’s
modulus of an isolated Si (Ge) nanowire increases as its
diameter decreases, and Young’s modulus of an isolated Si
(Ge) nanotube becomes large when its thickness is small
(Fig. 3). It seems that the Young’s modulus of the core and
the shell are competing each other to result in the
composition-dependent trends of the Young’s modulus
shown in Fig. 2(a). However, the core and the shell in a
considered core–shell nanowire do couple with each other,
and thus the Young’s modulus of either core or shell is not
the same as that of the isolated nanowire or the isolated
nanotube. On the other hand, the Young’s modulus of a
core–shell nanowire can not be expressed as a simple

summation of the Young’s modulus of the isolated
nanowires and the isolated nanotubes.
To explicitly reveal the relation of the Young’s modulus
between a core–shell nanowire and its core and shell, we
employ the ‘‘stress–strain relation’’ to serve our analysis.
As we know, the stress s
zz
is proportional to the loaded
strain e
z
along z direction for string-like materials [23],
s
zz
¼ Y e
z
. (2)
The proportional coefficient Y above is the Young’s
modulus along z direction. Here, the stress of the system
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0.0
Composition
144.5
145.5
146.5
147.5
148.5
Young’s modulus (GPa)
Ge−core/Si−shell
Si−core/Ge−shell
0.0

Composition
144.5
145.5
146.5
147.5
148.5
Young’s modulus (GPa)
Ge−core/Si−shell
Si−core/Ge−shell
0.1
0.2 0.3 0.4 0.5 0.6 0.7
0.8
0.2 0.4
0.6
0.8
Fig. 2. (Color online) Young’s modulus of Ge-core/Si-shell and Si-core/
Ge-shell nanowires as a function of composition, obtained by using (a)
formula (1), (b) the ‘‘stress–strain relation’’.
X.W. Liu et al. / Physica E 40 (2008) 3042–30483044
is evaluated by [18]
s
zz
¼
1
2V
X
i
X
N
jai

f
z
ij
r
z
ij
, (3)
where V is the volume of the system, N is the number of
atoms of a core–shell nanowire, f
z
ij
is the inter-particle force
along z direction between particles i and j, and r
z
ij
is the
spacing in z direction between the two particles. These
variables are a function of strain.
To calculate the stresses of the core and the shell under a
loaded strain, t he right-hand s ide o f for mula (3) i s r ewritten as
1
2V
X
N
core
i¼1
X
N
jðaiÞ¼1
f

z
ij
r
z
ij
þ
1
2V
X
N
i¼N
core
þ1
X
N
jðaiÞ¼1
f
z
ij
r
z
ij
The first term above contains the interaction between
core-atom and core-atom and the interaction between core-
atom and shell-atom, and the second term contains the
interaction between shell-atom and shell-atom and between
shell-atom and core-atom. The total stress of a core–shell
nanowire can be rewritten as
s
zz

¼
1
2V
X
core
f
z
ij
r
z
ij
þ
1
2V
X
shell
f
z
ij
r
z
ij
. (4)
With defining the volumes of the core and the shell, V
core
and V
shell
, we yield
s
zz

¼
V
core
V
s
zz
core
þ
V
shell
V
s
zz
shell
, (5)
where s
zz
core
and s
zz
shell
are the stresses of the core and the
shell, respectively. Considering expressions (2) and (5), we
have
Y
core2shell
¼
V
core
V

Y
core
þ
V
shell
V
Y
shell
. (6)
From this formula, total Young’s modulus Y
core2 shell
is not
only contributed from Young’s modulus of the core ðY
core
Þ
and the shell (Y
shell
) but also dependent on the fractional
volumes of V
core
=V and V
shell
=V. That is, Young’s modulus
of a whole core–shell nanowire is the weighted combination
of Young’s modulus of the core and the shell.
To evaluate Young’s modulus of the core and the shell
using the ‘‘stress–strain relation’’, it is necessary to
calculate their volumes. It is worth noting that the existence
of the interface region between the core and the shell in a
core–shell nanowire results in difficulty for defining

volumes of the core and the shell. When the volume is
taken to be the geometric volume, the calculated Young’s
modulus of the core (shell) decreases with increasing its
diameter (thickness) as plotted in Fig. 4, exhibiting the
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0
Radius (Ang)
Thickness of Nanotube (Ang)
144
146
148
150
152
Young’s modulus (GPa)
Ge nanowire
Si nanowire
10 20 30
40
158
156
154
152
150
148
146
144
4
81216
20 24
Young’s modulus (GPa)

Ge nanotube
Si nanotube
Fig. 3. (Color online) Young’s modulus of silicon and germanium (a)
nanowirs and (b) nanotubes, obtained by using formula (1).
0.0
Composition
Young’s modulus (GPa)
140
160
180
200
220
240
Ge−core
Si−shell
Si−core
Ge−shell
0.2 0.4 0.6 0.8
Fig. 4. (Color online) Young’s modulus of the cores and the shells
obtained by using the ‘‘stress–strain relation’’. The geometric volumes for
the cores and the shells are taken into account in calculations.
X.W. Liu et al. / Physica E 40 (2008) 3042–3048 3045
same trend as that of the isolated nanowire (nanotube) as
addressed above. The main discrepancy between Figs. 3
and 4 is the systematical shift-up of Young’s modulus
evaluated by the ‘‘stress–strain relation’’ relative to that
calculated by formula (1), this is essentially resulted from
the interaction between the core and the shell in the core–
shell nanowire. In fact, the volumes of the core and the
shell should be ‘‘physical volumes’’, and thus the geometric

volumes used above are not suitable for the case of the
core–shell nanowire. Unfortunately, the definition for
‘‘physical volumes’’ of the core and the shell in a core–
shell nanowire is somewhat uncertain, this is due to the
existence of the region around the interface between the
core and the shell of a core–shell nanowire.
Basically, it is unreasonable for the core or the shell to
include the volume of the whole interface region. A
possible way is to divide the whole interface region into
two parts according to the ratio of bondlengths of bulk Si
and bulk Ge, and the two parts, respectively, belong to the
core and the shell. In this case, Young’s modulus of Ge-
core decreases as its diameter increases strikingly, while
that of the Si-core goes up slowly (Fig. 5); The changes of
Young’s modulus of shells with increasing composition are
just opposite to the case of Ge-cores. This clearly indicates
that the embedded Si nanowire in a Ge nanotube exhibits a
unusual behavior in its Young’s modulus with respect to
the isolated Si nanowire, while the embedded Ge nanowire
in a Si nanotube just follows the normal trend in Young’s
modulus.
We emphasize that the behaviors above are essentially
originated from the interface effect between the core and
the shell, in which the volumes of the core and shell and
the caused stresses around the interface play important
roles. Firstly, let us pay our attention to the volume effect.
Fig. 6(a) displays the fractional volumes of cores and shells
as a function of the composition, from which we can find
that as the composition increases, the fractional volumes of
the cores linearly increase, while the fractional volumes of

the shells linearly decrease. According to these fractional
volumes and the obtained Young’s modulus (Y
core
and
Y
shell
), we immediately obtain the components, ðV
core
=VÞ
Y
core
and ðV
shell
=VÞY
shell
, of Young’s modulus of the core–
shell nanowires. Surprisingly, the two components of
Young’s modulus as a function of the composition almost
exhibit a linear trend (Fig. 6(b)), which is totally different
from the trends of Y
core
and Y
shell
, but strikingly similar to
the trends of the fractional volumes displayed in Fig. 6(a).
This observation strongly demonstrates that the ‘‘physical
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0.0
Composition
144

146
148
150
152
154
Young’s modulus (GPa)
Ge−core
Si−shell
Si−core
Ge−shell
0.2
0.4
0.6 0.8
Fig. 5. (Color online) Young’s modulus of the cores and the shells
calculated by using the ‘‘stress–strain relation’’. The ‘‘physical volumes’’ as
addressed in the text for the cores and the shells are taken into account in
calculations.
0.0
Composition
0.0
0.2
0.4
0.6
0.8
1.0
Fractional volume
Ge−core
Si−shell
si−core
Ge−shell

0.0
Composition
0
50
100
150
Young’s modulus (GPa)
Ge−core
Si−shell
Si−core
Ge−shell
0.2
0.4 0.6 0.8
0.2 0.4 0.6 0.8
Fig. 6. (Color online) (a) The fractional volumes of cores and shells vs the
composition. (b) The components of the total Young’s modulus as a
function of composition for each core–shell nanowire.
X.W. Liu et al. / Physica E 40 (2008) 3042–30483046
volumes’’, a kind of interface effect, between the core and
the shell critically govern the evolution of the components
of Young’s modulus for a core–shell nanowire indeed.
Secondly, we turn to the stresses arising from the
mismatch of lattice constants between the core and the
shell. As we know, the lattice constant of bulk Si is smaller
by about 4% than that of bulk Ge. For a given core–shell
nanowire, the surface atoms of the Ge-core are compres-
sively strained by the Si-shell, meanwhile the Si-shell atoms
are tensibly strained by the Ge-core. Moreover, the
distribution of the strain around the interface somehow
correlates with the composition of the core–shell nanowire.

These aspects are reflected in the averaged bond lengths of
Si–Si, Ge–Ge and Si–Ge varying with the composition as
shown in Fig. 7, from which we observe that all of the
averaged bond lengths decrease with increasing the
composition in Si-core/Ge-shell nanowires, but increase
with increasing the composition in Ge-core/Si-shell nano-
wires yet. As speculated above, such different strains
around the interface also correlate with the variation of
Young’s modulus of the core and the shell in a core–shell
nanowire. In order to illustrate this relat ion, we recall a
simple system consisting of two atoms, in which the two
atoms interact with each other. We know that a loaded
compressive strain around the equilibrium of the two-atom
system makes a larger stress than a loaded tensile strain
with the same amplitude. Combining this with formula
(2) an d (3), we may conclude that a compressive strain
makes a larger increment of Young’s modulus than a
tensile strain. Note that the interaction between any two
atoms in a concerned nanowire can be similarly described
by such a two-atom model. Hence, for our core–shell
nanowires, the Ge-cores that are compressed by the
connected Si-shells show larger values of Young’s modulus
than the corresponded Si-cores, even though Young’s
modulus of bulk Ge along h111i direction is lower by
about 2 GPa than that of bulk Si along h111i direct ion
[24]. The similar situation occurs for the Si-shells and the
Ge-shells when x40:35, as shown in Fig. 5.
Based on the calculated Y
core
, Y

shell
and the fractional
volumes, we easily obtain Young’s modulus of the core–
shell nanowires with using formula (6), which are plotted in
Fig. 2(b). As shown, the dispersion of Young’s modulus
of the core–shell nanowires matches that displayed in
Fig. 2(a), indicating that the evaluated Young’s modulus
by formula (6) is reliable qualitatively. We should point out
that (1) the definition of the ‘‘physical volume’’ for the core
or the shell in a core–shell nanowire as discussed above
does not affect the values of Young’s modulus of the core–
shell nanowire; (2) although the trends of the fractional
volumes varying with the composi tion are quite similar to
those of the ðV
core
=VÞY
core
and ðV
shell
=VÞY
shell
, the fact
that the summation V
shell
=V + V
core
=V ¼ 1 does always
keep at each composition, whereas the summation
ðV
core

=VÞY
core
þðV
shell
=VÞY
shell
shown in Fig. 2(b) are
dependent on the composition implies that the evolution of
the Y
core
and the Y
shell
with the composition plays a critical
role in the composition-dependent trend of Young’s
modulus for an entire core–shell nanowire. In addition,
as shown in Fig. 5, the trend of Young’s modulus of
the considered Si-shells roughly keeps pace with that of the
Ge-shells; but Young’s modulus of the Ge-cores show an
opposite trend against the Si-cores. Such distinct behavior
in Young’s modulus of Ge-cores a nd Si-cores are mainly
associated with the different trends of the Young’s
modulus for Ge-core/Si-shell and the Si-core/Ge-shell
nanowires.
4. Summary
In summary, we calculate Young’ s modulus of Ge-core/
Si-shell and Si-core/Ge-shell nanowires systematically. We
find that as the composition of the core–shell nanowire
ARTICLE IN PRESS
0.0
Composition

2.30
2.35
2.40
2.45
2.50
Bond length (Ang)
Si−Si
Ge−Ge
Si−Ge
0.0
Composition
2.30
2.35
2.40
2.45
2.50
Bond length (Ang)
Si−Si
Ge−Ge
Ge−Si
0.2 0.4 0.6 0.8 1.0
0.2
0.4 0.6
0.8 1.0
Fig. 7. (Color online) Variation of averaged bond lengths of Si–Si, Si–Ge
and Ge–Ge in (a) Si-core/Ge-shell and (b) Si-core/Ge-shell nanowires as a
function of composition.
X.W. Liu et al. / Physica E 40 (2008) 3042–3048 3047
increases, Young’s modulus of Ge-core/Si-shell nanowires
increases to a maximal value then drops down, while

Young’s modulus of Si-core/Ge-shell is increases. These
results are found to be tightly correlated with the mismatch
of the lattice constants between Si and Ge at the interface.
In addition, the relation between Young’s modulus and the
volumes for the core and the shell in a core–shell nanowire
is discussed in detail. We point out that the analysis about
the basic trends in Young’s modulus of the Si/Ge core–
shell nanowires can be helpful for understandings of the
mechanical properties for other kinds of core–shell
nanowires.
Acknowledgments
This work is partially supported by the Fund of
University of Science and Technology of China, the Fund
of Chinese Academy of Science, and by NSFC with code
number of 50121202, 60444005, 10574115 and 50721091.
B.C. Pan thanks the support of National Basic Research
Program of China (2006CB922000). We thank B. Xu, R.L.
Zhou and H.Y. He for valuable comments.
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[24] In order to recheck the reliability of this potential, we evaluate the
bulk modulus of Si and Ge to be 101.52 and 78.62 GPa, being in
excellent agreement to the reported values of 101.0 and 79.0 GPa
[K.A. Gschneidner, in Solid State Physics, vol. 16, Academic Press,
1964], and the Young’s modulus of bulk silicon and bulk ger-
manium along ½111 direction to be Y
Si
½111

¼ 143:37 GPa, Y
Ge
½111
¼
141:964 GPa.
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X.W. Liu et al. / Physica E 40 (2008) 3042–30483048