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

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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  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

r2008 Elsevier B.V All rights reserved

PACS: 61.46.w; 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 ½1 1 0 is higher than that of ½0 0 1 oriented single-crystal GaN nanotube [14] The similar

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1386-9477/$ - see front matter r 2008 Elsevier B.V All rights reserved.

doi: 10.1016/j.physe.2008.03.011

Corresponding author.

E-mail address: bcpan@ustc.edu.cn (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-Ge-core/Si-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 ½1 1 1

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 Rcore of a core as

shown inFig 1 For a Ge-shell with inner radius Rinshelland

outer radius Routshell, it is generated using the scheme

proposed in previous work [14] By filling a Ge-shell with

a suitable Si-core, we achieve 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-core/Si-shell nanowires For a

given core–shell nanowire, the composition, Ncore=ðNcoreþ

NshellÞ, where the Ncore and Nshell are the number of core

atoms and shell atoms, respectively, does actually reflect

the structural feature described by ðRcore/Rcore2shellÞ2 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 Rcore2shell of about

31 a˚, where the radius of the core (Rcore) and the thickness

of the shell ðTshell¼RoutshellRinshellÞ are adjustable with a limitation of ðRcoreþTshell¼Rcore2shellÞ.Tables 1 and 2list the structural features of the core–shell nanowires we considered

As listed inTables 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

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

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

The number of atoms of core–shell nanowires are fixed to be 2524 The unit of the length is in a˚.

alloys [22] More recently, 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

V0

q2E

qe2

 

e¼0

where E is the total energy, V0 is the equilibrium volume,

which is defined as the product of axial equilibrium length

ð‘0Þand the cross-section area S0 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 1.8% 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

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-decreasing 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

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 szz is proportional to the loaded strain ezalong z direction for string-like materials[23],

The proportional coefficient Y above is the Young’s modulus along z direction Here, the stress of the system

0.0

Composition 144.5

145.5 146.5 147.5 148.5

Ge−core/Si−shell Si−core/Ge−shell

0.0

Composition 144.5

145.5 146.5 147.5 148.5

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’’.

is evaluated by[18]

szz¼ 1

2V

X

i

XN

j ai

where V is the volume of the system, N is the number of

atoms of a core–shell nanowire, fzijis the inter-particle force

along z direction between particles i and j, and rz

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, the right-hand side of formula (3) is rewritten as

1

2V

X

N core

i¼1

XN

jð aiÞ¼1

fzijrzijþ 1

2V

XN i¼N þ1

XN

jð aiÞ¼1

fzijrzij

The first term above contains the interaction between atom and atom and the interaction between core-atom and shell-core-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

szz¼ 1 2V

X

core

fzijrzijþ 1 2V

X

shell

With defining the volumes of the core and the shell, Vcore

and Vshell, we yield

szz¼Vcore

zz coreþVshell

zz

where szzcore and szzshell are the stresses of the core and the shell, respectively Considering expressions (2) and (5), we have

Ycore2shell¼Vcore

V Ycoreþ

Vshell

From this formula, total Young’s modulus Ycore2 shellis not only contributed from Young’s modulus of the core ðYcoreÞ and the shell (Yshell) but also dependent on the fractional volumes of Vcore=V and Vshell=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

0

Radius (Ang)

Thickness of Nanotube (Ang)

144

146

148

150

152

Ge nanowire

Si nanowire

158

156

154

152

150

148

146

144

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

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.

same trend as that of the isolated nanowire (nanotube) as

addressed above The main discrepancy between Figs 3

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

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 (Ycore and

Yshell), we immediately obtain the components, ðVcore=V Þ

Ycoreand ðVshell=V ÞYshell, 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 Ycoreand Yshell, but strikingly similar to the trends of the fractional volumes displayed inFig 6(a) This observation strongly demonstrates that the ‘‘physical

0.0

Composition 144

146

148

150

152

154

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

Ge−core Si−shell si−core Ge−shell

0.0

Composition 0

50 100 150

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.

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 relation, 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) and (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 h1 1 1i direction is lower by about 2 GPa than that of bulk Si along h1 1 1i direction [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 Ycore, Yshell and the fractional volumes, we easily obtain Young’s modulus of the core– shell nanowires with using formula (6), which are plotted in

of the core–shell nanowires matches that displayed in

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 composition are quite similar to those of the ðVcore=V ÞYcore and ðVshell=V ÞYshell, the fact that the summation Vshell=V + Vcore=V ¼ 1 does always keep at each composition, whereas the summation

ðVcore=V ÞYcoreþ ðVshell=V ÞYshell shown in Fig 2(b) are dependent on the composition implies that the evolution of the Ycoreand the Yshellwith 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 and 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

0.0

Composition 2.30

2.35

2.40

2.45

2.50

Si−Si Ge−Ge Si−Ge

0.0

Composition 2.30

2.35

2.40

2.45

2.50

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.

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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