Mechanical Properties of Si/Ge Nanowire

12/10/2004

Eunseok Lee, Joohyun Lee, and Keonwook Kang

Final Project for ME346 Introduction to Molecular Simulation

Instructor: Wei Cai, Stanford University

1. Introduction Recently heterostructure of Si and Ge nanowires has been fabricatedi since it is

believed that they show fairly good semi-conducting propertyii, which can be used as

basic building blocks of molecular electronic devices or bio sensors.iii In this early stage

of developing 1D hetero-nanowires, mechanical property of the nanowires should also be

studied because, at nano scale, we cannot assume that the both mechanical and electrical

behavior are independent of each other.

In this term project, authors have performed several simulations for the tension tests of

Si, Ge, and Si-Ge core-shell nanowires to obtain their mechanical properties such as

Young's modulus and yield stress at 0K as a rudimentary step for future work. We also

observe how the deformation initiated and developed in the nano structures using

molecular simulation.

2. System Description

Si, Ge and Si-Ge nanowires of different diameters

(5nm, 10nm, 15nm and 25nm) are initially configured

like fig.1 to line up along [110] direction, which is a

typical growth direction. We stretch the simulation

box along z direction, and compute the average stress

as a function of strain.

Since the temperature at which simulation was done is

0K, it is the potential energy that determines system state and conjugate gradient method

was used to follow the local minimum of energy and therefore to find a relaxed state of

system. Periodic boundary condition is adopted along z direction so that nanowire is

considered to have infinite length along that direction. Along x and y directions, super

cell condition is used. Two different many-body potentials, Stillinger-Weberiv vand

Tersoffvi, are used to describe both Si and Ge atoms (see appendix). The simulations are

performed using the MD++ packagevii. More detailed on simulation geometry are listed

in table 1.

We also carry out Si-Ge core-shell nanowires of diameter 15nm to see what would

happen at the interface of Si and Ge under tension (fig 2). Different color in

Fig 1 Si Nanowire

x = [001]

z = [110]

y = [110]

fig.1 and fig.2 shows different

energy level of each atom. The red

atoms on the surface in fig.1 have

higher energy from that of bulk

atoms even if they both are Si.

Material Si Ge

Structure Diamond-cubic Crystal

Lattice Constant, a (Å) 5.430953 5.653621

Diameter, 5 NP=4908 4548

d (nm) 10 NP=18036 18072

15 NP=41676 40260

20 NP=73488 71556

Temperature, T (K) 0 0

Thickness, t (nm) 4.6 4.8

Potential SW, Tersoff SW, Tersoff

Computer Resource P-IV 3GHz X 2

3. Diamond Cubic Crystal Structure

Basically, Si and Ge are diamond- cubic crystal as shown in fig.3viii and it shows

different atomic arrangement when viewed along different direction. This explains why

many properties of Si and Ge are anisotropic. Different shrinkage ratio at (110) plane can

also be explained by the anisotropy. For example, the Si and Ge nanowire shrink

anisotropically in diameter while under uniaxial tension, as shown in Fig. 9.

Fig 2 Si-Ge core-shell nanowire

Si

Ge

5nm 15nm

5nm 15nm

Table 1. System Description

4. Data analysis and Discussion 1) Young’s Modulus i) Silicon by Stillinger-Weber Potential,

Young’s moduli are 124.807GPa for 5nm, 122.330GPa for 10nm, 121.601GPa for

15nm and 121.385GPa for 20nm. We obtained these values by selecting only the region

Fig 3 Diamond Cubic Structure (generated by Atomeye)

[001] direction

[111] direction [110] direction

Si

0.0E+00

2.0E+03

4.0E+03

6.0E+03

8.0E+03

1.0E+04

1.2E+04

1.4E+04

0 0.05 0.1 0.15

Strain

Stress

5nm

10nm

15nm

20nm

Si

0.0E+00

5.0E+03

1.0E+04

1.5E+04

2.0E+04

2.5E+04

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Strain

S

tre

ss

Fig. 4. The stress-strain curves for Si nanowires with different diameter are plotted. The curves in the red dotted square are enlarged in the right figure.

in which the stress changes linearly to the strain. In other words, we had chosen the

region that the strain is less than 0.1 (i.e. 10%). Young’s modulus has a small difference

for different diameter of the nanowires. But this fluctuation can be well accounted for by

the uncertainty in determining the diameter of the nanowire (due to the discreteness of

crystal structure). The diameter does not have a big effect on the Young’s modulus in

agreement with Liang et al.ix We find that the average Young’s modulus over the

different diameter is 122.531GPa.

ii) Silicon by Tersoff Potential

Young’s moduli are 135.769GPa for 5nm, 125.763GPa for 10nm, 119.723GPa for 15nm,

and 119.047GPa for 20nm. The average value is 125.076GPa

iii) Germanium by Stillinger-Weber Potential

Si

0.0E+00

5.0E+03

1.0E+04

1.5E+04

2.0E+04

2.5E+04

3.0E+04

0.000 0.100 0.200 0.300 0.400

Strain

Stress

Si

0.0E+00

2.0E+03

4.0E+03

6.0E+03

8.0E+03

1.0E+04

1.2E+04

1.4E+04

0.000 0.050 0.100 0.150

Strain

Stress

5nm

10nm

15nm

20nm

Fig. 5. The stress-strain curves for Si nanowires with different diameter are plotted using Tersoff potential.

Ge

0.00E+00

5.00E+03

1.00E+04

1.50E+04

2.00E+04

2.50E+04

0 0.1 0.2 0.3 0.4

Strain

Stress

Ge

0.00E+00

2.00E+03

4.00E+03

6.00E+03

8.00E+03

1.00E+04

1.20E+04

0 0.05 0.1 0.15

Strain

Stress

5nm

10nm

15nm

20nm

Fig. 6. The stress-strain curves for Ge nanowires with different diameter are plotted using SW potential.

Young’s moduli are 112.415GPa for 5nm, 105.117GPa for 10nm, 108.669GPa for 15nm,

and 108.669GPa for 20nm. The average value is 108.718GPa.

iv) Germanium by Tersoff Potential

Young’s moduli are 125.517GPa for 5nm, 127.040GPa for 10nm, 126.034GPa for 15nm,

and 120.326 for 20nm. The average value is 124.729GPa.

v) Si-Ge Coaxial by Tersoff Potential

Young’s moduli are 126.388GPa for the Si-shell/Ge-core and 126.066GPa for Ge-

shell/Si-core.

In summary, the Young’s modulus is about 123GPa by SW potential and 125GPa by

Tersoff potential for Silicon and 109GPa by SW potential and 125GPa by Tersoff

potential for Germanium. In Germanium case, the value is a little different between the

Core-Shell

0.0E+00

5.0E+03

1.0E+04

1.5E+04

2.0E+04

2.5E+04

0.000 0.100 0.200 0.300 0.400

Strain

Stress

Core-Shell

0.0E+00

2.0E+03

4.0E+03

6.0E+03

8.0E+03

1.0E+04

1.2E+04

1.4E+04

0.000 0.050 0.100 0.150

Strain

Stress

Ge core

Si core

Fig. 8. The stress-strain curves for Si-Ge core-shell nanowires with different diameter are plotted.

Ge

0.0E+00

5.0E+03

1.0E+04

1.5E+04

2.0E+04

2.5E+04

0.000 0.100 0.200 0.300 0.400

Strain

Stress

Ge

0.0E+00

2.0E+03

4.0E+03

6.0E+03

8.0E+03

1.0E+04

1.2E+04

1.4E+04

0.000 0.050 0.100 0.150

Strain

stress

5nm

10nm

15nm

20nm

Fig. 7. The stress-strain curves for Ge nanowires with different diameter are plotted using Tersoff potential.

one by SW potential and one by Tersoff potential. But Young’s modulus is obtained by

just selecting the region in which the stress changes linearly by the strain and finding the

slope simply. Considering how roughly the Young’s modulus was obtained, such

difference in Germanium is acceptable. This Young’s modulus seems quite reasonable

compared with the experiment resultx considering that the experiment was done under

300K and this simulation was done under 0K.

To see the qualitative comparison, we derived a simple formula of mixing law to get the

Young’s modulus of the coaxial nanowire. The detailed derivation is followed in

Appendix C.

2 21 1 2 2 1 2( / ) (1 ( / ) )E E d d E d d= + −

We apply the result from the simulation by Tersoff potential, because the coaxial Si-Ge

nanowire was computed by Tersoff potential. Young’s Modulus is 125.076GPa for Si and

124.729GPa for Ge. And 1d =5nm and 2d =15nm. Then, in case of Si-core/Ge-shell,

768.124))3/1(1(729.124)3/1(076.125 22=−+=E

This value agrees to the simulation result for coaxial Si-Ge, 126GPa.

In this simulation, we used conjugate gradient method which takes into account

thermal fluctuations. Hence, we simulated the material at 0K. However, the experiments

are always done at finite temperature such as 300K. So we need to simulate the material

using MD and this would be future work. The future step for this work would be to study

the deformation strength of these nanowires using molecular dynamics simulation.

2) Deformation Analysis

Fig. 9. (a) Si (d=15nm) at 20% strain highlighted with red above -3.8eV (b) Ge (d=15nm) at 20 % strain highlighted with red above –2.8eV.

13nm

15.2nm

(a) (b)

14nm

15.2nm

While the nanowire expands along z direction, it shrinks along x and y directions due

to its finite Poisson ratio. Because of elastic anisotropy, the shrinkage ratio along x and y

direction are different in fig. 10. It seems that the dimension along y direction remains

constant or increases during tension and most of contraction occurred in x direction. And

Si shrinks more in x direction than Ge at the same strain. And this difference looks like a

strong candidate for the generation of voids at the interface between Si and Ge at high

strain when Si is at the core and Ge is at the shell.

In case of single component nanowire, all the atoms start with high energy when high

strain, e.g. 30% strain, is initially loaded. During the CGR method, the high energy are

released through x direction and finally the fracture initiated from atoms at both ends of

major axis which still have high energy and propagated into the inner region. This

tendency gets stronger as the diameter decreases and it is stronger in Si than in Ge.

Fig. 10. (a) Si shell – Ge core nanowire broken at 30% strain highlighted with red above -2 eV

(b) Ge shell – Si core nanowire at 35% strain, highlighted above -2.96eV

5. Conclusion The Young’s modulus we obtained from the series of molecular simulations does not

show appreciable difference according to the potentials we used and the size of Si/Ge

nanowires. And those values are within the range of Young’s modulus for the bulk Si/Ge

material at 300K. And in the coaxial Si-Ge case, the result from the simulation is very

similar to the expected value using a simple mixing value. To see the temperature

dependence of Young’s modulus, our future work would be to perform MD simulation

with a finite temperature.

Due to the different value of the Poisson’s ratio along the different direction, the void

occurs at the interface between coaxial Si-Ge. The fracture, initiated at the interface, can

transmit to inward or outward in case of Si-shell/Ge-core or Ge-shell/Si-core respectively.

More rigorous analysis for these simulation result will be performed in the future.

Appendix

A. Tersoff Potential

Tersoff Potential is a many-body potential, although it looks like a modified 2 body

potential.(Ref. vi)

∑∑≠ ji

ij

i

i V=E=E2

1

where ( ) ( ) ( )[ ]ijAijijRijCij rfb+rfrf=V : bonding energy

and ijr : the distance between the atoms i and j

( ) ( )λrA=rfR −exp : repulsive pair potential

( ) ( )µrB=rfA −− exp : attractive pair potential

( ) ( ) ( )[ ]

−−

ij

ijijijijij

ij

C

S>r

S<r<R,RSRrπ+

R<r

=rf

0,

/cos2

1

2

1

1,

: smooth cutoff functions

( ) inn

ij

n

i

ij

ij

ζβ+

χ=b

2/11

: dependence

( ) ( )ijkikik

ji,k

Cij θgωrf=ζ ∑≠

: effective coordination number of atom i, taken into account the

relative distance of two neighbors r ij r ik and the bond-angle θ .

( )( )22

2

2

2

cos1

ijkii

i

i

iijk

θh+d

c

d

c+=θg

−−

( )θg has a minimum when θ=h cos , d determines how sharp the dependence on angle is, and c shows the strength of the angular effect.

C Si Ge

A (eV) 1393.6 1830.8 1769.0

B (eV) 346.7 471.2 419.230

λ (Å-1) 3.4879 2.4799 2.4451

µ (Å-1) 2.2119 1.7322 1.7047

β 1.5724E-007 1.1000E-006 9.0166E-007

C Si Ge

n 0.72751 0.78734 0.75627

c 3.8049E+004 1.0039E+005 1.0643E+005

d 4.384 16.217 15.6520

h -0.57058 -0.59825 -0.43884

R (Å) 1.8 2.7 2.8

S (Å) 2.1 3.0 3.1

0.9976=χ SiC−

1.00061Ge =χSi−

B. Stillinger-Weber (SW) Potential (Ref. iv)

( ) ( ) ( )jkikij

kj,i,

ij

ji,

r,r,rV+rV=N,Φ ∑∑ 322

11,2,L

where 2-body term ( ) ( ) ( ) ( )[ ]rφArφArf=rV c 212 − and

( )

≥

−

c

c

cc

Rr

R<rRr

µ

=rf

0

exp : cut-off function

( ) sλ

s r=rφ−

s = 1, 2

3-body term ( ) ( ) ( ) ( )iikijjkikij θgrψrZψ=r,r,rV3 ,

( ) ( )[ ]αc rf=rψ and ( ) ( )20coscos θθ=θg ii −

Si Si in md++ Ge Ge in md++

A1 189.360881 bb=A1/aa=11.60319226 185.3959382 bb=A1/aa=11.60319226

A2 16.31972277 aa=16.31972277 13.60564361 aa=13.60564361

1λ 4 rho=4 4 rho=4

2λ 0 0

aσ=RC (Å) 3.77118 acut=3.77118 3.9258 acut=3.9258

ελ=Z 48.61499998 plam=48.61499998 59.83 plam=59.83

α 1.2 pgam=2.51412 1.2 pgam=2.6172

µ (Å) 2.0951 pss=2.0951 2.181 pss=2.181

0θ (degree) 109.471221 109.471221

2.7

3.0

C. Effective Young’s modulus of coaxial material under tension d2 d1 Consider two materials 1 and 2 in a core-shell structure with Young’s moduli of E1 and E2, as chosen above. The total tensile force is f and the two material contribute f1 and f2 respectively. f A E Aσ ε= = , 1 1 1 1 1 1f A E Aσ ε= = , 2 2 2 2 2 2f A E Aσ ε= = To obtain Young’s modulus, we focus on the region in which the stress changes linearly

within the strain. So we can assume the strain is so small that the area change is

negligible. Therefore, the area for each element is as followed.

21 1 / 4A dπ= , 2 2

2 2 1( ) / 4A d dπ= − , 22 / 4A dπ=

In a tension test, both materials are deformed by the same strain, i.e.

1 2ε ε ε= = Then, 2 2 2

1 2 1 1 2 2 1 1 1 2 2 2 1/ 4 ( ) / 4f f f A A E d E d dσ σ ε π ε π= + = + = + −

2 2 2 22 1 1 2 2 1/ 4 / 4 ( ) / 4E d E d E d dεπ επ επ⇒ = + −

2 21 1 2 2 1 2( / ) (1 ( / ) )E E d d E d d⇒ = + −

Reference i L.J. Lauhon et al., Nature 420 (2002) 57 ii Y. Wu et al., Nano Letters 2(2002) 83 iii Y. Cui et al., Science 198 (2001) 1289 iv H. Balamane et al, Phys. Rev. B 46(1992) 2250 v K. Ding et al, Phys. Rev. B 34(1986) 6987 vi J. Tersoff, Phys. Rev. B 39(1989) 5566 vii MD++ simulation package, Wei Cai.

http://micro.stanford.edu/~caiwei/MD++ viii J. Li, Modelling Simul. Mater. Sci. Eng. 11 (2003) 173 ix W. Liang et al., Nanotech 2 (2003) 452 x J. J. Wortman and R. A. Evans, J. Appl. Phys. 36, 153 (1965).