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Thin Solid Films 492

Molecular dynamics study on size-dependent elastic properties of

silicon nanocantilevers

S.H. Park a,*, J.S. Kim a, J.H. Park a, J.S. Lee b, Y.K. Choi c, O.M. Kwon d

aDepartment of Mechanical and System Design Engineering, Hongik University, Seoul, 121-791 KoreabSchool of Mechanical and Aerospace Engineering, Seoul National University, Seoul, 151-742 Korea

cSchool of Mechanical Engineering, Chung-Ang University, Seoul, 156-756 KoreadSchool of Mechanical Engineering, Korea University, Seoul, 136-701 Korea

Received 27 May 2004; received in revised form 13 May 2005; accepted 22 June 2005

Available online 28 July 2005

Abstract

The motion of nanoscale structures made of pure crystalline silicon with different lattice conditions is simulated in vacuum by applying

the molecular dynamics technique with the use of the Tersoff potential. Elastic moduli for various sized specimens are obtained by simulating

flexural and longitudinal vibrations as well as simple tension tests. Compared with the bulk silicon, the elastic modulus decreases

monotonically by as much as 40% as the thickness of the specimen decreases, and the presence of voids in the specimen further decreases the

modulus by a significant amount. Estimation of thermal fluctuations and feasibility study of nanoscale cantilevers as molecular mass sensors

demonstrate that the continuum-theory-based analysis can still be used on nanoscale structures provided the dependence of the elastic

constants on dimensional scaling is accounted for.

D 2005 Elsevier B.V. All rights reserved.

PACS: 68.60.B; 62.20.F; 71.15.D

Keywords: Elastic properties; Nanostructures; Sensors; Silicon

1. Introduction

Nanoscale materials and structures have recently

gained considerable interest as sensors and devices. In

particular, the widespread availability of micro- and

nanoscale structures such as atomic force microscopes

has resulted in renewed interest in cantilever systems as

chemical, physical and biological sensors [1–3]. Apart

from small size, a number of unique properties arise from

relatively high surface area in nanoscale structures. For

accurate sensing, it is crucial to be able to evaluate the

elastic properties at different dimensional scales. The

present study investigates the feasibility of simulations

based on molecular modeling for evaluating the elastic

properties of pure crystalline silicon structures with

nanoscale dimensions.

0040-6090/$ - see front matter D 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.tsf.2005.06.056

* Corresponding author. Tel.: +82 2 320 1632; fax: +82 2 322 7003.

E-mail address: spark@wow.hongik.ac.kr (S.H. Park).

A number of efforts to evaluate the elastic properties

using molecular dynamics (MD) approach have been

reported. One approach employs a simple tension test to

statically determine the elastic properties of single-crystal

metals and semiconductors [4,5]. Another approach is to use

structural resonant responses. Experimental as well as

theoretical investigations of the dynamic behavior of ultra-

thin micro- and nano-structures have been carried out.

Studies have reported on the use of resonance variation in

detecting the presence of external force or mass [6,7], and a

significant size effect on the elastic modulus has been

reported [8,9]. Of particular interest in nanoscale structures

has been predicting the effect of thermal noise on the

cantilever flexural response [10–12].

In the present work, both the static extension and

resonant frequency responses are employed to determine

the elastic modulus of nanoscale structures by constructing

molecular simulation models. The sensitivity of the elastic

modulus on the number of voids is also investigated. Where

(2005) 285 – 289

Table 1

The Tersoff parameters for silicon

Parameters Tersoff (Si)

Aij 1830.8 eV

Bij 471.18 eV

k ij 0.24799 nm�1

�1

S.H. Park et al. / Thin Solid Films 492 (2005) 285–289286

possible, the simulation results are verified by comparing

with theoretical calculations based on the continuum model.

The elastic constants obtained are then applied to the

analysis of thermal fluctuations, and the applicability of

nanoscale structures as molecular mass sensors is also

demonstrated.

l ij 0.17322 nm

v ij 1.0

b ij 0.0000010999

n 0.78734

ci 100390

di 16.217

hi �0.59826

Rij 0.27 nm

Sij 0.30 nm

2. Molecular dynamics simulation

A classical MD simulation model for cantilever struc-

tures made of crystalline silicon is shown in Fig. 1. The

diamond structure of silicon can be constructed with the

help of the Tersoff three-body potential [13,14] that

incorporates the cut-off function into Morse potential for

two-body system and modifies the attractive term to account

for three-body interactions. The parameter values in the

Tersoff potential function for silicon are given in Table 1,

where subscript ij denotes neighboring silicon atoms i and j.

Atomic motion can be obtained from the Newton’s

equations of motion through a numerical solution that uses

the Gear Predictor–Corrector algorithm [15] with a time

step of 2 fs. The average temperature of the structure is

maintained at 300 K by the momentum scaling method. To

ensure consistency, the following procedure is applied for

all simulation runs: (i) placement of atoms in the cantilever,

based on the diamond lattice structure, (ii) equilibration of

the crystalline silicon at 300 K for a sufficient period (105

time steps), (iii) estimation of the properties at the same

temperature by processing data for time steps ranging from

2�105 to 5�105.

The elastic properties can be obtained by static simu-

lation runs in which an external tensile force is imposed on

the atoms at the end regions of cantilevers in the axial, i.e.,

z-direction and relative elongation in the same direction is

estimated. During tensile simulations the periodic boundary

conditions are applied in the x- (i.e., the width) direction and

the calculation domains in the y- and z- (i.e., the thickness

and length) directions are maintained at three times the sizes

of the cantilever in the respective directions. Each of the

Fig. 1. Molecular models for (a) static and (b) dynamic estimations of the

elastic modulus of crystalline silicon nanocantilevers.

lattice directions is aligned with the z-axis. The elastic

modulus can be obtained by applying a generalized Hooke’s

law of the form r =EVe where r, EVand e denote the stress,

elastic modulus for the periodic boundary condition in the x-

direction, and strain, respectively. The elastic modulus EVobtained in this manner can be expected to be greater than

the Young’s modulus EV, which denotes the elastic modulus

for the free boundary conditions (under the normal stress of

1 atm to be exact) in both the x- and y- (i.e., the width and

height) directions. For cubic structures such as silicon, there

exist three independent stiffness coefficients and EV(or EVforthat matter) can be calculated from those three stiffness

coefficients [16]. The following expressions can be derived

for EV in <100>, <110> and <111> crystallographic

directions, respectively:

E V<100>

¼ C11 1� C12

C11

�� 2#"

ð1Þ

E V<110>

¼ 1

2C11 þ C12 þ 2C44 � 2

C212

C11

��ð2Þ

E V<111>

¼ 1

3C11 þ 2C12 þ 4C44ð Þ

� 2

9

C11 þ 2C12 � 2C44ð Þ2

C11 þ C12 þ 2C44

ð3Þ

From the published data on the bulk stiffness coefficients

using the Tersoff potential [13], EVfor the bulk silicon have

been calculated to be 107.3, 142.3, 179.3 GPa in <100>,

<110> and <111> crystallographic directions, respectively.

Using equations similar to (1)–(3), the corresponding E

have been calculated to be 94.3, 142.3, 171.3 GPa,

respectively.

3. Results and discussion

As shown in Fig. 1(a), the elastic modulus can be

calculated from a simple tension via a generalized Hook’s

0 1 2 3 4 5 6 710-8

10-7

10-6

10-5

10-4

5th

4th3rd

2nd

x1011

0 1 2 3 4 5 6 7x1011

1st mode

MA

GN

ITU

DE

FREQUENCY [Hz]

(a)

10-9

10-8

10-7

10-6

10-5

10-4

2nd1st mode

MA

GN

ITU

DE

FREQUENCY [Hz]

(b)

Fig. 2. Resonant frequencies of (a) longitudinal and (b) flexural vibrations

of nanocantilevers.

1 2 3 4

1 2 3 4

50

100

150

200FOR BULK

STRUCTURE

<111>

<110>

<100>

EL

AS

TIC

MO

DU

LU

S [

GP

a]

THICKNESS [nm]

(a)

DIRECTION <100> <110> <111> TENSILE FLEXURALLONGITUDINAL

50

100

150

200

<111>

<100>

<110>

FOR BULKSTRUCTUREDIRECTION

<100> <110> <111>

YO

UN

G'S

MO

DU

LU

S [

GP

a]

THICKNESS [nm]

(b)

Fig. 3. Elastic modulus with respect to thickness for three crystallographic

directions: (a) with PBC in x-direction and (b) without PBC (Young’s

modulus).

S.H. Park et al. / Thin Solid Films 492 (2005) 285–289 287

Law. An alternative way of obtaining the modulus is to

induce flexural or longitudinal vibration (Fig. 1(b)) and

measure the natural frequencies of the vibration. For flexural

vibration analysis, the Euler–Bernoulli beam model [17]

can still be applied using EVinstead of E with the governing

equation given by

B2w z; tð ÞBt2

þ c2B4w z; tð ÞBz4

¼ 0; c ¼

ffiffiffiffiffiffiffiffiE VI

qA

sð4Þ

subject to clamped-free boundary conditions. Solving the

equation, the natural frequencies can be calculated from

fn ¼k2n2kl2

ffiffiffiffiffiffiffiEVI

qA

sð5Þ

where fn, EV, I, q, and A denote the natural frequency in

hertz, elastic modulus for the periodic boundary condition

case, moment of inertia, mass density, and cross-sectional

area, respectively, and kngbnl satisfies the characteristic

equation,

cosbnlIcoshbnl ¼ � 1 ð6Þ

For the longitudinal vibration, the governing equation

[17] is given by

EV

qB2w z; tð ÞBt2

þ B2w z; tð ÞBt2

¼ 0 ð7Þ

subject to clamped-free boundary conditions. The natural

frequencies can be computed from

fn ¼2n� 1

4l

ffiffiffiffiffiEV

q

sn ¼ 1ð Þ ð8Þ

By measuring the natural frequencies from the frequency

domain display of simulated motion of the specimen such as

given in Fig. 2 and knowing the specimen parameters I, q,and A, the corresponding EVcan be calculated.

Fig. 3 depicts the change in the elastic modulus for

various cantilever thickness. For each crystallographic

direction, EVand E are plotted as a function of the thickness

in Fig. 3(a) and (b), respectively. Notice that for the given

thickness of the cantilever, EVis always greater than E. For

all three crystallographic directions, both EVand E increase

monotonically as the thickness is increased. Also, the

modulus values obtained by applying the three different

simulation testing methods are in close agreement. Here, the

base width and length of the cantilever are 2.172 and 9.774

nm, respectively.

The results in Fig. 3 show that the elastic modulus for

perfect-crystalline silicon nanostructures has decreased from

the corresponding bulk modulus by as much as 40%. In the

0.0 0.5 1.0 1.565

70

75

80

85

EL

AS

TIC

MO

DU

LU

S [

GP

a]

VACANCY [%]

TENSILE FLEXURAL LONGITUDINAL

Fig. 4. Elastic modulus with respect to vacancy for <100> crystallographic

direction, obtained by MD simulation of extensional and vibration

responses.

S.H. Park et al. / Thin Solid Films 492 (2005) 285–289288

presence of defects, these values can be expected to

diminish even further. To examine this issue, the depend-

ence of EV(i.e., with one PBC) on point defects (vacancies) issimulated. As shown in Fig. 4, the presence of defects can

significantly affect the elastic property. We have created

models with various defect distributions, and have found

that EVdepends primarily on the total number of vacancies

present in the structure, and the effect of their spatial

distribution is at best secondary. And again, the three

different testing methods are in close agreement. Here, the

base width, length, and thickness of the cantilever are 2.172,

9.774, and 1.089 nm, respectively.

Thermal fluctuations constitute one fundamental source

of vibration in nanoscale sensors such as scanning probe

microscopes, and represent an important design consid-

eration. The integral of the thermal fluctuation over all

frequencies corresponds to an energy per degree of

freedom, 1 /2kT, which would be expected from the

equipartition theorem [10,18]. The equipartition theorem

states that if a system is in thermal equilibrium every

independent quadratic term in its total energy has a mean

value equal to 1 /2kT, where k is the Boltzmann constant

and T is the absolute temperature. For small flexural

motion, the potential energy of the cantilever is 1 /2Ky2,

1.0 1.5 2.0 2.5 3.0

0.1

0.2

0.3

TH

ER

MA

L N

OIS

E [

nm

]

THICKNESS [nm]

DIRECTIONMDTHEORY <100> <110> <111>

Fig. 5. Comparisons between theory and MD simulation of thermally

induced vibration amplitudes with respect to thickness for three crystallo-

graphic directions.

where K denotes the spring constant and y the vertical

deflection at the free end. Equating the two expressions,

we can write

1

2kT ¼ 1

2Ky2: ð9Þ

Here, the spring constant for the cantilever is given as

[18]

K ¼ 3E Vl

l3: ð10Þ

Rearranging Eq. (9) into the following form yields a

theoretical estimate of the amplitude of the thermal

fluctuations in the vertical direction at the free end.

ffiffiffiffiffiy2

p¼

ffiffiffiffiffiffiffikT

K

rð11Þ

For different thickness, thermal fluctuations are calcu-

lated using Eq. (11), in which the EVvalues of Fig. 3 have

been applied. The results are compared in Fig. 5 and good

agreement between the theory and MD simulations is

observed. To further examine the accuracy of the simulation

method, the dependence of thermal fluctuations on temper-

ature for <100> direction is shown in Fig. 6 for the same

specimen used in Fig. 5, with the thickness fixed at 2.172

nm. Also plotted is the corresponding EVat each temperature

needed to compute thermal fluctuations. For each temper-

ature, EV has been obtained by a tensile test simulation.

Again, close agreements between the theoretical and MD

simulation results are attained.

For sensing the presence of additional mass on the

cantilever, changes in the flexural vibration characteristics is

the most sensitive of the three methods employed in the

present study. Additional mass placed at the free end causes

the resonant frequencies of the flexural modes to decrease.

The effect of the concentrated mass on the natural frequency

200 300 400 500

0.06

0.08

0.10

0.12THEORY MD

TEMPERATURE [K]

TH

ER

MA

L N

OIS

E [

nm

]

80

85

90

95

100 EL

AS

TIC

MO

DU

LU

S [G

Pa]

Fig. 6. Comparisons between theory and MD simulation of thermally

induced vibration amplitudes with respect to temperature for <100>

direction with thickness of 2.172 nm together with the corresponding E V.

(a)

(b)

0 20 40 60 809

10

1121

22

23

1st

MO

DE

FR

EQ

UE

NC

Y [

GH

z]

NUMBER OF ADDED ATOMS

THEORY MD

THICKNESS: 1.086 nm

THICKNESS: 2.172 nm

Clamped Part Free Part

FlexuralVibration

Added Atoms

L

Fig. 7. (a) Molecular model for mass-loading detection and (b) shifts in the

first flexural resonant frequency due to added silicon atoms: comparison

between theory and MD simulation.

S.H. Park et al. / Thin Solid Films 492 (2005) 285–289 289

of the first flexural mode of the Euler–Bernoulli cantilever

beam can be calculated by [17]

f1 ¼1

2k

ffiffiffiffiffiffiffiffiffiK

meff

r; ð12Þ

meff ¼ mc þ 0:2427md ð13Þ

where K is given by Eq. (10), md is the cantilever mass, and

mc the concentrated mass (i.e., the additional mass). To

investigate the feasibility of the nanoscale cantilevers as

molecular mass sensors, silicon molecules are placed on top

of the cantilevers at the free end and the change in the first

flexural resonant frequency is simulated. Fig. 7 shows the

frequency as a function of the number of silicon atoms

placed on the cantilever. Here, the base width and the length

of the cantilever are 2.172 and 9.774 nm, respectively. The

simulation results and theoretical estimates based on Eqs.

(12) and (13) are in close agreement, with the thinner

structure showing superior sensitivity to the molecular mass

present.

4. Conclusions

The MD simulation based on the Tersoff potential model

has been conducted to investigate the size-dependence of

the elastic properties of the nano-scale crystalline silicon

cantilevers.

Three different estimation methods yield results that are

in close agreement and demonstrate strong dependence of

the elastic properties on dimensional scaling. The elastic

modulus decreases monotonically as the thickness of the

specimen decreases, and the presence of voids in the

specimen further decreases the modulus by a significant

amount.

Estimations of thermal noise and frequency shift due to

mass-loading demonstrate that the analysis method based on

the continuum mechanics modeling can still be used on

nanoscale structures provided the dependence of elastic

constants on dimensional scaling is accounted for.

Acknowledgments

The authors gratefully acknowledge the financial support

from the Micro Thermal System Research Center sponsored

by the Korean Science and Engineering Foundation.

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