amorphous silicon vdos

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Fitting the StillingerWeber potential to amorphous silicon

R.L.C. Vink a,*, G.T. Barkema b, W.F. van der Weg c, Normand Mousseau d

a Instituut Fysische Informatica, Utrecht University, P.O. Box 80195, 3508 TD Utrecht, The Netherlandsb Institute of Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands

c Debye Institute, Utrecht University, PO Box 80000, 3508 CA Utrecht, The Netherlandsd Department of Physics and Astronomy and CMSS, Ohio University, Athens, OH 45701, USA

Received 24 April 2000; received in revised form 26 October 2000

Abstract

Modications are proposed to the StillingerWeber (SW) potential, an empirical interaction potential for silicon.

The modications are specically intended to improve the description of the amorphous phase and are obtained by a

direct t to the amorphous structure. The potential is adjusted to reproduce the location of the transverse optic (TO)

and transverse acoustic (TA) peaks of the vibrational density of states (VDOS), properties insensitive to the details of

experimental preparation. These modications also lead to excellent agreement with structural properties. Comparison

with other empirical potentials shows that amorphous silicon congurations generated with the modied potential have

overall better vibrational and structural properties. 2001 Elsevier Science B.V. All rights reserved.

1. Introduction

In the past, various empirical potentials [19]

have been proposed to describe the dierent

phases of silicon. Several of these empirical po-

tentials [24,6] were tested and compared to each

other in review articles [1012]. The general con-

clusion of these reviews is that it is dicult to re-

produce accurately with a single empirical

potential the crystalline, amorphous and liquid

phases of silicon. Since atomic congurations are

readily obtained in the liquid and crystalline

phases of silicon, their structural and electronic

properties are typically used as input for tting the

parameters of the potential, with a consequence

that the amorphous phase is generally less well

reproduced.

Recent advances in the method of simulation

have made it possible to generate eciently well-

relaxed congurations of a-Si that are insensitive

to their history of preparation [13]. This allows us

to t a potential directly to the amorphous phase.

The aim of this paper is to present a new tting

procedure and use it to generate such an empirical

potential. This potential should accurately repro-

duce the structural and elastic properties of the

amorphous phase of silicon and serve as a basis for

direct comparison to experiments. To this end, we

start with the StillingerWeber (SW) potential [3]

and modify some of its parameters in order to

achieve good agreement with experimental data.

More specically, we concentrate on the location

of the transverse acoustic (TA) and transverse

optic (TO) peaks of the vibrational density of

states (VDOS) quantities that are found to be

Journal of Non-Crystalline Solids 282 (2001) 248255

www.elsevier.com/locate/jnoncrysol

* Corresponding author. Tel.: +31-30 253 3880; fax: +31-30

253 7555.

E-mail address: [email protected] (R.L.C. Vink).

0022-3093/01/$ - see front matter

2001 Elsevier Science B.V. All rights reserved.PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 3 4 2 - 8


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relatively insensitive to the experimental method of

preparation as well as on a number of structural

properties such as the radial distribution function

(RDF) and the bond-angle distribution. This ap-

proach is similar in spirit to the one followed by

Ding and Andersen for the generation of an em-

pirical potential for a-Ge [14]. As it turns out,

within our tting procedure, a correct description

of the positions of the TA and TO peaks is su-

cient to ensure good structural properties.

The outline of this paper is as follows. We

present the SW potential in Section 2 and outline

the procedure followed in the preparation of the

amorphous congurations in Section 3. Based on

these congurations, we adjust the parameters of

the potential to obtain the experimental positionsof the TA and TO peaks of the VDOS (Section 4).

Finally, the structural and elastic properties of the

modied potential are compared to experiment, to

the standard SW potential and the environment-

dependent interaction potential (EDIP) of Bazant

et al. [79].

2. The SW potential

Stillinger and Weber [3] proposed a potentialfor silicon with a two and three-body part:

V AXhiji

v2ij rij

"k

A

Xhjiki

v3jikrij; rik

#; 1

where the two-body part is given by

v2ij rij B

rij

r

ph 1

iexp

1

rij=r a

Ha rij=r; 2

and the three-body part by

v3jikrij; rik exp

c

rij=r a

c

rik=r a

cos hjik cos h

02Ha rij=r

Ha rik=r: 3

Here, Hx is the Heaviside step function and h0

the tetrahedral angle with cos h0 1=3. The

values of the original parameters are given in

Table 1.

The original SW parameters are tted to thecrystalline and liquid phases of silicon. For a-Si,

these parameters lead to a high-density liquid-like

structure which is characterized by a shoulder on

the second-neighbor peak of the RDF and a large

fraction of overcoordinated atoms [13,15]. A

common ad hoc modication to alleviate these

problems is to boost the three-body term by 50

100% (see, for instance, [13,15]). This modication

yields samples with a RDF in agreement with ex-

periments.

3. Generation of sample congurations

To generate amorphous congurations, we

proceed using the activationrelaxation technique

(ART) as in [13,16,17] and a SW potential with a

three-body interaction enhanced by 50%. This

follows a common ad hoc prescription to avoid the

large number of over-coordinated atoms produced

by the standard SW potential: boosting the three-

body term by 50100% generally yields samples

with structural properties in good agreement with

experiment [13,15].

The overall procedure is as follows:

1. Initially, 1000 atoms are placed at random in a

periodic cubic cell; the conguration is then re-

laxed at zero pressure.

2. The conguration is annealed using the ART.

One ART move consists of two steps: (1) the

sample is brought from a local energy minimum

to a nearby saddle-point (activation), and (2)

Table 1

Values of the parameters in the standard SW and the modied

SW potentials

Parameter Standard SW Modied SW

(eV) 2.16826 1.64833A 7.049556277 7.049556277

B 0.6022245584 0.6022245584

r A 2.0951 2.0951p 4 4

a 1.80 1.80

k 21.0 31.5

c 1.20 1.20

R.L.C. Vink et al. / Journal of Non-Crystalline Solids 282 (2001) 248255 249


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then relaxed to a new minimum with a local en-

ergy minimization scheme including volume op-

timization, at zero pressure. The new minimum

is accepted with a Metropolis probability at

temperature T 0:25 eV.

The annealing phase is stopped after approxi-

mately ve ART moves per atom, when the

energy has reached a plateau.

3. We continue to apply ART, storing one sample

every 250 ART moves, collecting a total of 20

congurations.

This procedure is repeated seven times, generating

seven statistically independent sets of 20 correlated

congurations. As was shown in previous studies

[13,16], this approach yields structures in goodagreement with experiment: the models display a

low density of coordination defects, a narrow

bond-angle distribution and an excellent overlap

with experimental RDF.

With this set of models, it is now possible to

ret the SW parameters to the amorphous phase.

4. The modied potential

Many properties of the interaction betweensilicon atoms are already captured by existing

empirical potentials such as the SW potential. It

should therefore be possible to optimize these

potentials to the amorphous phase with only mi-

nor modications.

Information on structural and elastic properties

of amorphous silicon can be obtained from a wide

variety of experiments, such as neutron and X-ray

scattering, Raman, electron-spin resonance, and

X-ray photo-absorption. Extracting a detailed

picture of the atomic interactions from these ex-

periments, however, is not straightforward. Most

of these measurements do not provide direct mi-

croscopic information but rather properties aver-

aged over a large number of atomic environments;

others are highly sensitive to the details of the

sample preparation, and can thus not be used,

since the experimental sample preparation cannot

be directly simulated.

A convenient quantity for our purpose is the

VDOS, especially the locations of the TA and TO

peaks, which are not sensitive to experimental

details and yet are highly sensitive to properties of

the potential used since they relate to higher terms

in the elasticity. In contrast to their positions, the

amplitudes of these peaks vary between experi-

ments [27].

To rst order, vibrational modes located in the

TA band correspond to bond-stretching modes

while those in the TO regime are associated with

bond-bending, or bond-angle, vibrations. Due to

the disordered nature of these materials, a signi-

cant amount of mixing makes the details of the

contributions to these peak more subtle in reality.

In the SW potential, the amount of energy re-

quired to stretch a bond is determined by the

strength of the two-body term (i.e. by A in Eq.(1)). Bond-bending motion is determined by the

strength of the three-body term (i.e. by k in Eq.(1)). By adjusting the two prefactors A and k, wecan t the locations of both the TA and TO peak

in the calculated VDOS to experimental values. Of

the three parameters A; , and k, one is redundant;it is sucient to vary k to set the relative three-

body strength and to choose to set the overallenergy scale.

Our rst aim is to adjust the ratio of the loca-

tions of the TA and TO peaks by tuning k. Fork=k0 1:00; 1:25; 1:50; 1:75 and 2.00, with k0 21:0, we follow the following procedure:1. All samples are re-relaxed at zero pressure us-

ing the steepest descent method, and the SW

potential with the original parameters (except-

ing k).

2. For each sample, the Hessian matrix is calculat-

ed and diagonalized, resulting in the VDOS.

The positions of the TA and TO peaks are re-

corded.

Figs. 1(a) and (b) show the location of the TA and

TO peak, respectively, as a function ofk=k0. Whilethe location of the TA peak is highly sensitive to

changes in k, the location of the TO peak changes

less than 2%.

Neutron scattering experiments [18] on a-Si

yield TA and TO peaks at xTA 184:8 andxTO 467:6 cm

1, respectively, which gives a ra-

tio xTO=xTA 2:53. Fig. 1(c) shows this ratio forthe computer simulation, as a function of k=k0.The dashed line is the experimental value. From

250 R.L.C. Vink et al. / Journal of Non-Crystalline Solids 282 (2001) 248255


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Fig. 1(c), we see that the choice k=k0 1:5 pro-vides a t with the experimental result and that the

original choice k=k0 1:0 does not reproduce ex-periments, as expected from earlier reports [13,15].

We can follow this procedure because of the

overall insensitivity of the peak location to the

detail of the local structure, i.e., the value ofk used

in the initial relaxation of the seven independent

sets of samples does not inuence the curve of Fig.

1. Using a set of samples generated with values of

k 1:40k0 and k 2:00k0, we found results iden-tical to those presented in Fig. 1. This conrms

once more the independence of the ratio of the

positions of the peaks to the structural details of

the models.

After tting the three-body parameter k, we set

the overall energy scale parameter in Eq. (1) asfollows: Fig. 1(b) shows that, for k=k0 1:5,xTO 536:3 cm

1. The experimental value is

xTO 467:6 cm1, a factor 0.87 lower. To obtain

agreement with experiment, the parameter istherefore multiplied by 0:872 0:76. The squareappears because the frequencies in the VDOS are

proportional to the square-root of the eigenvalues

of the Hessian. Thus, we have scaled the strength

of the three-body interactions, governed by k, bya factor of 1.14, and the two-body interactions,

governed by A, by a factor of 0.76.The set of parameters that we thus propose for

amorphous silicon is presented in Table 1. As

mentioned above, the 50% increase in the ratio of

the three-body to the two-body term has already

been proposed earlier for a-Ge by Ding and An-

dersen [14] and by many others as an ad hoc way

to obtain a realistic structure. We obtain a similar

ratio following a dierent route: by tting the

vibrational properties of the model to experiments.

5. Measurements and comparison

We now compare the structural and elastic

properties of congurations generated with the

modied SW potential with experiment, as well as

with those of the standard version of this potential

and that of the EDIP of Bazant et al. [79], the

best empirical potential now for the crystalline and

liquid phase. The properties of other empirical

potentials, such as BiswasHamann [6] and Terso

[2], have been reviewed elsewhere [11] and shown

to reproduce poorly the experimental properties of

amorphous silicon.

To test these potentials, we generate a new,

1000-atom conguration following the method

outlined in Section 3 and using modied SW in-

teractions. The structural properties of this sample

are reported in Table 2 and Fig. 2 and are found to

be in agreement with experiment. In order to oer

Fig. 1. The TA peak position xTA (top) and the TO peak po-

sition xTO (middle), both in cm1, as a function of k=k0. The

bottom gure shows the ratio xTO xTA as a function ofk=k0obtained by computer simulation. The dashed line is the ex-

perimental value [18]. The best agreement with experiment oc-

curs for k=k0 1:5.



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a better comparison to the elastic and vibrational

properties of the three potentials, we use the nal

conguration presented here as a seed for ART

using standard SW and EDIP. With each of these

potentials, an additional 5000 ART moves are

generated; this allows the conguration to evolve

signicantly away from its starting point and

yields a structure optimized for that potential.

Table 2 lists energetic and structural properties

of these three samples. The list of nearest-neigh-bors is obtained by setting the cut-o distance at

the minimum between the rst- and second-

neighbor peaks in the RDF.

It is clear from Table 2 that modied SW gen-

erates the best structure, followed by EDIP, which

leads to a wider bond-angle distribution and a

higher number of defects (mostly over-coordinated

atoms). These two factors are important, since the

width of the bond-angle distribution is considered

a ne measure of the quality of a model and ex-

periments have shown no indication of overcoor-

dination in a-Si.

The top graph of Fig. 2 shows the RDF for the

conguration generated using modied SW inter-

actions, together with the experimental RDF ob-

tained from high quality a-Si samples prepared by

ion bombardment [24,25]. To our knowledge, this

experimental RDF has the highest resolution to

date. The middle and bottom graphs show the

RDF obtained using standard SW and EDIP in-

teractions, respectively.

The RDF generated with standard SW shows a

shoulder on the left side of the second neighbor

peak and thus fails to reproduce experiment; this

shortcoming of standard SW is well known. EDIP

reproduces the experimental RDF fairly well,

though not as well as modied SW does: there are

substantial discrepancies with experiment in the

heights of the rst and second neighbor peaks. The

RDF generated with modied SW interactions has

the best overall experimental agreement: it repro-duces the experimental rst and second neighbor

peak-heights and does not produce a shoulder on

the second neighbor peak. It does, however, fall

below the experimental curve on the left of the

second neighbor peak.

As an additional check, we determine the area

between the calculated and experimental RDF

curves. In case of perfect experimental agreement,

this area should be zero. We consider the interval

from r 2:0 to r 8:0 A since the uctuations inthe experimental RDF for r< 2:0 A are nite-sizeartefacts [24,25]. For modied SW, standard SW

and EDIP, we nd areas of 0.63, 0.67 and 0.84,

respectively, showing that modied SW is indeed

closest to experiment.

Fig. 3 compares the experimentally obtained

VDOS for a-Si to vibrational state densities ob-

tained using modied SW (top), standard SW

(middle) and EDIP (bottom). While standard SW

and EDIP correctly reproduce the experimental

TA peak frequency, the TO peak frequency

Table 2

Comparison of energetic and structural properties of a-Si obtained using SW, modied SW and EDIPa

Experiment SW Modied SW EDIP

q 0.0440.054 0.049 0.047 0.049

E )4.137 )3.072 )4.451

hri 2.342.36 2.38 2.37 2.37hhi 108.6 107.99 109.16 108.57Dh 9.411.0 15.08 10.02 13.68

3-fold 0.00 0.50 0.20

4-fold 83.50 98.00 91.90

5-fold 15.60 1.50 7.80

6-fold 0.90 0.00 0.10Z 3.903.97 4.17 4.03 4.08

a Also given are experimental values [1923], when available. Shown are the density q in atoms per A3, the energy per atom Ein eV, the

mean hri of the rst neighbor distance in A, the mean h and the standard deviationDh of the rst neighbor bond angle in degrees. Alsogiven are the coordination defects in % and the average coordination number Z.



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exceeds the experimental value. Naturally, the

modied SW potential does not suer from this

shortcoming, as it was explicitly tted to repro-

duce the experimental values.

Fig. 3 also shows that none of the potentials

capture the experimental shape of the VDOS. In

particular, there is large discrepancy between

simulation and experiment in the TA and TO peak

amplitudes. However, several studies [26,27] have

shown that the TA and TO peak amplitudes de-

pend on defect concentration and the degree of

relaxation in the a-Si samples. The most likely

explanation for this discrepancy is therefore that

the degree of relaxation in the experimental sample

diers from that of the computer generated sam-

ples used in our simulation.

As a nal test, we calculate the elastic constants

of a-Si using the various potentials. The results are

summarized and compared to experiment in Table

3. The experimental value for the bulk modulus B

is obtained from sound velocity measurements

[28], where the relation B qv2=31 2m was

Fig. 3. Vibrational state densities for a-Si. The solid curves in

the graphs show the VDOS obtained in computer simulations

using modied SW (top), standard SW (middle), and EDIP

(bottom). In each of the graphs, the dashed curve represents the

experimentally obtained VDOS [18]. Frequencies are in cm1,

the area of the VDOS is normalized.

Fig. 2. Radial distribution functions for a-Si. The solid curves

in the graphs show the RDF obtained in computer simulations

using modied SW (top), standard SW (middle), and EDIP

(bottom). In each of the graphs, the dashed curve represents the

experimentally obtained RDF [24,25]. Distances are in A.



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used, with Poisson's ratio m 0:22, sound velocityv and density q. To our knowledge, no experi-

mental values for the constants Cij for a-Si exist.

The bulk modulus for modied SW is signicantly

lower than for the other two potentials, in better

agreement with experiment.

6. Conclusions and future work

In this paper, we have presented a new set of

parameters for the SW potential, obtained by di-

rectly tting to the amorphous phase, and com-

pared the structural, elastic and vibrational

properties of congurations obtained with this

potential to those generated using the standard

SW potential and the EDIP, two common empir-ical potentials.

The modied SW potential was tted to re-

produce the locations of the TA and TO peak of

the experimental VDOS, quantities independent of

the method of preparation. This is achieved by

scaling k and by factors of 1.5 and 0.76, respec-tively, while keeping all other parameters xed.

Thus, we have scaled the strength of the three-

body interactions, governed by k, by a factor of1.14, and the two-body interactions, governed by

A, by a factor of 0.76.Without any additional tting, the modied SW

potential gives structures for a-Si in excellent

agreement with experimental results: the RDF is

correctly reproduced, and the number of coordi-

nation defects and angular deviations of the gen-

erated structures are in agreement with

experiment. Both the structural properties and the

bulk modulus obtained with this potential are in

better agreement with experiment than those gen-

erated using EDIP or the standard SW.

It has been well established that empirical po-

tentials cannot be tted with accuracy to all the

phases of silicon. The modied SW potential pre-

sented here is designed to accurately describe the

vibrational as well as the structural and elastic

properties of a-Si. This makes the potential par-

ticularly useful to model Raman and neutron

scattering experiments. In the future, we will use it

to study the relation between structural properties

of amorphous silicon and features of the Raman

spectrum, such as the narrowing of the TO peak

width during relaxation, and to investigate the

relation between Raman and neutron-scattering

spectra.

Acknowledgements

N.M. acknowledges partial support from the

NSF under grant number DMR-9805848.

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Elastic constants for a-Si (in Mbar) calculated using various

potentialsa

Experiment SW Modied SW EDIP

C11 1.31 1.23 1.30

C12 0.71 0.47 0.81

C44 0.32 0.48 0.56

B 0.6 [28] 0.94 0.73 0.97

aB is the bulk modulus.



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