comparison of semi-empirical potential functions for silicon and germanium

PHYSICAL REVIEW 8 VOLUME 47, NUMBER 13 1 APRIL 1993-I

Comparison of semi-empirical potential functions for silicon and germanium

Stephen J. Cook* and Paulette ClancySchool of Chemical Engineering, Cornell University, Ithaca, New York 14853

(Received 7 July 1992)

We report the results of investigations into the behavior of the crystalline and liquid phases of siliconand germanium as they are modeled by the Tersoff and modified-embedded-atom-method (MEAM) po-tentials. The structure of the solid-liquid interface produced by each potential is determined, as are themelting points corresponding to each potential model. The equilibrium properties of both crystallineand liquid silicon and germanium as modeled by the Tersoff and MEAM potentials are then presented.The Tersoff potential is found to overestimate greatly the melting point of both silicon and germanium,while the MEAM underestimates the melting point of silicon by approximately 15%%uo. The MEAM doesnot yield a stable liquid phase of germanium. Both potential models produce a (100) solid-liquid inter-face for both semiconductors that is rough, while the (111) interface is found to be atomically smooth.The Tersoff potential yields an amorphous phase for both silicon and germanium, but efforts to producean amorphous MEAM phase were unsuccessful. Rapid solidification of MEAM silicon produced un-

physical regrowth velocities.

I. INTRODUCTION

Laser annealing techniques are being considered onceagain as viable processing techniques for the productionof compound semiconductor materials, such as alloys ofsilicon and germanium. Silicon-germanium alloyspresent a materials processing challenge in that misfitdislocations will form if the material is held at elevatedtemperatures for an extended period of time. This prob-lem has been remedied at least in part by the introductionof a technique that combines molecular-beam or vapor-phase epitaxy with laser annealing. ' This method,which is still under development, is receiving increasingattention and holds promise for the production of com-pound semiconducting materials. Mechanisms whichmight mediate this growth or limit the germanium con-centration, however, are still completely unknown.

Although great progress has been made toward devel-oping ab initio simulation methods, such as the Car-Parrinello method, for atomic-scale studies for semi-conductors, these methods are still far too computation-ally demanding to allow large-scale studies. This limita-tion is particularly important in atomic-scale investiga-tions of materials processing phenomena such as liquid-phase epitaxy. This has led several groups to proposecomputationally efficient empirical potential models forsilicon that could be used for this or similar pur-poses. ' Although there have been many efForts to pro-duce a reliable empirical potential for crystalline silicon,only the Stillinger-Weber potential was parametrized byalso considering liquid-phase properties. Of the poten-tials that were not fitted to liquid-phase properties, onlyTersofF' has reported any results relating to the liquidphase. Terso6; moreover, only reported an estimate ofthe melting temperature and the structure of the liquid asdetermined by the radial distribution function. Althoughthere are many potential models for silicon, only the Ter-soff and modified-embedded-atom-method (MEAM) po-

II. POTENTIAL MODELS

The Stillinger-Weber potential utilizes both two-bodyand three-body interaction terms to stabilize the diamondcubic structure of crystaHine silicon and is given as

P(r) = g $2(r J )+ g $3(r J, r k, cos8&k ),

i,j i,j,k

j&i k& j&iq

0 0 0p2( r ) = A e I3

r r exp r —a

p, (r;, , r;k, coso;,k ) =1 «xp QO

r; —a

X expr;k —a

1COSH;jk +

'2

(3)

where 3, B, p, q, A, , and y are adjustable parameters anda is a cutofF' radius at which the potential goes smoothlyto zero. The parameters in the potential were adjusted toreproduce the properties of both crystalline and liquid sil-icon.

The Tersoff potential is written as a Morse pair poten-tial and includes many-body efFects by specifying that the

tentials have been extended to germanium. Ding and An-dersen suggested that germanium might also be modeledby the Stillinger-Weber potential with appropriatelyscaled length and energy parameters. The quality ofthe germanium potentials is almost completely untested.

In order to investigate the suitability of these potentialsfor studies of materials processing phenomena involvingdisordered phases, we have performed investigations ofthe phase behavior of the Terso6' and MEAM potentialsusing equilibrium molecular-dynamics techniques.

47 7686 1993. The American Physical Society

47 COMPARISON OF SEMI-EMPIRICAL POTENTIAL FUNCTIONS. . . 7687

coeKcients of the Morse terms be dependent on the localenvironment of a specific atomic pair. Tersoff gives hispotential as

where

V,"=f, (r, )[ A;"exp( X,—r, )""b,—B,""exp( p, r, —)]""and

b; =y; (1+@ g;j')

g;, = g f, (r;, )g(&;,k)exp[&3(r j r'k)],kwi, j

C.I

d; +(h; —cos0;j.k )

(7)

Tersoff later modified his potential slightly by suggestingthat setting A, 3

=0 had little effect on the quality of the po-tential. ' This simplification has been adopted in thiswork. The function f, (r, )is a c"utoff function thatchanges smoothly from unity to zero within a thin,specified cutoff shell. The adjustable parameters 3, B, A, ,

p, P, n, c, d, h, R, and S were fitted to a large database ofenergies determined by ab initio calculations for a varietyof real and hypothetical structures, and the subscript I'j

indicates that the parameter depends on the identity ofatoms i and j. Tersoff apparently did not fit to anyliquid-phase data. Parameters are given for carbon, sil-

icon, germanium, and their alloys.The modified-embedded-atom method is a more gen-

eral form of the embedded-atom method (EAM), which isa very successful potential model for face-centered-cubic

where p, is the total electron density at atom i due to allother atoms in the system, I' is the embedding function,and P," is the pair potential correction term. EAM stud-ies of metals assume that the electron density p; can beexpressed as a pairwise summation of radially symmetricelectron densities from the other atoms in the system

EAM y a

I,JWI

(10)

Baskes, Nelson, and Wright have shown that this as-sumption is invalid for silicon and that higher-ordermultibody terms must be included in the electron-densitycalculation to account for bond-bending interactions:

p rip (rj)+ Xrfp (r'j rik rjk)j j k

where p and p signify the two-body and three-body con-tributions to the electron density. Baskes, Nelson, andWright expand the three-body term as a sum of Legendrepolynomials and arrive at the expression

metals. The EAM was developed by Daw and Baskes forpure metals ' and extended to alloy systems by Foileset al. Embedded-atom-like potentials have alsobeen offered by Oh and Johnson, Chen, Srolovitz,and Voter, and Finnis and Sinclair. The EAMmethodology expresses the energy of a system of interact-ing atoms as the sum of the energy required to embed"an atom in an electron gas whose density is determinedby the other atoms in the system, and a pair potential,which provides a correction term related to ion-ion in-teractions. The EAM total energy is given as

F-= QF(p, )+ g P,, (r,, ),

pi Q pj(r'j )+X X [+j~kcos~'jk +j ak( 1 —3 cos'~ j )]p;(«j )pk( ik )

j j k(12)

The constants a ' and a are adjustable parameters, 0; kis the angle between atoms i, j, and k subtended at atomi, and p is the radially symmetric two-body electron-density expression that is termed p in Eq. (11)above.

The authors of the MEAM further restrict the totalcohesive energy of the crystalline phase of silicon or ger-manium to be given by an equation of state developed byRose et al. , which is similar in form to that used in thedevelopment of the Tersoff potential. Unlike EAM mod-els for metals, the MEAM pair potential is calculated byassuming a functional form for the embedding functionand then calculating the pair potential necessary to bringthe system energy to the value required by the equationof state. No liquid-phase properties were used to fit thepotential.

The Tersoff and MEAM potentials for silicon and ger-manium have been used to obtain information about boththe bulk phases of solid and liquid silicon and germani-um, as well as to characterize the nature of both the equi-librium and rapidly crystallizing solid-liquid interfaces.These data, which will be presented in the following sec-

tions, have been compared to data obtained from theStillinger-Weber potential, as well as to existing experi-mental data.

III. EQUILIBRIUM SOLID-LIQUID INTERFACESOF SILICON

The phase behavior of silicon and germanium asmodeled by the Tersoff and MEAM potentials has beenexamined using a variety of molecular-dynamics (MD)codes. The melting point and solid-liquid interfaces werestudied using a method similar to that used by Landmanet al. to study the solid-liquid interface of Stillinger-Weber silicon. The periodic boundary condition in thedirection normal to the desired surface was removed andreplaced by a free surface that prevented the superheat-ing commonly found in bulk simulations of melting be-havior. Several atomic planes on the bottom of the cell,corresponding to a distance slightly greater than thecutoff of the potential in use, were fixed in their bulk posi-tions to emulate the presence of an underlying bulk

7688 STEPHEN J. COOK AND PAULETTE CLANCY 47

phase. Periodic boundary conditions were applied in thetwo transverse directions and a time step of 1 fs was usedfor the Stillinger-Weber and MEAM potentials. A timestep of 0.2 fs was used in simulations of the Tersoff poten-tial. Although the larger time step of 1 fs was sufficientto maintain energy conservation during simulation ofcrystalline silicon using the Tersoff potential, it was notsufficient for simulations of the liquid phase. Energy con-servation using the specified time steps was excellent,with a variation of no more than 10 eV/atom in totalenergy throughout the approximately 150 ps simulationtime necessary to obtain equilibrium solid-liquid inter-faces for each potential. The simulation cell used tostudy the (100) orientations of each potential contained50 atoms/plane, for a total of 2000 atoms, while that usedto study the (ill) orientations contained 160 atoms per(111) doublet. The total system size for studies of the(111) orientation was 2400 atoms. The atoms above thefixed lattice were either thermostated for isothermal(NVT) conditions or allowed to interact via Newtoniandynamics under conditions of constant total energy(NVE). The method of Landman et al. can be used tofind the melting point of a material in the following way.An interfacial configuration exposing the desired crystal-line orientation is first created with a system density cor-responding to zero applied pressure at a temperature thatis expected to be below but near the melting point of thematerial. The configuration is equilibrated at this tem-perature for a short time, the temperature scaling isturned off, and energy is added to the cell so that approx-imately half of the solid melts. The resultingconfiguration is an adiabatic system comprised of approx-imately equal parts of solid and liquid that are in equilib-rium at the melting temperature. If the final temperatureis very different from the original estimate, the density ofthe cell may be adjusted. A further simulation periodwould then be necessary to allow the system to reachequilibrium.

The procedure described above was used to determinethe melting points of MEAM, Tersoff; and Stillinger-Weber silicon. No data yet exist for the liquid phase ofthe MEAM semiconductors, and it is of interest to deter-mine how well these potentials might reproduce disor-dered semiconductor phases. Although Tersoff estimatedthe melting point of his potential to be 3000+500 K, hedoes not describe the method used to make this deter-mination. It is of interest, therefore, to calculate thisvalue independently. The melting point of the Stillinger-Weber potential is already well known. Recalculation ofthe melting point, therefore, provides a check on the ac-curacy of the computer program used and provides dataon the Stillinger-Weber solid-liquid interface that may beused for comparison with those produced by the otherpotentials.

Since it was desirable to have Stillinger-Weber interfa-cial configurations with which to compare the results ob-tained from other potential models, and also to check ourmethod and simulation code for errors, the method de-scribed above was used to produce an equilibrium solid-liquid interface on both the (100) and (111)orientations ofthe Stilling er-Weber solid. This simulation was per-

formed by Landman et a/. in 1986, who obtained an es-timate of the melting point of 1665 K. Later work byBroughton and Li showed the true melting point to be1691 K, which is in very good agreement with the experi-mental value of 1683 K. The value obtained by Land-man et al. thus had an error of l%%uo. Study of theStillinger-Weber system began here with the creation of acrystalline configuration whose density was set to that re-ported by Broughton and Li for a reduced temperature of0.065, which may be compared to the reduced meltingtemperature of 0.0672. Melting was initiated on bothorientations using the method of Chokappa, Cook, andClancy. Melting initiated at the crystal-vacuum surfacein both systems and the solid-liquid interface proceededinto the solid until the average reduced temperature inboth simulation cells was 0.0671+0.0011 (1688+26 K).This process took approximately 150 ps (150000 MDtime steps) for both surface orientations studied. The ex-cellent agreement between the value obtained here andthat reported by Broughton and Li validates the methodand simulation code that we have employed.

The melting point and interfacial morphology of Ter-soff silicon was also studied using the method of Land-man et al. The system size employed was equal in size tothat used for the Stillinger-Weber (SW) potential. An in-terfacial configuration exposing the (100) orientation wascreated and equilibrated for 50000 MD time steps (10 ps)at a temperature of 1500 K. The system density was setto that of the crystal at 1500 K under conditions of noexternally applied stress, i.e., zero pressure. This valuewas obtained using an isothermal-isobaric (NP'r)molecular-dynamics simulation code. Energy was addedto the system using the method of energy carriers de-scribed in Ref. 39. Melting initiated at the vacuum sur-face and proceeded into the solid until the system temper-ature fell to a value of 2744+42 K. Since the steady-statetemperature was so different from the original startingtemperature, a new interfacial configuration was createdand equilibrated at a temperature of 2500 K. The melt-ing process was repeated and a lengthy simulation run of700000 MD time steps (140 ps) revealed the equilibriummelting point to be 2547+22 K. The large change in theestimate of the melting point was caused by aninsufficiently long simulation time for the first attempt.Although the number of time steps required to reachsteady state has increased by a factor of 4 over SW, theamount of "real" time required has remained constant.The additional effort is only required because the timestep used in studies of the Tersoff system is smaller. The(100) solid-liquid interface of Tersoff silicon is verydiffuse, and shows the presence of interplanar defects allthe way down to the static lattice. Figure 1 shows the na-ture of the (100) solid-liquid interface of Tersoff silicon,along with that of the Stillinger-Weber potential. Al-though the actual melting point of the potential, 2547 K,is somewhat better than that estimated previously by Ter-soff; it is still almost 900 K higher than the experimentalmelting point of silicon. The value is in error by 51%%uo.

Due to the large error in the melting point, no study wasmade of the Tersoff (111)solid-liquid interface.

The melting point of MEAM silicon was calculated on


O

O

o~OI~O

O

(100) OrientationTersof f Silicon

OOO

~OI~O

O

c &UIiLLUL LLUiUII

(100) OrientationS% SiliconT = 1691K

, L. ULULII, LUJII

0 10 20 30 40Height (A)

50 60

FICx. 1. Density profiles comparing the morphology of the(100) solid-liquid interface of silicon as predicted by themodified- TersofF and Stillinger-%'eber potentials.

both the (100) and (111) orientations using the samemethod as was used for the Stillinger-Weber and Tersoffpotentials. An initial estimate of 1000 K was used as thestarting point for the determination of the melting point,and the initial density of the atoms in the simulation cellwas set to the zero-pressure density for crystallineMEAM silicon at 1000 K. The cell was then equilibratedat constant temperature for 50000 time steps (50 ps), thetemperature scaling was turned off, and then enough en-ergy was added to the cell to cause approximately half ofit to melt. Melting initiated at the surface and continueduntil enough of the crystal had melted to cause the sys-tem temperature to fall to a constant value. The averagetemperature for both orientations stabilized at 1445+25K after a simulation time of 120 ps. The zero-pressuredensity of MEAM crystalline silicon at 1445 K was thencalculated and the size of the simulation cell was adjustedto this value. Equilibration of the (100) solid-liquid inter-face was then continued for another 120 ps. Observationof the average system temperature during the three 40 pssections of this latter 120 ps indicated that the systemhad relaxed to the new temperature of 1475+25 K afteronly the first 40 ps of this latter equilibration period. Al-though this procedure could be repeated yet again withthe density for the higher temperature, calculation of thezero-pressure density at 1475 K showed that the systemwas only under a 0.05%%uo strain when using the density ap-propriate for 1445 K at the higher temperature. Thismay be compared to a mismatch of 0.6% between the lat-tice constants appropriate for 1445 and 1000 K. Sinceany feature change in the temperature was likely to have

O

O

~OI~O

O

OOOOO

i It I I i I i I I i I I I I I J L L

(100) Or ientationMEAM SiliconT = 1475K

~OI~O

O

(100) OrientationSW SiliconT = 1691K

O0

, L. I, LILIIIL, »10 20 30 40

Hei ht (A)50 60

FIG. 2. Density profiles comparing the morphology of thesolid-liquid interface on the (100) orientation of silicon as pro-duced by the MEAM and Stillinger-Weber potentials.

been less than the statistical error of the simulation data,the procedure was stopped at this point. The value of theMEAM melting point for crystalline silicon was then tak-en to be 1475+25 K. The underestimation of the meltingpoint of silicon observed here is consistent with previousinvestigations of the melting points of transition metals asmodeled by the EAM. Foiles and Adams found thatthe EAM usually predicted the melting points of thetransition metals to be at or below the values that are ob-served experimentally.

An attempt was made to use the same procedure ofchanging the system density for the (111) solid-liquid in-terface as had been used for the (100) orientation, but asimple change of the width of the simulation cell pro-duced no subsequent change in temperature after 80 ps.Subsequent attempts to produce interface motion byturning on the "heat bath" portion of the simulation cellshowed that the solid-liquid interface was immobile onsimulation time scales at system temperatures at least aslow as 1350 K, which represents an undercooling of al-most 10%. This phenomenon is caused by the nature ofthe (111) solid-liquid interface, which is atomicallysmooth. Regrowth on this interface, like on the (111)in-terface of the Lennard-Jones potential, requires the initialnucleation of atoms onto existing crystal to form a newlayer. Our finding does not indicate a specific deficiencyof the MEAM potential, only that nucleation is a slowprocess that is difFicult to capture in a molecular-dynamics simulation. This is especially true for the (111)interface. The equilibrium (111)solid-liquid interface wasobtained by simply scaling system velocities back to 1475K and equilibrating the system for 80 ps before collecting


O

O

~O~O

O

(111) OrientationMEAM SiliconT = 1475K

OOOOO

~Ot~O

O

ill, I i Lil

(111) OrientationSW SiliconT = 1691K

OO

O

O0

&~i ~i I „ I &. i t il lr

10 20 30 40Height (A)

I

50 60

FIG. 3. Density profiles comparing the morphology of thesolid-liquid interface on the (111) orientation of silicon as pro-duced by the MEAM and Stillinger-Weber potentials.

low temperature with a perfect diamond lattice as an ini-tial atomic configuration. The system was equilibratedfor at least 50000 time steps (50 ps for the MEAM) andaverages were then collected for at least another 50000time steps (50 ps). Simulation of the solid at a highertemperature was then begun with the resulting atomicconfiguration of the previous lower-temperature study.This process was continued until melting of the crystaloccurred. The resulting liquid configuration was thenequilibrated for at least 100000 time steps at a tempera-ture 500 K above the temperature at which melting oc-curred, thus assuring that no residual solid structureremained. Studies of the liquid phase began at this point.Each liquid state point was obtained by the same equili-bration and averaging process as for the solid, except thatthe system temperature was systematically reduced fromthe highest temperature configuration studied. Studies ofthe crystalline phase of both Tersoff and MEAM siliconwere started at a temperature of 500 K, and each statepoint in the solid phase was obtained by instantaneouslyincreasing the system temperature in the manner just de-scribed. Each liquid state point was obtained by instan-taneously decreasing the system temperature below theprevious configuration temperature. All simulations wereperformed under conditions of zero externally appliedpressure.

A. Equilibrium phases

averages.The spatial variation of the MEAM and Stillinger-

Weber interfaces is shown in Figs. 2 and 3, where thesinglet density profiles obtained from the MEAM arecompared to those obtained from SW. The densityprofiles are remarkably similar, showing a broad andrough interface on (100) and the sharp, abrupt interfacecharacteristic of (111). There is also no evidence of "lay-ering" or ordering in the liquid region adjacent to the(111) solid, with the exception of the very pronouncedpeak showing ordering of the liquid atoms immediately incontact with crystal on (111).

The densities and configurational energies of crystallineand liquid Terso6' silicon are shown in Figs. 4 and 5,while data for MEAM silicon are shown in Figs. 6 and 7.The crystalline phase produced by each potential exhibitsa nearly linear decrease in density and increase in energyuntil it finally reaches the limit of superheating and melts,which occurs between 1750 and 2000 K for the MEAM

OOO

IV. PHASE DIAGRAMS FOR TERSOFFAND MEAM SILICON

Information concerning the crystalline and liquidbranches of the equilibrium phase diagram of both theTersoff and MEAM semiconductors was obtained usingan isothermal-isobaric (NPT) molecular-dynamics simula-tion code that employed the modified leap-frog algorithmof Brown and Clarke. ' The simulation cell was cubicand periodic boundary conditions were applied in allthree Cartesian directions. Ensemble averages were cal-culated using a system comprised of 1000 atoms. Studiesof the MEAM semiconductors were conducted using atime step of 1 fs, while the time step in simulations of theTersoff systems was 0.2 fs. Studies of the crystallineconfigurations for each semiconductor system begun at

O

~ WCQ

g(D

A co

OO

coo 0 1000 2000 3000

Temperature (K)

FIG. 4. The density of silicon in the solid and liquid phasesas represented by the modified-Tersoff potential. Data were col-lected along the isobar corresponding to zero average pressure.


P&oI

lm

1000 2000 3000Temperature (K)

FIG. 5. The configurational energy per atom of silicon in thesolid and liquid phases as represented by the modified-Tersoffpotential. Data were collected along the isobar correspondingto zero average pressure.

and between 2500 and 3000 K for Tersoff. Thisrepresents a superheating of at least 21/o for the MEAMand at most 17% for Tersoff. This may be compared tothe 34% superheating that was observed by Broughtonand Li during their study of the Stillinger-Weber poten-tial. The liquid configuration that resulted from melt-

1000 2000Temperature (K)

FIG. 7. The configurational energy per atom of silicon in thesolid and liquid phases as represented by the modified-embedded-atom Method. Dashed lines indicate instantaneousquenches to metastable states from an initial temperature of 900K. Data points on the solid line, which represents the liquidbranch at temperatures below 900 K, represent values calculat-ed by cooling the liquid at the rate of 10' K/s, followed byequilibration and averaging. Data were collected along the iso-bar corresponding to zero average pressure.

MEAM Silicon

1000 2000Temper. at.ure (K)

FIG. 6. The density of silicon in the solid and liquid phasesas represented by the modified-embedded-atom method.Dashed lines indicate instantaneous quenches to metastablestates from an initial temperature of 900 K. Data points on thesolid line, which represents the liquid branch at temperaturesbelow 900 K, represent values calculated by cooling the liquidat the rate of 10' K/s, followed by equilibration and averaging.Data were collected along the isobar corresponding to zeroaverage pressure.

ing the Tersoff crystal at 3000 K was used as input for thesimulation run at 3500 K, and work on the liquid branchof Tersoff silicon began there. Likewise, the MEAMliquid configuration at 2000 K was used as input forsimulation at 2500 K. It should be noted that the aver-age densities and energies calculated for the liquidconfigurations at 3000 K (for Tersoffl and 2000 K (forMEAM) did not depend on whether the initialconfiguration was solid or high-temperature liquid, indi-cating that the equilibration times employed for each po-tential were sufficiently long.

The phase diagrams show that both the Tersoff andMEAM yield a liquid silicon phase that is denser thanthe crystal. This phenomenon, which is observed experi-mentaHy, is also reproduced by the Stillinger-Weber po-tential. The calculated densification at the melting pointof MEAM silicon, 1475 K, is 5.4%, which is identicalwith the experimental value reported in Ref. 38. The cal-culated densification at the melting point of Tersoff sil-icon, 2547 K, is 2.0%%uo. Previous work on the Stillinger-Weber potential by Broughton and Li has shown SW togive the densification as 7.7%, which is somewhat toohigh. The heat of melting may also be obtained from thephase diagrams as the difference in configurational energybetween the crystalline and liquid phases at the meltingpoint. This may be done because all systems are at anaverage pressure of zero. The calculated value of the la-tent heat for the MEAM is 35.0 kJ/mol, while that of theTersoff potential is 41.8 kJ/mol, which compares with

47TTE CLANCYC~OK pAU LETTSTEpHEN J.7692

ber

ct,1on

4Radius (A)

r li uid silicon ob-d amies using h

d tthtained from m

rve was obtaint represent sys-'l H thes ective potential. en

temperature.tems at the same temp

the temperature e is in-n rows as the ey MEAM silicon grty ofreaching a v

r silicon as r pthe

Th values fothotentia i

tentials. od MEAMh h T ff

i' - eber an

tential yie sthermal exp

of each phasegation of the potentia .

cit at constan t pressure o eThe heat capacity a

ExperimenntlQ~ooM ~&oQ o

C5

(D co

~ O

CQ ~wo&o

II

1500Temperature

OO 0 2000

oft eh density of crys-'

l9. The temperaature p

three semiemp' incaresented by t ras repretalline sihconmodels.

the Stillinger,alue «50.6 kJ/mol anderjmenta va

andf 309kJ/mol.

bbetweensixWeber value o

t the liquid tot with the

redictsa reemen

The ME

0

coordinated, with experiment.

eightfoldt nt;al and w

h t dependen«nthe definit

d fined as anyf the first- an

'ion that »

atom

0

near igd to be the

t the same te

bpr isra e off shell define o

in the crystal ayf the

econd-neigdination is r p

d to the first

0

erature, thef ction is in g

k then the' hb«pea

radial distrithe first-neig

shallow, how-follow~ng

jnjmumminimum

i ht The mh Tersoff poten-

value is jusvalue pf 0

'hbor coordina-

abPve eig846. T eever, only r

' 'd to have ad 5 0, althpug

reaching afirst-neig

'l redicts th q '

f between 4. .h high~~

tia p1 g ppint o

1 is muction at th™

he Tersoff poteh t the Tersoff

ointoft eefound t a .

r)

the melting p ' t 1 value. We'mum in g "

than the explj uid with a

.is somewhat

erimentafirst min™

smaller t ad sjlicpn pre

t redicted usre of 'qu

tot atpe structure

] comparebt ined from

otentia s is'

and that oand MFAM P

ber pptential a; . 8. Each curvethe Stilling -

jments in g. . f r the pp-

ing'

ction experimropriate pr

neud at the me tin

in genera

tron-diffrac1. g point appr p

1 that thewas obtained

results shpwle agreementStillinger-

lthough sjgnificff potential also

w(,r) are app

th of the rsrst peak» g

h h the deptsly report-

oughas previou

does fairly w, as Tersoff;d using the

what too deepfunction pbtain

js somew'

1 d trjbution u'

h xperimot ~gree wet to the ex-

11

e .ential does no

with respecMEAM poten

'

A ho ldlline and liqui

pthe relative p

are satis a' f ctory, how-b th MEAMroduced ybranches p

elting is pre is-

thedensi ca '

e the densitye

interesthet re sh e potentia sd b h

'hthtobt'1

and

considere

1'extreme y w

ro ressure p 'ls

yr stalline si'

nd MEger-Weber,tained by w

The data showmethods.p

b 1eber ensiand is in e

S i}linger Web pature cur

ot redict tof the densi y

ntials do no p

curvature o

le is in errcrystal

Ptia .of 500 K. etemperature o


TABLE I. The properties of silicon at zero pressure as represented by the TersofF, MEAM, andStillinger-Weber potential models, as well as the values obtained by experiment. All simulation valuesare reported at the melting point of the respective potential model.

Property

Melting point (K)liquid

Latent heat, I (kJ/mol)Crystalline density (g/cm )

Liquid density (g/cm )

Liquid coordinationCrystalline Cp (J/gK)Liquid Cp (J/gK)Amorphous density (g/cm')Latent heat, , (kJ/mol)Amorphous coordination

Tersoff

2547

4.182.2182.263

4.6-5.01.001.302.291

27.04.16

MEAM

14751.16X 10-"

35.02.1842.302

6.0-8.21.081.152.387'

23.5'4.65'

SW'

16916.94X 10-'

30.92.2832.459

5.5-6.20.9991.2562.341'

iS.4'4.2i'

Experiment'

1683—10

50.62.402.53

-6 4'1.041.04

1.71—2.2g

11 9"4.0—4.25

'Reference 38, except where otherwise noted."Taken at melting temperature of potential model.'Reference 42."Value at a temperature of 300 K.'It is not clear that the MEAM actually produces a-Si.'Reference 47.Reference 43.

"References 54 and 55.

produced by the two potentials has been calculated by aleast-squares fit of the enthalpy data obtained by NPTmolecular dynamics. The values obtained are reported inTable I. Although the MEAM predicts a crystalline den-sity that is much lower than that predicted by either Ter-soff or Stillinger-Weber, its description of the heat capaci-ty is comparable to the other two potentials.

The dynamical properties of MEAM liquid siliconhave been investigated by calculating the diffusioncoefficient at a variety of temperatures between 1000 and2500 K. The diffusion coefficient of Tersoff liquid siliconwas not calculated due to the large difference between theexperimental melting point and that predicted by theInodel.

The diffusion coefficient of MEAM liquid silicon wascalculated using a constant volume-constant temperature

(XVT) molecular-dynamics simulation code. A systemsize of 100 atoms and a time step of 1 fs were used in thisstudy. Values of the diffusion coefficient were calculatedusing mean-squared displacement data averaged for 50 pswith averages calculated over 100 time origins to reducestatistical error. The values of the diffusion coefficientsso obtained are displayed in Fig. 10, along with a curve fitobtained assuming Arrhenius behavior for the liquid.Least-squares analysis yielded a value of the pre-exponential of 1.54X 10 cm /s and an activation ener-

gy of 0.33 eV. The fit is quite satisfactory. This may becompared with the Stillinger-Weber values of 3.5X10cm /s and 0.56 eV, respectively. 6

B. Amorphous phase

ransition

h ooo 0.001 0.002t /Temperature (K ')

FICx. 10. The temperature dependence of the diffusioncoefficient of MEAM liquid silicon.

The densities of the liquid phase of both the Tersoffand MEAM models for silicon exhibit sudden decreasesat temperatures well below the calculated melting pointsof the potentials. These decreases have been examinedwith an emphasis on uncovering possible transitions intoa phase that approximates amorphous silicon.

The Tersoff potential has been shown previously toyield a phase that resembles amorphous silicon. Thisphase was attained by instantaneously quenching a liquidconfiguration at approximately 3000 K to room ternpera-ture using a continuous-space Monte Carlo method. Thisconfiguration was then annealed for an extended, butunspecified, period of time. The resulting phase had afirst-neighbor coordination of approximately 4.25. Con-tinued annealing at negative applied pressures reducedthe coordination to 4.09. Tersoff also reports a"significant" reduction in system energy, but does notgive the exact value obtained, nor does he report a value


for the density of the amorphous phase produced by hispotential. Calculations performed here show that thedensity of Tersoff silicon decreases substantially as theliquid is instantaneously quenched from 2500 to 2000 K.Further cooling below 2000 K produces a disorderedphase that has a density less than that of the solid, but anenergy only slightly less than that of the liquid. Exam-ination of the radial distribution function of this phaseobtained at 300 K, which is shown in Fig. 11 along withthat obtained by Tersoff, shows that the structure of thesystem obtained here and that obtained by Tersoff arenearly identical. The first-neighbor coordination of thesystem obtained in this work was obtained by integratingthe radial distribution function up to the first minimum.The value obtained at 300 K is 4.16, which lies betweenthe values obtained by Tersoff. The amorphous phasethat is obtained is approximately 1% less dense than thecrystalline phase at 300 K, and the amorphous phase isapproximately 0.28 eV/atom (27.0 kJ/mol) higher in en-ergy than the crystalline phase at 300 K.

An interesting phenomenon occurs in the liquid phaseof MEAM silicon at low temperatures in the form of adensity maximum that is found at a temperature of 900K. Instantaneous cooling of the liquid from 1000 K to atemperature of 500 K produced a state point that was aplausible extrapolation of the liquid branch to 500 K.Simulation of a liquid instantaneously quenched from1000 to 750 K, however, showed a distinct drop in systemdensity below the extrapolation to 500 K. Thesequenches are shown by dashed lines in Figs. 6 and 7.Broughton and Li had also observed a density maximumwhen studying the supercooled state of liquid Stillinger-Weber silicon. Their conclusion in 1987 was that themaximum corresponded to a glass transition, wherebythe system retained the atomic structure of the liquid butattained a diffusion coefficient appropriate for the solidphase. Luedtke and Landman later suggested, however,

l

(['

II

(I

l(

(1

(I(I

II

I

I

i~/

This Work

Ref. 19

4Radius (A)

FIG. 11. The radial distribution function of amorphous sil-

icon as represented by the modified-Tersoff potential. Data arepresented for a system temperature of 300 K.

that this density maximum actually corresponded to afrustrated phase transition into the amorphous phase.Luedtke and Landman showed that Broughton and Lidid not obtain the amorphous phase because they cooledthe liquid too quickly, preventing the exploration ofphase space necessary to nucleate the amorphous phase.

The possibility of obtaining a representation of amor-phous silicon using the MEAM was examined by per-forming a series of simulations whereby the liquid phaseat 1000 K was slowly quenched to lower temperatures.The configuration at 1000 K was quenched to 900 K at arate of 10' K/s, and then equilibrated at constant tem-perature for 50 ps before collecting averages for a further50 ps. This process was then repeated in intervals of 100K until the system was at room temperature. This pro-cess revealed that the decrease in density occurred be-tween 900 and 800 K, and that further cooling below 800K produced a smooth, gradual increase in the densityfrom its value at 800 K. These values are shown by thesolid line extending to temperatures below 900 K in Figs.6 and 7. The density at 750 K, obtained from the slowerquenching procedure, was found to be lower than thatobtained using the instantaneous quench by a modest0.1%, although the density of the liquid at 500 K predict-ed using an instantaneous quench procedure was 1.35%higher than that obtained from a slower quenching of theliquid.

The transition between 900 and 800 K was further ex-amined by decreasing the quench rate to a value of 10"K/s. Equilibration and average collection were per-formed as with the higher quench rate. Although thequench rate had been reduced by an order of magnitude,the resulting system density at 800 K was identical,within the error of the simulation, to the value that wasobtained using the higher quench rate. The reversibilityof this transition was further probed by heating the 800K configuration to 900 K at a rate of 10' K/s. Onceagain, this resulted in identical densities andconfigurational energies as were obtained from the firstquench to 900 K. The stability of the atomicconfigurations at 800 and 900 K was also probed by hold-ing the systems at their respective temperatures andequilibrating them each for 1 ns of simulation time (10MD time steps each). System densities and energies werecalculated in 50 ps segments and as a cumulative averageover the 10 time steps. Neither system exhibited energyvariations between 50 ps intervals greater than 10 meVper system atom, and the density variation between inter-vals was less than 10 A for both temperatures. Fur-thermore, no systematic trend in energy or density wasfound at either temperature.

The amorphous phases obtained by quenching Tersoffand MEAM liquids may be compared to results that havebeen obtained for the amorphous phase of the Stillinger-Weber potential. It should be made clear from theoutset that the existence of a MEAM amorphous siliconis somewhat in doubt due to the small changes in densityand configurational energy that are observed. Table Ilists some of the relevant information. Although none ofthe potentials studied reproduce the experimental resultsfor amorphous silicon accurately, specifically with regard


Tersoff Germanium

MEAM Crystal

~ WCQ m

(DaO

4Radius (A)

C)O p

I m


FIG. 12. The radial distribution function of amorphous sil-icon produced by the Tersoff, MEAM, and SW potentials. Allcurves represent systems at a temperature of 300 K.

FICx. 13. The density of germanium in the crystalline andliquid phases, as represented by the modified-Tersoff potential.Data were collected along the isobar corresponding to zeroaverage pressure.

to the density, the Stillinger-Weber potential does predictthe energetics of the amorphous phase with considerablymore accuracy than either Tersoff or the MEAM. Theamorphous-crystal latent heat produced by the MEAM,in fact, approaches that predicted by the MEAM for thecrystal to liquid transition. The structure of each amor-phous phase is presented in Fig. 12, where the radial dis-tribution function obtained from each potential is corn-pared. The first-neighbor coordination for each amor-phous phase may be calculated by integrating the radialdistribution function up to the first minimum. This pro-cedure has been applied to the functions obtained herefor the Tersoff and MEAM potentials to obtain values of4.16 and 4.65, respectively. Luedtke et al. report a coor-dination for amorphous SW silicon of 4.21. Although theSW and Tersoff results are in reasonable accord, the valueobtained from the MEAM potential is somewhat toohigh. This provides more evidence that the low-temperature disordered phase produced by the MEAM isnot representative of amorphous silicon.

V. PHASE DIAGRAMS FOR TERSOFFAND MEAM GERMANIUM

obtained using the isothermal-isobaric (NPQ simulationcode that was developed for the silicon systems. Theconfigurational energies and densities of Tersoff germani-um in the crystalline and liquid phases are shown in Figs.13 and 14. The modified-Tersoff potential predicts aliquid germanium phase that is denser than the crystal atthe melting point by approximately 6.8%. This compareswell with the experimental value, which is approximately7.1%, ' although the melting temperature of the Ter-

OO

I

~cu

~ I

+O

Procedures like those described above were also usedto characterize the solid and liquid phases of germaniumas described by the Tersoff and MEAM potentials. Re-sults were generally less favorable than those obtained forsilicon.

Tersoff germanium was studied using the same methodthat was used for Tersoff and MEAM silicon. The melt-ing point of Tersoff germanium was determined using themethod of Landman et al. to be 2554+38 K, which is ap-proximately twice the experimental value of 1210 K.Data for the various phases of Tersoff germanium were

OO

I p

I I


4000

FIG. 14. The configurational energy of germanium in thecrystalline and liquid phases, as represented by the modified-Tersoff potential. Data were collected along the isobar corre-sponding to zero average pressure.


soff system is far above the experimental value. The la-tent heat of melting was found to be 0.465 eV/atom, or44.9 kJ/mol. This may be compared to the experimentalvalue of the latent heat, 36.9 kJ/mol. Finally, theroom-temperature heat capacity at constant pressure forcrystalline germanium was calculated with the Tersoffpotential to be 0.37 J/gK. This is in general accord withthe experimental value of 0.322. '

An attempt was made to determine the melting pointof MEAM germanium using the method of Landmanet al. A crystalline system exposing the (100) orientationto vacuum was created and equilibrated for 50000 MDtime steps at a temperature of 1000 K. Energy was addedto the system using the method of energy carriers de-scribed in Ref. 39 over a time period of 2 ps. The systemtemperature rose to approximately 2000 K after the addi-tion of energy, but no melting occurred. Further simula-tion of this system showed that the crystal was stable atthis temperature. An attempt was made to add more en-

ergy into the system and the system temperature roseabove 2500 K. Significant evaporation of germanium oc-curred at this temperature without the establishment of astable liquid phase. It was unknown whether the failureto establish a liquid was due to a weakness of the MEAMgermanium potential, or whether energy was added to thesimulation ce11 at a rate that was too great to allowthermal equilibration of the deposited energy. The simu-lation was, therefore, repeated with a pulse length of 5 pswith a simulation cell whose temperature was 2000 K.The system once again did not melt, but rather, evaporat-ed. A final effort to produce a MEAM germanium sys-tem with a solid-liquid interface was carried out by usingthe equilibrium solid-liquid MEAM silicon system as in-put. This system had an equilibrium temperature of 1475K. The identity of the atoms was changed and aconstant-energy simulation performed. The liquid quick-ly began to solidify, causing the system temperature to in-crease. The temperature reached a value between 2000and 2500 K in a time of 50 ps. The simulation was haltedat this point and efforts to obtain an interfacial MEAMgermanium system were abandoned.

The phase diagram of MEAM germanium was studiedusing the isothermal-isobaric (APT) simulation code thatwas described in Sec. IV. Heating of crystalline MEAMgermanium began at 500 K and continued in incrementsof 500 K until a temperature of 2000 K was reached.Heating of the crystal was halted at this point, since thesystem had already superheated significantly with respectto the experimental melting point of germanium, and,since simulations of the interfacia1 system indicated an in-stability near 2500 K.

Since previous efforts to produce liquid MEAM ger-manium failed, an attempt was made to produce theliquid by changing the identity of a bulk MEAM liquidsilicon system at a temperature of 1500 K. This systemwas equilibrated for 50 ps and averages were collected fora further 50 ps. The density and energy that were ob-tained were plausible values for the liquid, except that thedensity of the liquid was predicted to be less than that ofthe diamond crystalline phase. This system was instan-taneously heated to 2000 K and the equilibration and

00Gl

00

~O~oM

0G3F300

MEAM Germaniu

~ Crystal

~ Liquid

00 0 1000 2000

Temperature (K)

FIG. 15. The density of germanium in the crystalline andliquid phases as represented by the MEAM potential. Datawere collected along the isobar corresponding to zero averagepressure.

00

~ I

+0& ca

MEAM Germaniu

~ Crystal~ Liquid

00I p 1000 2000

Temperature (K)

FIG. 16. The configurational energy of germanium in thecrystalline and liquid phases as represented by the MEAM po-tential. Data were collected along the isobar corresponding tozero average pressure.

averaging process were repeated. Liquidlike values wereonce again obtained, with the exception of the densitychange already noted. Upon instantaneous heating of thesystem at 2000 K to a temperature of 2500 K, a spon-taneous transition to a gaseous state was observed withina time of 100000 MD time steps (100 ps). In order to re-move the possible effect of significant volume fluctuationscaused by instantaneously heating the system, data for

COMPARISON OF SEMI-EMPIRICAL POTENTIAL FUNCTIONS. . . 7697

LQ~o-o

ca ~Fo(D o

XXXXX

~ O~ oG5

CQ ~4 O

o 0 500 1000 1500Temperature (K)

2000

FIG. 17. The temperature dependence of the density of crys-talline germanium as represented by the Tersoff and MEAM po-tentials.

VI. RAPID SOLIDIFICATION OF MEAM SILICON

The results presented in Sec. VI generally indicate thatthe MEAM predicts the properties of the solid to liquidtransition in silicon fairly well. Although the modelseems to be somewhat "underbound, " resulting in solid

liquid MEAM germanium above 2000 K were obtainedby alternately heating the liquid at a rate of 10' K/s fora period of 100 ps, followed by equilibration and averag-ing at constant temperature for a further 100 ps. Infor-mation was then obtained at intervals of 100 K. Slowheating of the liquid resulted in stable configurations at2100 and 2200 K, but the system spontaneously vapor-ized during heating to 2300 K. Inspection of the result-ing phase diagrams shown in Figs. 15 and 16 suggeststhat the system at 2200 K might also turn to vapor if asu%ciently long simulation were performed. The ap-parent difference in quality between the MEAM descrip-tions of silicon and germanium is puzzling. Thisdifference may be related to the fact that the pair poten-tial portions of MEAM silicon and germanium are notconformal, that is, they are not similar in shape. Al-though the similarily between silicon and germaniumwould lead one to expect similarly shaped pair potentials,this is not the case. The origin of this difference in thepair potentials was not investigated.

The density of crystalline germanium predicted by theTersoff and MEAM potentials can also be compared toexperiment. The data, displayed in Fig. 17, show in gen-eral that the agreement is no better for germanium thanfor silicon. The crystalline density given by the MEAM,for example, is in error by —2.2% at 500 K, and in-creases at higher temperatures. Although the Tersoff po-tential is more accurate than the MEAM in this regard, itis in error by —1.3% at 500 K.

and liquid densities and a melting point that are too lowwith respect to experiment, the error in most properties isless than 20%. It is of interest then to determine howwell the MEAM reproduces aspects of the nonequilibri-um solid-liquid interface during a simulation of rapidsolidification.

A MEAM silicon system exposing the (100) system wasprepared for melting and solidification by equilibrating itat a constant temperature of 1000 K for a period of 50 ps.The temperature scaling was turned off and the systemwas heated by subjecting it to a beam of energy carriersusing the method described in Ref. 39. The intensity ofthe beam was Gaussian in time, with a density of 0.4J/cm and a total length of 15 ps. The total absorbed en-ergy density was 3.21 eV/A (5.14 mJ/cm ). A total of2.43 keV (3.89X10 ' J) was absorbed. Melting initiatedat the vacuum surface and proceeded into the crystal un-til a maximum melt depth of approximately 32 A was at-tained 25 ps after the beginning of the heating process.The interface temperature rose from its initial value of1000 K during this heating process, reaching a maximumvalue of approximately 2000 K after 10 ps of heating, orshortly after the maximum intensity of the energy beamwas reached at a time of 7.5 ps. The interfacial su-perheating drove the melting process, causing the inter-face to cool. The interfacial temperature data were fittedwith a least-squares parabola to facilitate analysis. Thebest-fit temperature at the maximum melt depth, whichcorresponds to zero interfacial velocity, was found to be1467 K, which is very close to the equilibrium meltingtemperature of 1475 K. This indicates good agreementbetween the melting points determined by the equilibriumsimulation methods. As energy was withdrawn from thesimulation cell by the heat bath, solidification of theliquid layer began. The liquid was completely crystal-lized after a time of 115 ps. The melt and temperaturehistory of the sample is shown graphically in Fig. 1S,where the melt depth and interfacial temperature areshown as functions of time. The results of two differentmethods of analysis are shown for the melt depth. Thedata points, shown as filled circles, are the result of usingthe radial distribution function and singlet density profileto determine the average position of the interface. Thecurve is the result of a method whereby the dynamicnumber of liquid atoms in the system is calculated. Thecriterion used here to define a "liquid atom" was simplyto declare an atom without fourfold coordination to beliquid. Although this definition is not unique, it is anoth-er route to the interfacial velocity. The temperature datawere acquired using the melt depth calculated from theradial distribution function and density profile. It is ap-parent from the data that the interface continued to coolduring solidification, i.e., no steady-state undercoolingwas established.

The interfacial velocity was calculated by fitting themelt depth data with a least-squares parabola. The re-sulting fit is shown in the top graph of Fig. 18 as a solidline. Although the solidification velocity increases con-tinuously as the interfacial undercooling increases, it isclear that the crystal is growing at a rate that is unreal-isitically high for (100) silicon. The regrowth velocity at


~o

&D nA

ion

o

g) og O

Q

~oE O

OC

0o0

I

P5

(100) OrientationMEAM Silicon

Ts„,.„.„= 1000K

50 V5Time (ps)

I

100 125

FIG. 18. The melt depth and temperature history of a none-quilibrium molecular dynamics (NEMD) simulation of meltingand rapid solidification for MEAM silicon. The temperatureshown corresponds to the temperature of those atoms in the in-terfacial region between solid and liquid.

a time of 80 ps, for example, is 43 m/s. The interfacialtemperature at this time is approximately 1275 K, whichrepresents a 200 K or 14% undercooling with respect tothe equilibrium melting temperature. At the same per-centage undercooling, the Stillinger-Weber potential pre-dicts an interfacial velocity of approximately 15 m/s. "Furthermore, the maximum interfacial velocity at whichsolidification will yield crystalline silicon has been experi-mentally determined to be 15 m/s. ' Experiments haveshown that regrowth above this rate yields the amor-phous phase, rather than crystal. Solidification in theMEAM system produced defect-free crystal at a velocityof 43 m/s, which represents a serious error.

The unphysically large solidification velocities ob-served in the MEAM system may be due to the fact thatMEAM silicon has a diffusion coefficient of 1.16X 10cm /s at a temperature of 1475 K, while the correspond-ing value for Stillinger-Weber is 6.94X10 at a temper-ature of 1691 K. The MEAM silicon diffusion coefficientis, in general, twice as large as that predicted by theStillinger-Weber potential. Since the regrowth velocity isdirectly proportional to the diffusion coefficient in theWilson-Frenkel formulation, this accounts for much ofthe discrepancy between the regrowth velocities using theMEAM and Stillinger-Weber potential model. It shouldbe noted that the larger atomic mobility found in MEAMliquid silicon may be related to the low liquid density pre-dicted by the MEAM.

VII. SUMMARY

The properties of silicon and germanium as modeled bythe modified- Tersoff and modified-embedded-atom

method have been examined. The properties of crystal-line and liquid silicon were found to be adequatelymodeled by the MEAM, although the level of agreementwith experiment was not as good as was obtained previ-ously by Broughton and Li using the Stillinger-Weber po-tential. Both the crystalline and liquid phases were foundto be less dense than is observed experimentally, and themelting point of 1475 K is approximately 14% lowerthan the experimental value of 1683 K. The latent heatof melting was found to be 35.0 kJ/mol, which comparespoorly with the experimental value of 50.6 kJ/mol, but isslightly better than the value of 30.9 kJ/mol predicted bySW. The liquid-phase diffusion coefficient was found tobe approximately twice that predicted by the Stillinger-Weber potential, which led to unrealistic resolidificationvelocities during a nonequilibrium molecular-dynamicssimulation of laser melting and regrowth. The MEAMwas found to model the amorphous phase of silicon rath-er poorly. A maximum in the liquid density was found ata temperature of 900 K, but the lower densities that werefound by slowly quenching below this value were stillsubstantially greater than those of the crystal at the sametemperature. The density of this disordered phase at atemperature of 300 K, for example, was 2.387 g/cm,which is too large in comparison to the range of 1.71—2.2g/cm that is reported for amorphous silicon at this tem-perature. The coordination of MEAM amorphous siliconis approximately 4.65, which is too large compared withthe experimentally observed coordination range of4.0—4.25.

Although the properties of crystalline silicon aremodeled well by the modified-Tersoff potential, those ofthe liquid phase are not reproduced well. The major er-ror was found to be the predicted melting point of 2547K, which is approximately 850 K too large. The coordi-nation of Tersoff liquid silicon was found to be approxi-mately 4.6, which is lower than the experimental value of6—8. The liquid was found to spontaneously nucleate theamorphous phase upon cooling from 2500 to 2000 K. Noeffort was made to further resolve the transition tempera-ture. The amorphous phase that is obtained upon furthercooling to room temperature has a structure that is ingeneral agreement with that produced by the Stillinger-Weber potential. The latent heat of the crystal to amor-phous transition at room temperature, however, is 27.0kJ/mol, which does not compare well with the SW valueof 15.4 kJ/mol, nor with the experimental -value of 11.9kJ/mol.

The MEAM potential was found to model the liquidphase of germanium very poorly. Efforts to obtain astable solid-liquid interfacial system failed, resulting in-stead in evaporation of the disordered solid. BulkMEAM liquid germanium was obtained by switching theidentity of a moderate temperature MEAM liquid siliconconfiguration. This system was stable to a temperature of2200 K, but vaporized during a slow heating of 10' K/sfrom this value. Although the Tersoff potential wasfound to produce a stable solid-liquid system for germani-um, the predicted melting point of 2554 K comparespoorly to the experimental value of 1210 K. The ratherpoor description of germanium afforded by these two po-


tentials casts serious doubt on their ability to successfullymodel disordered phases of silicon-germanium alloys.

ACKNOWLEDGMENTS

The authors are pleased to acknowledge the financialsupport of the National Science Foundation (Grant No.

DMR-8915333) and the Cornell Materials ScienceCenter, which is supported by the National ScienceFoundation (through Grant No. DMR-9121654). We arealso pleased to acknowledge a very generous allocation ofcomputer time on a Cray Y-MP from the PittsburghSupercomputer Center. S.J.C. wishes to acknowledgehelpful discussions with Mike Baskes.

*Current address: Department of Chemical Engineering, Uni-versity of California, Berkeley, CA 94720-9989.

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comparison of semi-empirical potential functions for silicon and germanium

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