PHYSICAL REVIEW B VOLUME 48, NUMBER 19 15 NOVEMBER 1993-I

Path-integral simulation of crystalline silicon

Rafael Ramirez and Carlos P. HerreroInstituto de Ciencia de Materiales, Consejo Superior de Investigaciones Cientigcas,

c/ Serrano 115 duplicado, E 280-06 Madrid, Spain(Received 6 July 1993)

A Monte Carlo path-integral simulation of crystalline silicon has been performed with the empir-ical interatomic potential developed by Stillinger and Weber. Several Gnite-temperature properties(potential energy, radial distribution function, and quantum delocalization) have been calculatedand compared with experimental data and with classical simulation results. The employed quan-tum method leads to an adequate description of quantum eKects such as zero-point vibrations, andreproduces the crossover to the classical limit at high temperatures. Deviations of the simulatedvibrational energies from those derived from experiment are due to the limitations of the potentialmodel, which overestimates the vibrational frequencies of the solid.

The interest of silicon in basic science and technol-ogy of semiconductors has motivated a large number ofcomputer simulation studies on clusters, surfaces, andcrystalline and amorphous bulk silicon. To exemplifythe wide scope of these investigations we recall some re-cent reports on structural and dynamical properties ofSi adatoms on surface steps, on the formation and in-teraction of crystal defects, on epitaxial crystal growthfrom the melt, on dynamical and electronic propertiesof amorphous Si, or on low-energy crystal structure ofsilicon. All these investigations rely on molecular dy-namics (MD) or Monte Carlo (MC) simulations as nu-merical approaches to study finite-temperature proper-ties. An essential point in simulation studies is the elec-tion of an adequate model for the microscopic interac-tions. For silicon, empirical potentials including two-and three-body terms have been developed by Stillingerand Weber (SW), Tersoff, 7 and Biswas and Hamann.The SW potential was parametrized on the basis ofsolid- and liquid-phase data for Si and yields an ade-quate description of these aggregation states, althoughsome failures of this potential when modeling surfacereconstructions have been reported. Besides, since theappearance in 1984 of the milestone paper by Car andParrinello, it has become feasible to perform condensedmatter simulations, in which the energy of the systemis derived from standard electronic structure methods(density-functional theory, Hartree-Fock). In this ap-proach, both electronic states and nuclear coordinatesare treated as dynamical variables that evolve simultane-ously in time. The dynamics associated to the electronsis fictitious and represents a way of keeping the electronsclose to the Born-Oppenheimer surface. In spite of thequantum mechanical nature of this method, nuclei aretreated as classical particles and therefore typical quan-tum eR'ects such as zero-point vibrations or tunneling arenot directly accessible.

The Feynman path-integral formulation of quantummechanics represents a powerful approach to the statis-tical mechanical properties of quantum systems. Thisformulation allows us to map the partition function ofa quantum system onto a classical one, upon whichclassical techniques (MD, MC) can be applied. A sum-

mary of some applications of this method in condensedmatter physics can be found in Ref. 12. The purposeof the present report is to study by Monte Carlo path-integral simulation, finite-temperature properties of crys-talline silicon. The atoms are treated as quantum parti-cles interacting through the eA'ective Stillinger-Weber po-tential, which depends only on the nuclear coordinates.The interest of this work is justified by the extensive useof the SW potential in classical simulations, althoughto our knowledge no work has been reported so far fora quantum mechanical study using this potential. Fur-thermore, this approach can be the starting point forfuture work centered on properties which are essentiallyof quantum nature, e.g. , specific heat of amorphous Si atlow temperatures or tunneling of light atoms or muoniumin a silicon matrix.

The details of the path-integral approach to the ther-modynamical properties of nonrelativistic quantum sys-tems can be found elsewhere. Therefore, we sum-marize only the formulas necessary for the presentationof our results. Given a system of P quantum particles,described by a Hamiltonian H, the partition function Zat temperature T is equal to the trace of the statisticaldensity matrix p(R, R'; P):

Z = Tr [e t'I = Tr Ip(R, R'; P)], (I)

where P is (k~T), k~ is the Boltzmann constant, andR and R,' refer to the 3P particle space coordinates. Thepath-integral evaluation of Z can be undertaken througha discretization of the density matrix along a cyclic pathcomposed of N steps:

pl ( pl t' p)dR~. p

/

Rq, R2, —/

p /Rz, Rs., —

/

. . p /

RN. , Rg., —/'~)&

' '~)where the integral is extended to the whole coordinate space. For sufticiently large N the density matrix

0163-1829/93/48(19)/14659(4)/$06. 00 48 14 659 1993 The American Physical Society

14 660 BRIEF REPORTS 48

p(R~. , R~+q, P/N) can be evaluated within the high-temperature approximation by a free-particle propagator, leadingto the following expression for the partition function of the quantum system:

( N ) 3PN/2

I 2~@$2 )I ~ I

j=l

N

dR~ exp &—P) Izk(R~+g —R~) + N V(R~)) (3)

whereP

(R~+~ —R~)' = ) .(rp(~+~) —r~~)'p=1

Nm2/2

(4)

The vector r„j indicates the position of particle p and jlabels the coordinate along the path, rn is the particlemass, and V is the potential energy part of the Hamil-tonian. The condition RN+q ——Rq holds as a conse-quence of the cyclic decomposition of the density matrixin Eq. (2). The last expression for Z is formally identi-cal to the partition function of a classical system madeup of cyclic chains of particles coupled by harmonic in-teractions. Quantum particle paths are then representedby cyclic chains formed by N beads. Within a givenchain, bead j is harmonically coupled to the j + 1 and

j —1 beads with the spring constant k. Beads in dif-ferent chains interact through the potential V only ifthey have the same index j. The interaction is the sameas that corresponding to the quantum particles V(Rz)but reduced by a factor N . The approximated expres-sion for Z [Eq. (3)] becomes exact in the limit N ~ ooand it is valid for distinguishable particles. Exchange ef-fects arising from the symmetry of the density matrix inthe case of indistinguishable particles can be included inthe formulation of Z. However, although Bose systemscan be readily treated by this formalism, the exten-sion to many-fermion systems is the subject of currentinvestigations.

In this work, we have applied the classical Metropo-lis Monte Carlo sampling to obtain finite-temperatureproperties of crystalline silicon. The canonical partitionfunction corresponds to Eq. (3), where V(R~) is givenby the SW potential. The simulation cell is a 2 x 2 x 2superlattice of the fcc unit cell, and contains 64 Si atoms.The cell parameter amounts to 10.86 A. A Monte Carlostep (MCS) consists of two different samplings. The firstone proceeds by moves of individual beads, and attemptsto move are performed sequentially for each bead of eachatom. The second one is a random move of the center ofgravity of the cyclic chain associated with each Si atom.At each studied temperature, the maximum distance al-lowed for random moves was fixed in order to obtain anacceptance ratio of about 50% for each kind of sampling.The physical quantities calculated in this work are theaverage potential energy (U), the radial pair distribu-tion function (rdf) g(r), the center of gravity (r) of thecyclic paths associated with the Si atoms, and their meansquare deviation (Ar ), which is a measure of the quan-tum delocalization of the particle. The angular bracketsindicate ensemble averages over the whole MC trajectoryat temperature T.

In Fig. 1, the convergence of the potential energy (U)for temperatures in the range 50—500 K is shown as afunction of the number of beads N, employed in the dis-cretization of the cyclic paths. Each data point corre-sponds to a simulation run consisting of 10 MCS pre-ceded by an equilibration of 10 MCS. At 500 K a smallnumber of beads (N 4) is enough to obtain good con-vergence in the potential energy. At 50 K the numberof beads needed is about ten times larger. The conver-gence of (U) as a function of N is related to the goodnessof the high-temperature approximation [Eq. (3)] at tem-perature NT. From the simulations we found that theproduct NT must be larger than 1300 K (i.e. , about twotimes the Debye temperature of Si, 8~ = 640 K) in orderto obtain adequate convergence in the potential energy.

The temperature dependence of the average potentialenergy (U) is shown in Fig. 2. Classical results cor-responding to the SW potential are displayed by opensquares. Note that within the employed path-integralformalism, the classical limit is easily obtained by settingN = 1 in Eq. (3). The numerical results can be comparedwith the average potential energy of a three-dimensional(3D) classical harmonic oscillator ((U) = 3k~T/2), rep-resented by a dotted line. The deviation of the clas-sical simulation from this straight line at temperaturesabove 500 K is due to the anharmonicity of the inter-atomic interaction. The results obtained by quantumpath-integral simulation are shown by filled squares. At

80

500 K

60-300 K

c5~ W

C

I40 I

/ Q

I

/

20- (/

/

/

200 K

100 K

50K

I

10I

20 30

Beads per Si atom

FIG. 1. Average potential energy (U) of crystalline siliconobtained by quantum Monte Carlo path-integral simulationfor a different number of beads N and several temperatures.The zero of energy corresponds to the classical model at tem-perature T = 0 K. Error bars are approximately the size ofthe symbols.

BRIEF REPORTS 14 661

120(U) = -', d~ Ru n(~) + — P,„p(w), (6)

o 100

60

40

20

r~~ r ..'

r)r 9'

0 psr

gr rP

~rg'

~ ~

,cf

0 i' I

0I

200I

400I

600

Temperature (K)FIG. 2. Temperature dependence of the potential energy

(U) of crystalline silicon. Filled and open squares are resultsobtained by quantum and classical simulations, respectively.The broken line is derived from the experimental phonon den-sity of states p, „~(cu), while the dotted line is the classicalexpectation in the harmonic approximation ((U) = 3knT/2).Error bars are approximately of the size of the symbols.

low temperatures, the energy converges to a constantvalue of about 35 meV/atom, associated to zero-point vi-brations. The broken line in Fig. 2 has been calculated byusing the phonon density of states p,„~(w), derived fromneutron scattering measurements on crystalline silicon, "and within the harmonic approximation:

where the average number of excited phonons of fre-quency w, n(w), is given by the Bose-Einstein statistics:

n(ur) = exp~ ~

—1fRu)

The simulation results have a systematic deviation fromthe values obtained by Eq. (6). The zero-point energyis shifted by about 5 meV/atom, which converted intofrequencies translates into a shift of ~ 40 cm towardshigher wave numbers. It is interesting to note that thephonon density of states calculated for crystalline sili-con by MD simulations with the SW potential, ps'(cu),agrees reasonably well with p,„~(w), and the main dif-ference between both is an almost rigid shift of psvv(cu)by ~ 50 cm towards higher frequencies. Such a shiftincreases the zero-point energy by about 5 meV/atom,in agreement with our results. Also Car-Parrinello sim-ulations of the phonon spectrum of amorphous Si gave ashift close to 50 cm in comparison to experiment.Therefore, we conclude that the quantum simulation re-produces well the temperature dependence of the poten-tial energy of crystalline Si, apart from the constant shiftinherent to the employed SW potential.

The center of gravity (r„) of the paths sampled for agiven atom p is

(rp) =% ' ) rp,

and the corresponding mean square deviation is given by

CCI

~ ~

~ ~v5VC

75

1

60 ~i

'o g

~ y

3P

15

10

0

10

800 K

200 K

15-

CV

00

I I I I

100 200 300 400 500

Temperature (K)

0

20

15

10

~l

I II II II

II I

I

50 Kquantumc= lawaic=al

FIG. 3. Mean square deviation of the center of gravity ofthe paths associated to a quantum particle as a function of thetemperature. Filled squares are quantum simulation resultsfor crystalline silicon. The dotted line shows the analyticalresults for the SW potential. The broken line is derived fromthe experimental phonon spectrum, while the broken-dottedline is the value for a free Si atom.

0 R

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7distance (A)

FIG. 4. Radial distribution function (rdf) for crystallineSi at three different temperatures. Full lines are the resultsof quantum simulations. Broken lines represent the classicallimit ~

14 662 BRIEF REPORTS 48

(+r„') = (rr') —(r„)' .

The value (Zr„) measures the delocalization of the parti-cle due to quantum fluctuations. In Fig. 3, the results ob-tained for the delocalization (Ar ) in the quantum sim-ulation of crystalline silicon are shown for several tem-peratures (filled squares). The broken-dotted line is thevalue of a &ee particle (fp) with the mass correspondingto a Si atom:

ph2(Ar„) (10)

For a 3D harmonic oscillator (ho) with frequency w, thedelocalization (Er„) can be worked out analytically:iz

2mar i,2k' T ) mw

The dotted line in Fig. (3) was obtained by weightingthe value derived from Eq. (11) with the phonon den-sity of states ps'(ar). The broken line corresponds toa similar calculation, but using the density of states de-rived from experiment, p,„~(ar). The simulation resultsfollow closely the values expected for the SW potential.Their systematic deviation from the broken curve cal-culated by using p,„~(ar) is due again to the rigid shifttowards higher frequencies of the phonon spectrum asso-

ciated with the SW potential. Above 400 K the quantumdelocalization of the Si atoms is almost the same as thatexpected for a free particle.

Finally, we compare the results obtained for the rdfg(r) by using the SW potential in quantum and classi-cal simulations. In Fig. 4, the g(r) curves are presentedfor three temperatures (50, 200, and 800 K), and for dis-tances corresponding to nearest neighbors. The differ-ences between the classical and quantum results are re-markable at low temperature, while at 800 K, above theDebye temperature of Si, both curves are nearly identical.

Summarizing, we have undertaken the study of sev-eral finite-temperature properties of crystalline Si inter-acting through the SW potential by means of Feynmanpath-integral simulation. Differences between the quan-tum mechanical treatment and the classical limit havebeen quantified. Our results are encouraging enoughto try to extend this study to more complicated sys-tems (e.g. , doped and amorphous silicon), for whichfinite-temperature properties are not readily calculableby other computational methods.

This work was supported by CICYT (Spain) underContract No. MAT91-394. We thank Victor Velasco forcritically reading the manuscript.

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