determination of parameter-free model potentials for the molecular dynamics simulation of amorphous...

E L S E V I E R Journal of Molecular Structure (Theochem) 313 (1994) 173-188

THEO CHEM

Determination of parameter-free model potentials for the molecular dynamics simulation of amorphous

semiconductors - - application to silicon

M i c h a e l S c h r e i b e r *'a, B e r n d L a m b e r t s b

alnstitut ff~'r Physik, Technische UniversitiT"t Chemnitz-Zwickau, Postfach 964, D-09009 Chemnitz, German)' blnstitut fti'r Physikalische Chemie, Johannes-Gutenberg-Universitiit, Jakob-Welder-Weg 11, D-55099 Mainz, German),

Received 3 February 1994; accepted 3 March 1994

Abstract

We present a molecular dynamics simulation of amorphous silicon. Emphasis is put on a new method for the determination of the interaction potential which is calculated self-consistently by means of a density-functional approach without external input parameters. This procedure is repeated during the simulation in order to optimize the potential. The quality of the results thus obtained is demonstrated by a comparison with experimental results for the structure and the dynamic behaviour of amorphous silicon, in particular the pair correlation function, the bond angle deformation, and the density of phonon states.

I. Introduction: Amorphous materials

With respect to many physical properties, amorphous semiconductors occupy an intermediate position between the fluid and the crystalline state [1]. In contrast to crystalline systems, for example, one does not find a uniquely characterized structure. On the other hand, there is no unambiguously defined transition temperature for the transition between the liquid and the amorphous phase. In this respect amorphous systems are similar to glassy materials, in which a generally accepted definition of the glass transition temperature is also lacking. This analogy between the glassy and the amorphous state is not only restricted to this feature. More important and characteristic for

* Corresponding author.

these systems is the statistical distribution of most physical quantities, which follows from the above mentioned fact that the amorphous (as well as the glassy) structure cannot be characterized uniquely, but only statistically. Such a distribution can be more or less pronounced. For example, the arrangement of the atoms or molecules in the first or second coordination sphere, i.e., the short-range structural features are governed by rather narrow distributions of the interatomic distance, of the bond angle, and of the number of nearest neighbours, close to the respective values of the crystalline configuration. But beyond next nearest neighbours it is nearly impossible to predict the atomic positions with respect to each other [2].

Of course, these structural properties are reflected in the electronic behaviour of the systems. The density of states for amorphous silicon which

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174 M. Schreiber, B. Lamberts/J. Mol. Struct. (Theochem) 313 (1994) 173-188

0.5 e -

0.4

0.3 t

0,2

0.1

0 -15 -10 -5 0 5

E(eV)

Fig. 1. Calculated electronic density ofstatesn(E) for a-Si(thick line), compared with that for c-Si (thin line), from Ref. [35].

is displayed in Fig. 1 shows peak positions which are nearly unchanged as compared to the crystalline silicon. But the peaks are widened and, moreover, different amorphous probes show different widths. Additionally, one observes an unregular occupation of the band gap which can be attribu- ted to the local deviations of the coordination number from the crystalline value. In the valence bond picture [3] the occurring threefold coordinated atoms feature an unpaired electron in an orbital which is not hybridized. Accordingly, the orbital energy of these so-called dangling bonds is located within the band gap.

This is only one example which shows the close relation but also important differences between the amorphous and the crystalline modifications. With respect to silicon, the technological relevance of the crystalline material is undisputable. This leads naturally also to an interest in the electronic behaviour as well as the structural properties of amorphous silicon. That interest is certainly enhanced by the lower price which seems to favour the manu- facturing of devices like solar cells of amorphous rather than crystalline silicon. But the already mentioned existence of occupied states within the band "gap" in amorphous silicon means that doping with the aim of controlled alteration of the

conduction properties which can be easily performed in the crystal is almost impossible here unless one finds a way to reduce the number of silicon states in the gap drastically. The number of these states is usually of the same order of magnitude as the number of excess electrons or holes which are created by doping. This explains the significance of theoretical as well as experimental investigations of the electronic properties in the band gap region.

Although it will not be the subject of the present investigation, we note in passing that one way to decrease the number of states in the band gap is to saturate the dangling bonds of the atoms with "false" coordination (i.e., those without four bonds) by hydrating the probes. Such a compensa- tion with hydrogen atoms leads to a binding of the electrons of the dangling bonds so that the respective states are shifted into the valence band region. In experiments, this is usually at least partially the case, because hydrogen is often present in common preparation techniques like in the glow discharge method [4].

The significance of the dangling bonds already points out the importance of the short-range order. In these covalent materials the interaction is governed by the nearest-neighbour bonds which are strongly oriented into certain directions. The influence of more distant particles is much smaller. Consequently one expects that differences between the crystalline and the amorphous modifications are due to different short-range order and not to the long-range order which is missing in the amorphous system. This is confirmed by the density of phonon states, shown in Fig. 2. As observed for the density of electron states, the phonon peaks are also nearly unchanged but wider in the amorphous as compared to the crystalline system. Therefore it is reasonable to expect that a good description of the short-range order, say up to third or fourth nearest neighbours, is sufficient for a realistic picture of these materials. This argument already determines the size of the system which we have to analyze below.

Experimentally obtained spectra however, e.g. of the static structure factor of amorphous semiconductors, are slightly different for different probes. This occurs for all properties like the

M. Schreiber, B. Lamberts/J. Mol. Struct. (Theochem) 313 (1994) 173 188 175

O- • ," " e ~

K 20 40 60 80

hw (meV)

Fig. 2. Experimental vibrational density of states for a-Si (a), compared with the calculated vibrational density of states for c- Si (b) and a-Si (c), from Ref. [36].

density of states or transport properties which depend on the local characteristics and thus on the short-range order of the samples. It reflects the above mentioned fact that there is no unique amorphous system, but rather a statistical ensemble. As a consequence, the theoretical description requires an ensemble average over a set of appropriately chosen samples.

Of course, additional and usually larger deviations occur if one compares samples made by different preparation techniques, which can lead to qualitatively different probes. A trivial example is the mentioned hydrogen which always appears in chemical vapour deposition but does not necessarily show up in samples made by laser quenching [1,5]. Therefore additional care has to be taken with respect to these experimental details, when one compares experimental spectra with numerical results of amorphous materials.

But it is an advantage of computer experiments that (at least in principle) all these details are fully controllable and can be independently varied, so that the experimental conditions can be reproduced. Such a computer experiment shall be described in the following, namely a molecular

dynamics (MD) simulation of amorphous silicon. An important feature of our approach is the self- consistent generation of the interaction potential which yields the forces for the MD calculation. This is achieved by a local-density-functional (LDF) computation which does not need empirical input data. Thus we perform an ab initio LDF calculation of the electronic system for the actual atomic configuration which follows from the MD. The LDF results are in turn used as input for the continuation of the MD procedure. In this way the self-consistency is achieved.

Although we employ a parametrization of the interaction potential, the numerical expense is con- siderable because of the ab-initio program. There- fore we cannot consider system sizes of the order of 10 000 atoms, which are feasible in purely empirical MD simulations. We are restricted to approxi- mately 100 atoms, but in view of the above arguments this should be sufficient because of the dominant short-range characteristics. In principle we have to average the results over a large number of samples thus taking into account the statistics of the distribution of the physical quantities as discussed above. But according to the ergodic hypoth- esis that the ensemble average is equivalent to the time average over the phase space, it is sufficient to average over many configurations during the MD simulation of the same sample. We conclude that a reasonably long MD run of a relatively small system provides a realistic description of amorphous silicon.

In addition, the MD method provides a micro- scopic picture of the dynamics of the system, allow- ing observation of the movement of the individual particles. This aspect of our investigation, however, shall be pursued in another paper. In this paper we explain our approach. After a more general discus- sion of computer models in the next chapter, we present the main aspects of the density-functional formalism and the MD simulations, including the parametrization of the ab-initio potentials in the following two Sections. Then Section 5 contains technical details of the simulations. Results for amorphous silicon (without any hydrogen) are described in Section 6 in comparison with experiments to demonstrate the feasibility of our approach. Finally, the summary and an outlook

176 M, Schreiber, B. Lamberts/J. Mol. Struct, (Theochem) 313 (1994) 173 188

to fl~rther investigations with this approach as well as possible improvements are contained in Section 7.

2. Computer models for silicon

Due to the technological relevance of silicon and due to the relative simplicity of the material (con- taining only one kind of atom, which in addition is not too large) there has been great interest in computer simulations of silicon in recent years. Various modifications have been studied and many different aspects and physical properties have been analyzed. It is beyond the scope of the present paper to give a review of these works, we mention rather a few typical investigations which are of particular interest in the context of our subse- quently described approach.

Stillinger and Weber [6] have successfully described the structural properties of liquid silicon by means of an empirical three-body potential. Similarly, Biswas and Hamann [7] performed an MD simulation, but they fitted the potential to ab initio calculations of the total energy of different lattice structures. In a different ansatz Sankey and Niklewski [8] took the valence electrons into account in a simplified density functional calculation, and derived the potential and the MD forces via the Hellman-Feynman theorem. The Car-Parrinello method [9] is also based on the density functional formalism, so that an ab initio calculation of the interaction is possible. In that approach the self-consistent solution of the electronic problem is directly combined with the dynamics of the atoms by means of a Lagrangian formalism. As a motivation for the characteristics of our approach, we discuss briefly some principal aspects of these and similar calculations. The computer models for covalent semiconductors may be divided into two fundamentally different classes. On the one hand, classical models use effective potentials [6] which are more or less deduced from experimental data. Quantum mechanical approaches [8] can be based on ab initio calculations so that the results are independent of input parameters. But even on the fastest corn-

puters which are today available, a complete quantum mechanical description of the dynamics of many-particle systems is only possible for systems with very few (<< 10) particles. Thus one has to compromize. For reasonably large systems a full configuration interaction method would require far too many Har t ree-Fock matrices. Even a complete self-consistent treatment of all electrons and nuclei is not feasible for all realized configurations. Thus in general a classical equation of motion is assumed for the atoms.

One can further distinguish static and dynamic models. In the former [10] a given geometry is investigated, but not as a function of time, although it might be optimized during the simulation [11,12]. For dynamic models the time develop- ment of the system is explicitly studied [13,14]. A natural choice in this case is the MD method because the equations of motion are solved numerically and thus the trajectories of the atoms in phase space are explicitly evaluated, as is performed in the present work.

To estimate the reasonable size of the system which has to be investigated we recall the domi- nance of short-range order. This is corroborated by an analysis of the radial pair distribution function, i.e. the Fourier transform of the experimentally available static structure factor. The plot in Fig. 3 does not show a significant structure beyond the third-nearest neighbour peak. Thus the important information should be contained within this range, longer range correlations do not appear due to the disorder. A further argument is pro- vided by the overlap integral which becomes negli- gibly small between orbitals on sites beyond third nearest neighbours. Although this overlap depends on the chosen basis and should also be corrected with respect to the occupation numbers, the con- clusion is in agreement with the above observa- tions: we need at least a system size of 64 atoms. We assume a diamond lattice initially, with these atoms in the supercell and employ periodic boundary conditions.

The time span which we have to cover in the simulation depends on the physical effects we want to simulate. The fastest process would be the relaxation of the electrons. But the description of these would require a complete quantum

M. Schreiber, B. Lamberts/J. Mol. Struct. (Theochem) 313 (1994) 173-188 177

Fig. 3. Schematic illustration of the structural origin of certain features in the radial pair distribution function p(r) = J(r) /r 2 for an amorphous solid. The lower picture shows a two- dimensional disordered system. The shaded areas around the atom at the origin (large black circle) demonstrate how the atoms can be associated with shells of first and next nearest neighbours. This is reflected in the radial pair distribution function shown in the upper picture.

mechanical treatment of the system, which is beyond our computational means. As mentioned, we describe the dynamics of atoms classically and apply the Born-Oppenheimer approximation so that the electrons follow the movement of the nuclei instantaneously. Thus the relaxation of electrons cannot be studied here.

The lattice vibration or in the amorphous systems analogously local vibrational modes and rotational modes determine the smallest time scale in the simulation. From the phonon density of states its order of magnitude can be estimated as 10-15s, which constitutes the time step for the numerical integration of the equations of motion.

The speed of diffusion or the mean quadratic displacement of the atoms limit the reorientation

of the atoms during the structural relaxation. Accordingly the simulation time should be longer than the time needed for an atom to diffuse through the system size of 64 atoms, yielding about 10-12 S.

An even longer time scale of the order of 10 l0 s is given by the fastest cooling rate that can be experimentally achieved, e.g. by laser quenching [15]. In order to simulate the structural relaxation of the system over such a long time, one would need 10 5 MD steps (of 10 -15 s as estimated above).

But even with the most advanced quantum mechanical method of Car and Parrinello [9] it is not feasible to perform more than a few thousand time steps for the discussed system size of about 10 2 atoms. Therefore we separate the ab initio quantum mechanical calculation of the electronic system from the classical dynamics of the atoms. Experi- ence from test runs has shown that it is not necessary to update the MD potential by means of the LDF calculation after every MD step. Thus the time-consuming quantum mechanical part of the program has to be performed very rarely, about every 10 2 MD steps only. To exploit this advantage even more, we implement a parametrization of the potential so that the evaluation of the forces in each MD step is even easier. This parametrization is improved after each LDF calculation, so that during the entire simulation run more and more configurations contribute information. In this way the rather crude initial parametrization which is based on the diamond lattice as the starting configuration is optimized during the MD walk through the configuration space of the system. Thus at each MD step we can use a parameter- free approximation to the potential, which is already rather good at least for the actual configuration, because this configuration cannot yet be too different from the last LDF configuration which entered the parametrization. Of course, it is necessary to continue the simulation at least until the potential parametrization has converged. But during the simulation the number of possible time steps between updates of the potential by means of the time-consuming LDF procedure is growing. Consequently, our approach allows us to perform the simulation for comparatively long time scales up to 10 -1° s.

178 M. Schreiber, B. Lamberts/J. Mol. Struct. (Theochem) 313 (1994) 173 188

3. The local density functional calculation of the electronic system

The density functional formalism is the state of the art of electronic structure calculations for covalent semiconductors as well as other materials. It is therefore not surprising that a variety of imple- mentations and modifications of the algorithm can be found in the huge amount of literature on this method. We do not attempt to give a review here but rather briefly resume the foundations in order to explain the details of our implementation.

The formalism was based by Hohenberg and Kohn [16] on the theorem that all ground state properties of a system of electrons and ions in an arbitrary external potential Vext are functionals of the electron density n(r). This is valid in particular for the total energy E = E[n(r)]. Due to a variational principle, this energy is always larger than the ground state energy except for the corresponding ground state density. Following Kohn and Sham [17] the energy can be written as

+ Idrn(r)(Vext(r) + ½b(r)) E[n(r)] Tin(r)]

+ E~c[n(r)] (1)

Here T denotes the kinetic energy of the electrons, is the Coulomb potential and Ex~. represents

the exchange-correlation energy comprising the exchange interaction and the correlation between the electrons.

The main problem of the density functional approach is the determination of this exchange- correlation energy. The exact functional dependence is unknown, one can even imagine pathologi- cal examples for which no such functional exist. Therefore reasonable assumptions have to be made for this functional [18]. For solids it is most commonly approximated locally by the exchange- correlation energy density ~xc of a homogeneous electron gas of the respective density n = n(r):

E,. LD = J dr n(r)¢xc(n(r)) (2)

In this local density approximation the variation of the total energy with respect to the density, i.e.

6E[n(r)]/6n(r) leads to a single-particle equation

( - 1 V 2 + V(r))t),(r) = ei~,i(r ) (3)

with a particularly simple interaction potential

V(r) = V,x,(r) + O(r) + %,(n(r)). (4)

The charge density in Eq. (4) is given by the N eigenfunctions ~i of Eq. (3) as

N

n(r) = I ;,(rll 2 !51 i I

so that Eqs. (3), (4) and (5) have to be solved self- , consistently, which is performed by iteration.

We represent the eigenfunctions 2~i in a basis of Gaussians and (real) spherical harmonics $/,,:

~,(r) = ~-~ g'~('l/__, ~'j,m ~_S,m(r,r,)lr - ral / / Im a

x exp(-o4(r - r~) 2) (6)

Each orbital of an atom (located at r,) is given by a linear combination of two such functions ( j = 1,2). The advantage of this basis set is that many of the integrals which are required in the computation (except the exchange-correlation contribution and the non-local three-centre pseudopotential, see below) can be solved analytically. In contrast to the basis of plane waves which was used by Car and Parrinello [9,14] only very few basis functions are needed and it should also be easier to interpret the results in our local basis, if we want to describe local characteristics of the amorphous material.

In order to reduce the number of orbitals in the computation further, we employ non-local pseudo- potentials [19] so that only the valence electrons are described explicitly by one-particle functions (6), while the core electrons are incorporated in the pseudopotential screening the nuclei. In comparison to the full potential this screened potential is rather smooth so that the number of functions in the linear combination for the orbitals of the valence electrons can be rather small. It is well known [20-23] that this method yields excellent results for the cohesion energy as well as the structural properties.

M. Schreiber, B. Lamber ts / J. Mol. Struct. ( Theochem ) 313 (1994) 173-188 179

However the simplification of the computation due to the reduction of the basis set has to be paid for: the disadvantage of the method is that the pseudopotential is given by a non-local operator with different components for each orbital momentum I. The respective matrix elements can be expressed by means of the projection operator P / a s

2

{il Vps[Jl = Z Z I drOi(r - rb)l/ps(r -- rc) I = 0 a,b,c

× P,(r - r,)Oj(r - ra). (7)

Usually only the two or three lowest values of l have to be taken into account, because the expan- sion series converges rapidly. We note that the components of the pseudopotential, which we have used in the following investigation, are also expanded in Gaussians [20], which simplifies the evaluation of the integral (7).

Nevertheless, the calculation of the non-local three-centre integrals for the matrix elements of the pseudopotential is one of the most time- consuming parts of the program, beside the Coulomb interaction and the exchange-correlation contribution. The determination of the local contribution of the pseudopotential and the diagona- lization of the secular equation are less expensive and the effort for the kinetic energy is even smaller. All further parts of the computation including sto- rage and transformation of the data require less than 5% of the overall computer time. Therefore we discuss the first three terms in more detail to show how the numerical effort can be reduced.

In the calculation of the three-centre integrals we distinguish those terms for which the projection operator Pt and the orbital function are located on the same atom, i.e. a = c in Eq. (7). Then the projection is trivial and the matrix elements can be analytically integrated. In the general case we express the spherical harmonics of the basis (6) as Fourier integrals. The respective exponentials exp (iqra,) are expanded into Bessel functions and spherical harmonics. Then the integrations with respect to the annular coordinates can be performed exploiting the orthogonality of the spherical harmonics to simplify the expressions. With respect to the radial coordinates two two-centre

integrals appear. An analysis shows that only nearest neighbour and next nearest neighbour contributions of the potentials are significant. We note that the number of different integral terms can be drastically reduced by symmetry arguments using the Clebsch-Gordon coefficients. These terms are numerically computed and the results are fitted by polynomials. Finally the remaining product of two such polynomials and a Gaussian can be analytically integrated with respect to the radial coordinate. The decisive advantage of this procedure is that there is no numerical integration necessary any more during the self-consistent LDF iteration of Eqs. (3)-(5), because the mentioned determination of the polynomials has to be performed only once for a given configuration of the atoms.

The Coulomb potential is expressed [24] as a discrete Fourier series in terms of the Fourier- transformed overlap integral which can be obtained analytically and the Fourier-transformed charge density. This is efficient because the fast Fourier transformation requires only Nlog N operations [25] instead of the N 3 operations of a three dimensional numerical integration (on a mesh of N 3 points in both cases). The computational amount is further reduced by determining the charge density on a regular mesh which allows us to treat most of the calculation as a quasi-one-dimensional problem.

The numerical computation of the exchange- correlation contribution would require a three- dimensional integration. This can be avoided by the fast Fourier transformation of ext. The resulting two-centre integrals over Gaussians and the exponentials exp (iqra,) can again be solved analytically. The main advantage of this procedure is that in this way the exchange-correlation contribution is analogous to and can therefore be treated together with the Coulomb term, thus reducing the numerical effort.

A well-known problem of LDF calculations especially for large systems is the convergence of the charge density. Often fluctuations occur. The usual strategy to avoid these fluctuations is to reduce the alteration of the charge density in each LD F iteration by mixing the input and output densities of the iteration to obtain the input for


Ecoh [a~b.u.]

-1

-1.02

-1.04

-1.06

-1.08

-1.1

-1.12

-1.14

-1.16

+

I I I I i I I I

LDF calculations MD potential +

+ ÷ ©

I I I I t I I I L

4 4.1 4.2 4.;] 4.4 4.5 4.6 4.7 4.8 4.9 nearest neighbour distance [a.u.]

i +

©

Fig. 4. Comparison of the cohesion energy resulting from our ab initio LDF calculations and our potential parametrization. The values are calculated for different lattice constants of a diamond lattice.

the next iteration:

hi. (r) old , ew = + (1 - ( 8 )

Much more efficient is a modified Newton algorithm [26,27] in which the new input density is determined from the old output and the previous alteration.

The other parts of our LDF program use stan- dard methods. Due to the non-orthogonality of the Gaussian basis functions centred on different atoms, Eq. (3) yields a generalized eigenvalue problem. It is solved by a Cholesky decomposition and Householder tridiagonalization with implicit shift [28].

4. The molecular dynamics simulation

The time evolution of a many-particle system is most commonly calculated by means of a molecular dynamics simulation [29,30], which means the explicit numerical integration of Newton's equations of motion. Various algorithms exist for this purpose [31]. We use one of the simplest schemes, the Verlet algorithm [32], which can be comprehen- sively written as

r.(t + 7-) = 2r.(t) - r.(t - T) + Fa(t)TZ/m.. (9)

Here the new position r. (of the atom a with mass ma) after the time step 7- is determined from the

previous positions and from the forces F~ acting on the atom at time t.

To derive these forces, we need the interaction potential. In order to obtain this potential, we have tried to map the interaction terms of the LDF calculation onto the atoms in MD simulation. This was, however, not satisfactory, because it is not uniquely possible to attribute a certain contribution to a distinct atom. This is particularly obvious for the Coulomb integral and the non- local three-centre contributions of the pseudopotential. As a consequence, locally similar configurations might lead to significantly different interaction terms. Thus the parametrization turned out to be very difficult and converged badly.

Therefore, we have chosen another possibility, namely to fit a parametrized ansatz for the potential to the cohesion energies computed for different configurations of the MD simulation by means of the LD F routine. As an ansatz for the potential

V = ~ ~(r,, rb)O,b (10) a~:b

we use a two-centre function ¢) to describe the unperturbed covalent interaction in a regular diamond lattice and a many-body part 0 to include the influence of the disordered structure, i.e. the deviation from the lattice. In this way the two-centre part can be separately fitted to the ab initio results for crystals with various densities. (We choose the


Table 1 Cohesion energy of disordered configurations of 64 silicon atoms, taken from independent simulations after 50 LDF calculations which were separated by 10 MD steps in turn, starting each simulation from an already thermalized amorphous configuration. The MD values are determined from the parametrized potential and compared with the ab initio LDF results

MD (eV) -4.61 -4.24 -4.92 -4.74 -4.85 -4.72 -4.71 -4.75 LDF (eV) -4.50 -4.34 -4.88 -4.78 -4.85 -4.83 -4.84 -4.84

A (eV) -0.11 +0.10 -0.04 +0.04 ±0.00 +0.11 +0.13 +0.09

diamond lattice because it yields the lowest cohesion energy.) The results in Fig. 4 demonstrate the quality of the fitted parametrized potential

O(ra, rb) :=_ Araff eXP(ra~_p) _ BrJexp ( k, rab"/2_ P/

(11)

with rab --= Ira -- rb[ and a cut-offradius p to enforce the short range of the interaction. This is important for the numerical efficiency of the ansantz, because it guarantees that the calculation of the interaction between N particles scales with N for large N.

For the weight function 0 we use the following expression:

Oab =exp (-- Z [f(ra'rb'rc) + f(rb'ra'rc)])

f(ra, rb, r,.)=Cexp(~)Zc,Pl(coSf~abc) l

(13)

where Pl denotes the Legendre polynomial and f~abc the angle between the bonds (a-b) and (b-c). The sum in Eq. (12) covers all neighbouring atoms within the cut-off radius p, and the sum in Eq. (13) was restricted to the three lowest polynomials. The coefficients Co = 4/9 and c I = c2 = 2/3 have been chosen in such a way that the repulsive contribution of" the potential is minimized for the tetra- hedrat angle 12= 109.47 °. Thus the average influence of the surrounding atoms c on the potential between the atoms a and b is taken into account. The averaging is certainly a significant approximation, but the good agreement between the LDF results and parametrized-potential results for the cohesion energies of several

arbitrarily chosen disordered configurations which can be seen in Table 1 support the validity of our parametrization.

We note that the separation of 0 and 0 has simplified the fitting of the parameters considerably. The quality of the fit is better, because ambiguities are avoided and because the number of independent variables in the separate fits is smaller.

The above ansatz for the functional dependence of ~b and 0 is a rather arbitrary choice. We have obtained it empirically, taking into account several conditions which have to be fulfilled, namely: the weight function 0 should be unity for the diamond lattice due to the definition of 0, but the cohesion energies for other lattice structures should also agree well with experimental values. The function 0 should decrease with increasing number of atoms within the cut-off radius, because the single covalent bonds become weaker if more atoms compete for the valence electrons. On the other hand, 0 should not be dominated by the contribution of a single neighbour c. To guarantee a proper numerical behaviour of the MD simulation, 0 and 0 should be continuous and differentiable within the cut-off radius, the contribution of each atom should vanish continuously at the cut-off. Of course, the potential should also be translationally invariant, as we do not invoke any external potential Vex t. It is straightforward to see that all these conditions are fulfilled by the above ansatz. A typical set of parameters, taken arbitrarily from one of our simulation runs, is the following (in atomic units): A = 1332, q = 4.011, ")'1 = 4.067, B = 8.419, p = 0.00045, 9'2 = 3.867, C = 2.934, 9'3 -- 4.391. Of course, these values depend on temperature and pressure of the simulated system and are included here only to illustrate the ansatz. The cut-off radius p = 7.2 has been chosen in such a way that the


~tart configuration: any lattice ) potential: any empirical ~ tart potential

emplrical MD simulation )

amorphous configuration

slstent LDF calculation

~ proved parametrlsatlon the Interaction potential

~ D calculation Ith parametrlzed potential

)

ab Inltlo electronic I ~- structure of an

I amorphous configurat on

] Inon-emplrlcal potential I

Icharacterlstic I amorphous conflauration

Fig. 5. Flow diagram of the simulation packet, indicating the different self-consistency circles. The various types of output are specified on the right hand side.

third nearest neighbours in the respective crystal would be just excluded so that the potential parametrization without four-centre interactions leads to a unique choice of the c l in Eq. (13).

5. Initialization and convergence of the simulation

In the following we present the details of the actual implementation of our simulation program, as depicted in the flow diagram in Fig. 5. The simulation begins with an initial configuration of

the atoms, which is arbitrary although we usually use a crystalline one. Employing an empirical potential [6] we generate an amorphous configuration by heating the sample up to a temperature (e.g. 2000 K) significantly above the melting point and then cooling it slowly (i.e. in 5000 steps) down to the intended simulation temperature (e.g. room temperature) below the melting point. For this amorphous configuration we perform an LDF calculation iteratively until the value for the cohesion energy has converged. This yields the first value for the fit of the weight function 0. (The two-centre


J(r) [k -1]

35

30

25

20

15

I0

5

0

• ¢

3 4 5

I I I i , .

M D s i m u l a t i o n " .. E x p e r i m e n t •

• ,, 4'

f \ : j ¢

i i

6 7

Fig. 6. R a d i a l p a i r d i s t r i b u t i o n f u n c t i o n J(r) o f a-Si f r o m o u r s imu la t ion , c o m p a r e d to expe r imen t a l resul ts m e a s u r e d by n e u t r o n

sca t t e r ing f r o m L a n n i n a n d F o r t n e r [37]. The s i m u l a t i o n p a r a m e t e r s a re T = 300 K , N = 216 a t o m s , L = 31.2 a .u .

integral ~ has been fitted already from test runs for respective crystalline configurations.) A conjugate gradient method [25] is used to fit the parameters of the potential weight function 0, so that the cohesion energy is reproduced by the parametrized potential.

This is now used to continue the MD run, per- forming about 10 2 steps before the next LDF iteration is initiated. That number of steps should be large enough to obtain a significantly different configuration, otherwise we would waste computational effort in the LDF procedure. But it should also be not so large that we are in a completely different region of the phase space of the system, because then the parametrization of the potential would become much more difficult. Therefore the reasonable number of MD steps between two LDF cycles depends strongly on the external conditions like temperature and pressure.

The MD cycle (the outer cycle in Fig. 5) is con- tinued until satisfactory convergence of the potential is achieved. Obviously, the obtainable precision depends on the external conditions. If one is less interested in generating a parametrized potential which is a good approximation for all possible configurations, but if one rather aims at improving the MD simulation, then another procedure is advantageous: the parametrization of the potential should be restricted to the last 30 or 50 LDF data, because the previous configurations no longer resemble the actual configuration. In this way a

much better fit of the potential is possible, which is, however, only valid in a rather small part of the entire phase space of the system. Whether convergence has been reached, is now decided with respect to the physical properties which are determined from the MD runs. Examples will be given in Section 6.

6. Results for amorphous silicon

In order to demonstrate that the described approach allows us to simulate amorphous silicon with reasonable accuracy, we show some typical results of our calculation. These results have been obtained for a system with N = 216 atoms, per- forming 20000MD steps with a time step ~-= 10-16s. The temperature was 300K and the box length of the supercell was L = 31.2a.u., corresponding to atmospheric pressure.

The radial pair distribution function, which measures the probability to find an atom at a certain distance from another atom, is displayed in Fig. 6. The agreement with the experimental curve which is determined from the static structure factor by Fourier transformation is good. Similarly we compare in Fig. 7 the static structure factor which is directly measurable by neutron scattering and the Fourier transformation of our numerically obtained radial pair distribution function. Again, agreement is found, confirming that the static

184 M. Schreiber, B. Lamberts/J. Mol. Struct. (Theochem) 313 (1994) 173-188

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behaviour of amorphous silicon is well reproduced in our simulation. One has to be aware, however, that these spectra do not contain any angle- dependent information, because of the averaging over the angular coordinates.

In any case, from previous investigations [13] we know that dynamic properties are much more sen- sitive to the quality of the simulation so that the respective comparison between experiments and numerical results is more significant. Therefore we calculate the density of phonon states by Four- ier transformation of the velocity autocorrelation function (implicitly assuming a harmonic approximation for the vibronic levels [33]). The results are compared in Fig. 8 with the respective neutron

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scattering data. The agreement is satisfactory, the deviations can at least partly be explained by the size of the simulated system, because only time intervals shorter than the ratio of half the box length and the sound velocity can be taken into account for the spectrum. Otherwise one would include vibrations of the entire system as an arte- fact of the periodic boundary conditions. Simula- tions of systems of different size show that this finite size effect influences especially the TA band. On the other hand, the shape of the TO band depends strongly on the degree of relaxation of a particular sample, as shown previously [13]. This renders the comparison between experiment and simulation particularly difficult. Nevertheless, we

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500 600 700

M. Schreiber, B. Lamberts/J. Mol. Struct. (Theochem) 313 (1994) 173-188 185

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conclude from Fig. 8 that the simulation is good also with respect to dynamic features.

In the following we compare our approach and a frequently used empirical M D potential [6] for silicon. For this purpose we have performed a conventional M D simulation with the Stillinger- Weber potential to obtain an (almost) relaxed configuration. This was then used as the initial configuration for our approach with a parametrized potential which had been optimized in a previous M D run with identical external parameters. The subsequent M D simulation in which the temperature was fixed for the first 500 steps and the total energy was kept constant thereafter lead to a significant further relaxation of the configuration as shown in Fig. 9. This means that the (local) minima of the potential are not identical, although some of the static properties (like the pair distribution function) of the resulting configuration are as good for the empirical potential as for the LDF-based ab initio potential.

Differences appear, however, for the distribution of the number of nearest neighbours, which can be defined in a unique way by means of Voronoi polyhedrons: let the connecting line between each pair of atoms be bisected by an orthogonal plane. I f a plane contributes to the smallest polyhedron that can be constructed from all such planes around a given particle, then the two respective atoms are nearest neighbours. (In a crystal this

definition yields the Wigner cell.) The resulting distribution in Fig. 10 is narrower and shifted to smaller numbers of Voronoi neighbours for our potential (average distance 7.5 4- 1.6 a.u.) in contrast to the empirical potential (8.3 4- 1.9 a.u.), indicating that the local environment is stiffer. This agrees with experimental findings [34], corroborat- ing the quality of our approach. The distribution of the bond angle, which we present in Fig. 11 is also narrower than in conventional M D simulations, again in agreement with experimental findings [34].

7. Concluding remarks

We have presented an approach for the simulation of amorphous silicon which combines the precision of an ab initio LDF-calculat ion of the electronic system with the speed of conventional M D simulations for the classical dynamics of the atoms. An important feature of the approach is the parametrized potential which is generated from the L D F data and more and more optimized during the simulation. This potential does therefore not depend on empirical parameters and nevertheless allows us to perform a fast and sufficiently accurate calculation of the forces during the M D run. The comparison with experiments corroborated that the obtained potential yields a realistic description

186

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M. Schreiber, B. Lamberts/J. Mol. Struct. (Theochem) 313 (1994) 173 188

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of amorphous silicon. It was further possible to show that the widely used empirical Stillinger- Weber potential leads to a less stiff configuration.

Here we have presented only results of prelimin- ary test runs to confirm the validity of our approach. Systematic investigations are planned to describe the influence of the external conditions (like temperature and pressure) on the parameters of the potential. This should lead to conclusions about how local properties are changed and it should allow us to study modifications of the local bond order or orientational changes in the

local order. Respective changes of the density of electronic states (in particular the dangling bonds) as well as the density of phonon states are also to be analyzed. These features should be most characteristic for a covalently bonded semiconduc- tor, in particular for the deviation of the properties of the amorphous material from those of the crystal. Moreover, we intend to perform the same simulation for liquid silicon, and compare the mentioned features, too.

The presented approach shall then be applied to other covalently bonded semiconductors. The

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investigation of germanium would be particularly interesting, because our approach would enable a comparison with silicon without being hampered by differences in the ansatz for the potential which are inherent in the empirical MD procedures.

The approach is flexible enough to treat hetero- geneous systems, too. We intend to extend our calculation first to gallium arsenide, because of its present and future importance in semicon- ductor applications, and to hydrogenated silicon, because of the significance of hydrogen for the saturation of the dangling bonds in connection with technologically motivated doping of the amorphous silicon.

Moreover, several methodological improvements of the approach can be implemented and shall be investigated in the future. The inclusion of four- centre interactions, for example, allows us to take longer-range interactions and the dihedral angle interaction into account. The latter is important, because different dihedral configurations are known to yield different energies. But with the above employed three-centre interaction those configurations cannot be distinguished. It is therefore not surprising that the MD results for the dihedral angle do not coincide with experiment. But the numerical effort is much higher for such a potential so that only smaller systems and/or shorter time spans can be simulated. A possible way to overcome this disadvantage would be to start the calculation with four-centre terms only from some already (almost) relaxed configuration. First test runs of such an approach yielded promising results for the dihedral angles, but these runs also emphasized the problem, how to attribute the total interaction to the different three- and four-centre terms, avoiding possible ambiguities in the parametrization of these terms. A projection technique with respect to the contributions of different angular momentum is presently under investigation.

We also intend to compute the MD forces directly from the ab initio LDF calculations by means of the Hellman Feynman theorem. In this way the parametrized potential can be tested and possibly improved. It should be noted, however, that the Hellman Feynman theorem is valid only for equilibrium configurations. Assuming that the relaxed configurations are close to equilibrium, it is

thus possible to compare the minima of the potentials, which are of course most important. But this computation of the forces from the LDF program is very time consuming and has been postponed to future studies.

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determination of parameter-free model potentials for the molecular dynamics simulation of amorphous...

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