PHYSICAL REVIEW B VOLUME 41, NUMBER 6 15 FEBRUARY 1990-II

Computer models for amorphous silicon hydrides

Normand Mousseau and Laurent J. LewisDepartement de Physique et Groupe de Recherche sur les Couches Minces,

Universite de Montreal, Case Postale 6128, Succursale A, Montreal, Quebec, Canada H3C3J7(Received 3 October 1989)

A procedure for generating fully coordinated model structures appropriate to hydrogenatedamorphous semiconductors is described. The hydrogen is incorporated into an amorphous matrixusing a bond-switching process similar to that proposed by Wooten, Winer, and Weaire, which en-sures that fourfold coordination is preserved. After each inclusion of hydrogen, the structure is re-laxed using a 6nite-temperature Monte Carlo algorithm. The method is applied to a-Si:H at varioushydrogen concentrations. The resulting model structures are found to be in excellent agreementwith recent neutron-scattering measurements on a sample with 12 at. % H. Our prescription, whichis essentially nonlocal, allows great Aexibility and can easily be extended to related systems.

I. INTRODUCTION

In the absence of theory, structural models have beenof paramount importance in our understanding of thefundamental properties of amorphous matter. ' Whilemodels were originally built and relaxed "by hand, " itwas rapidly realized that, in order for large structures tobe generated in a systematic fashion, one had to resort tothe computer. ' Computer models are of two types: (i)"static, " where models are constructed using a set ofpredetermined geometrical rules, and relaxed using anenergy-minimizing procedure such as conjugate gradientsor Monte Carlo, and (ii) "dynamic, " where the structureis obtained by, for instance, rapidly quenching from themelt in a molecular-dynamics simulation. The latter,however, is a rather computer-intensive solution to theproblem, especially in the case of low-coordinationsystems —such as amorphous semiconductors —wherethree-body potentials are usually required to stabilize thestructure. Thus, there is still a need for efficient model-construction algorithms.

In this article, we discuss in detail a new procedure forcomputer generating hydrogenated covalent random net-works with periodic boundary conditions, which will sub-sequently be employed as input to more elaboratemolecular-dynamics simulations. Though our model caneasily be extended to other systems, we concentrate hereon the silicon hydrides (a-Si:H). Because of its impor-tance in the area of microdevice technology and fabrica-tion, this material has attracted considerable interest inrecent years. Comprehension of its structure, however,is still in its infancy, and a number of questions remainunanswered: floating-bond-mediated H diffusion, colum-nar formations, hydrogen microbubbles, etc. Surprising-ly, attempts at modeling the structure of a-Si:H havebeen relatively few, be they static or dynamic, thoughseveral models exist for the pure material.

Computer-built a-Si models provide an excellent start-ing point for generating a-Si:H structures, given a properprescription for incorporating the hydrogen. The firstsystematic procedure for computer generating periodic

covalent random networks —with or withouthydrogen —was proposed by Guttman: starting with abcc structure of appropriate dimensions, bonds and "non-bonds" are interchanged until the system's strain energydeclines to as small a value as possible. These ideas wereextended and refined by Wooten, Winer, and Weaire(WWW). They introduced the concept of "local bondswitches" in order to randomize a cubic diamond lattice,and also employed a finite-temperature Monte Carlo al-gorithm to minimize the energy. They found excellentagreement with measured radial distribution functionsfor pure Si, but did not consider hydrogenation. Recent-ly, their method was used to model the a-Si/c-Si inter-face.

In our approach, which is described in detail in Sec. II,we first use the algorithm of WWW to construct a fullycoordinated a-Si structure, then incorporate the hydro-gen by a bond-switching process similar in spirit to thatof WWW. We have used our method to generate a num-ber of model structures containing amounts of hydrogenvarying between 4 and 16 at. %. As will be demonstratedin Sec. III, we find excellent agreement with recentneutron-scattering measurements.

II. MODEL CONSTRUCTION

As mentioned earlier, the starting point of our ap-proach is the algorithm of WWW for generating a-Sistructures. This so-called "sillium" model has been re-viewed at length by Wooten and Weaire, and we shallnot discuss it in detail here. However, for the benefit offurther discussion, it is useful to mention its main attri-butes. Sillium is defined as follows: (i) each atom is four-fold coordinated; (ii) the total energy is the sum of bond-bending and bond-stretching terms; and (iii) bondswitches of the type defined by WWW are the only de-grees of freedom allowed. Thus, starting with a diamondlattice of pure Si, one performs a series of bond switchesat high temperature in order to randomize the structure,and then minimizes the energy by carrying out asimulated-annealing relaxation at successively lower tem-

41 3702 1990 The American Physical Society

41 COMPUTER MODELS FOR AMORPHOUS SILICON HYDRIDES 3703

peratures, until the ground state is reached at T =0.Before discussing our prescription for incorporating

the hydrogen, a few remarks are in order. First, if four-fold coordination is to be preserved (i.e., if no danglingbonds are to be induced by the inclusion process), it isnecessary for hydrogen atoms to be added by pairs(though molecular hydrogen is a priori excluded)Second, only monohydride and dihydride bonding groupsare permitted: trihydrides are known to be present withonly very low probability —if not zero. ' Third, polysi-lane groups are also allowed, i.e., nearest-neighbor Siatoms are permitted to each bond to one or two H atoms.All these rules, of course, can be relaxed if necessary.

Figure 1 illustrates in a schematic way our procedurefor incorporating a pair of H atoms. This is similar inspirit to the bond-switching process employed byWWW, ' and to a criterion suggested by Guttman.However, it is essentially "nonlocal, " and thus allowsgreat flexibility. In detail, we proceed as follows.

1. Choose, at random, a set of 4 to 8 (or more) atomsin a configuration such as that illustrated in Fig. 1(a).

2. Perform a series of bond breakups and transposi-tions, as shown in Fig. 1(b), and affix two H atoms to theresulting two threefold-coordinated Si atoms.

3. Relax the new structure under an appropriate set ofinteratomic potentials (see below).

Just as with the WWW algorithm, however, a numberof conditions must be verified in order for the chosenconfiguration to be valid. In particular, the length ofnewly created Si—Si bonds should not exceed 1.7 timesthe equilibrium bond length and the new configurationshould not induce excessive strain in the structure. Fol-lowing a suggestion by WWW, " in order to acceleratethe convergence, no restriction is imposed on the forma-tion of fourfold rings: these are expected to relax outduring the energy-minimization stage. After each addi-tion of hydrogen, the system is Monte Carlo relaxed us-ing a finite number of WWW-type bond switches. Thisprocedure is repeated until the desired concentration isachieved.

The potentials used to describe the local tetrahedralconfigurations in the present simulations are of theStillinger-Weber (SW) type. ' In spite of its knowndeficiencies, ' the SW potential provides a more realisticdescription of the energetics of such materials than does,

FIG. 1. Schematic representation in a six-atom cluster of thebond switching process used to incorporate the hydrogen in thea-Si matrix (a) before and (b) after addition: open circles, Si;solid circles, H.

for instance, the Keating model, ' used by WWW (Refs. 7and 9) and by ourselves' in an earlier version of thepresent model. In addition, we now include terms thatdescribe nonbonded Si-H interactions as well as atomicH-H interactions, in order to avoid the unphysical hard-core overlaps that may otherwise result.

The bond-stretching Si-Si and Si-H interactions arethus of the following form

V; =eA, (B; p, ' —1)expEJ

where p; =~p," =r, /cr; . The numerical values of theparameters for this and other potentials are given inTable I. For Si-Si, we adopt the values given by Stillingerand Weber, ' whereas for Si-H, we fit the functional form(1) to the potential recently determined from ab initioquantum chemistry calculations by Selmani, Salahub, andYelon. '

The bond-bending interactions are also of the SWtype here, however, we write the potential in the formdeveloped by Biswas and Hamann' (see also Ref. 18),which results in a substantial economy of computer time:

2

g Q el/)(pj 1=0

TABLE I. Parameters of the model potentials, Eqs. (1)-(5), used in the present calculation, whichare either of the Stillinger-Weber (SW) or of the Slater-Buckingham (SB) type.

e (eV/atom)

B

0 (A)

Cp

Si-Si (SW)

2.16827.04960.602 2242.09511.8

21.01.2

Si-H (SW)

2.16824.78100.296 875.37341.36421.8

21.01.2

Si-H (SB)

0.1974

3.18

H-H (SB)

0.1470

3.261

14.19 15.63

3704 NORMAND MOUSSEAU AND LAURENT J. LEWIS 41

with

and

(P, ) =X NI(P )Y. . I (P )..jWi

(3)

2-2

E xpt.

26.3

$1(p, . )=expV

L

(4)

Here, Y& (P;1 ) is the spherical harmonic of order Im withargument P; =p; /p, ". The parameters of this potential,also listed in Table I, are the same for the three three-body interactions considered in our calculation, namelySi-Si-Si; Si-Si-H, and H-Si-H.

For the nonbonded Si-H and H-H interactions, finally,we found it convenient to employ a Slater-Buckinghammodel:

41

MI

'rn 0Cg

2

12.2

8.5

4 4

V, (r)=1 —6/a;J a;J

exp[a;, (I —p;, )]— . (5)

1

P)l

The parameters for these two interactions were looselyadapted from corresponding fits of integral elastic crosssections of hydrogen scattered by rare gases, ' and empir-ically adjusted to provide an appropriate description ofour system; the values used in our calculations are listedin Table I.

0 2 4 6r(A)

FIG. 2. Computed partial pair correlation functions gs; s, (r)for samples with 4.4, 8.5, 12.2, and 16.3 at. % H, together withthe corresponding results extracted from the neutron-scatteringdata of Menelle et al. for a sample with 12 at. % H (Ref. 20).Zeroes are displaced vertically for clarity.

III. RESULTS

In accord with our earlier discussion, the starting a-Sistructure was obtained by using the WWW algorithm toamorphize a diamond lattice consisting of 216 Si atoms ina cubic box of length 16.288 A; the results for this part ofthe calculation are entirely consistent with those reportedby WWW. Pairs of hydrogen atoms were then incor-porated into the amorphous a-Si matrix such as to yieldconcentrations in the range 0—16.3 at. % H. In the fol-lowing, we present a detailed structural analysis of foursamples containing 4.4, 8.5, 12.2, and 16.3 at. % H, re-spectively. Table II gives, for each sample, the energy(per atom) along with other structure-related parametersobtained in our simulations. Also given for reference arethe corresponding data for a-Si and c-Si.

Figures 2-4 show the Gaussian-broadened radial dis-

tribution functions (RDF) for the four samples underconsideration. Also shown on these plots are the corre-sponding functions extracted from the neutron-scatteringdata of Menelle et al. for a sample containing 12 at. %H; the latter have also been filtered in order to reducethe spurious contributions resulting from the Fourier in-version of the scattering intensity. Agreement is clearlyexcellent, when due account is made for the difhculties in-herent in this type of calculation, which we now discuss.

First, the apparent discrepancies between calculatedand measured H-H correlations should not be taken tooseriously. They are in large part due to the relativelysmall concentration of hydrogen, which renders the taskof extracting this function troublesome both from themodel and from experiment. (The relative contributionfrom each ij partial correlation to the total scattering in-tensity is proportional to the product x;x of relative con-

TABLE II. Computed structure-related properties for the four samples under consideration, togeth-er with the corresponding values for a-Si and c-Si. E, „f is the configurational energy per atom, Z;, isthe first coordination number for ij correlations, 6 and 68 are the mean bond angle and its root-mean-square deviation, respectively, and R„ is the number of n-membered rings per atom.

at. % H a-Si 44 8.5 12.2 16.3 c-Si

E„„f(eV/atom)zsi-siZSi-H

ZH-St

ZH-H8 (deg)68 (deg)R4 (atom ')

R5 (atom ')

R6 (atom ')

—4.004.09000

108.314.80.060.361.44

—3.934.130.051.000

107.118.00.100.411.26

—3.904.110.091.000

106.819.40.120.461.17

—3.884.100.141.000

106.620.60.130.410.99

—3.844.050.191.000

106.221.60.170.370.86

—4.324000

109.50002

COMPUTER MODELS FOR AMORPHOUS SILICON HYDRIDES 3705

~14

I

in 0Cg

P

0

4r(A)

Expt.

16.3

12.2

8.5

4.4

than bonded Si-H contributions (first peak). No pro-visions are made for such features in our calculation; thisexplains the mismatch in width, but not in position, of thesecond Si-H peak.

Within these limitations, we find the agreement be-tween model and experiment to be entirely satisfactory,and our structures should therefore provide an excellentstarting point for more detailed dynamical calculations.It is quite remarkable to note the relative independence ofthe various partial contributions with concentration (ex-cept, of course, for the problems mentioned above for theH-H contribution). This is also clearly visible in thestructure factors obtained by Fourier transforming thecalculated g;~(r), shown in Figs. 5 —7. Here again, thediscrepancies originate from finite-size and truncationeffects, but also from the relative inadequacy of the po-tentials used in the calculation, and in particular the factthat they are short ranged.

A quantity of interest which can be extracted from theRDF is the first coordination number,

FIG. 3. Same as Fig. 2, but for Si-H.

centrations; for a sample with 12 at. % H, for instance,x„=0.014, while xs;=0.77.) In fact, we note that theagreement between model and experiment improves withH concentration. Second, finite-size effects in the modelstructure and truncation effects in the Fourier transformof the experimental data introduce additional sources oferror. In particular, the "prepeak" in the experimentallydetermined gH H(r) is spurious. Finally, it is difficult toreproduce the Debye-Wailer broadening of the real distri-bution functions without explicitly taking dynamicaleffects into account, even though we mimic the effect tosome extent by Gaussian smoothing our data. For in-stance, we expect the nonbonded Si-H correlations[second peak in gs; „(r)] to be substantially "floppier" '

Z'j 47Tr x)nog; r r (6)first peak

where no is the number density. Z; gives the averagenumber of j-type atoms nearest neighbor to an i-typeatom. Results for this quantity are given in Table II.Our analysis indicates that, in all samples, H atoms areperfectly well coordinated: we find ZH s; =1, i.e., each Hatom is singly bonded to a Si atom —as it should —while

ZH H=O. Si atoms, on the other hand, are on averageslightly overcoordinated, with Zsj Zsj sj+Zsj H rangingfrom 4.18 to 4.24. (In "ideal" a-Si:H, this number shouldbe exactly 4.) Indeed, we found a small number of Siatoms not having perfect tetrahedral coordination, which

2-

2-

00-

0

MoM

M

14

0Ig

001

0

0-

0=

0-0 4 6

k(A i)10

0I I I I I

0 2 4 6r(A)

FIG. 4. Same as Fig. 2, but for H-H.

FIG. 5. Computed static structure factors Ss; s, (k) for sam-ples with 4.4, 8.5, 12.2, and 16.3 at. % H, together with thedeconvoluted neutron-scattering data of Menelle et a1. for asample with 12 at. % H (Ref. 20). Zeroes are displaced vertical-

ly for clarity.

3706 NORMAND MOUSSEAU AND LAURENT J. LEWIS 41

2-2

16.3

A

0~II'

M

0

1 - 12.2

~0tdD0 1 8.5F4

e0~ M

f 4 4

0

xXg

00

I I I I I I I I I I

2 4 6 8 ioi (A-')

I I I I I I I I

40 80 120 1608(deg )

FIG. 6. Same as Fig. 5, but for Si-H.

indicates that relaxation is incomplete or, perhaps morelikely, that the potentials are not perfectly optimized forthe problem under study. A similar situation has in factbeen noted in corresponding molecular-dynamics studiesof a-Si based on the SW potential; ' this seems to indi-cate that the bond-bending term in this model is notsufBciently strong. Recent electron paramagnetic reso-nance measurements also appear to confirm the overcoor-dinated nature of point defects in a-Si.

We have also computed the bond and dihedral angledistributions from our model structures, as well as thestatistics of coordination rings. The average bond angleand its root-mean-square deviation are given explicitly inTable II, while the actual distributions are displayed in

FIG. 8. Distribution of bond angles, in arbitrary units, forthe four samples under consideration. Zeroes are displacedvertically for clarity.

Fig. 8 for our four samples. To within the accuracy ofthe calculation, these distributions can be described by aGaussian line shape. Though their widths are a littlelarge, they compare reasonably well with the correspond-ing data for pure a-Si; discrepancies are mainly related tothe deficiencies of our potential functions. The distribu-tion of dihedral angles, on the other hand, shows an in-teresting evolution as a function of hydrogen concentra-tion (Fig. 9). Though it roughly approximates a Gaussianshape at low H concentration (in c-Si, this function is

0

XI

0

1

0

~ 0a$

0F4

Q~H

0gf

I)1 ~

44

12.2 444444d ddd

4444 ddddd 4

pp

0pp000

8.5 0 0000 pppp 0

0000 0000

XXX

XXX

4.4 xxxX XX

X XXX

X X XX X XXXX X X

16.3 ++++++ + +++

+++ +++++ + +

0

0I I I I I I I I I I

2 4 6 8 10k(A-')

FIG. 7. Same as Fig. 5, but for H-H.

g I I I I I I I I I I I i I00 20 40 80

8(deg )FIG. 9. Distribution of dihedral angles, in arbitrary units, for

the four samples under consideration. Zeroes are displacedvertically for clarity.

41 COMPUTER MODELS FOR AMORPHOUS SILICON HYDRIDES 3707

sharply peaked about 60'), the distribution becomesessentially linear —and almost flat —at high H concen-trations; this, incidentally, agrees we11 with the predic-tions of the Beeman-Bobbs computer model for a-Si,and it would be interesting to obtain an experimentalverification of our observation. Finally, our analysis ofring statistics, Table II, indicates that four-memberedrings are few, as expected. While six-membered rings arenaturally prevalent, there nevertheless exists a non-negligible number of five-membered rings, of about 0.4per atom, a value which appears to be independent of thehydrogen concentration. Note also that the number ofsix-membered rings diminishes constantly with hydrogenconcentration. Clearly, therefore, such structures, whichwe believe approximate quite closely the real material,contain numerous intrinsic structural defects.

IV. CONCLUDING REMARKS

Using as a starting point the WWW algorithm for gen-erating a-Si structures, we have presented a procedure forincorporating hydrogen into them. Our prescription,which is versatile and eScient, is found to yield RDF andstatic structure factors in good agreement with experi-ment. Large models can be constructed using this

method, and a number of them have been generated atvarious H concentrations. These should prove extreme-ly useful in the study of structural and other properties ofa-Si:H. Currently, we are using our structures as input tomolecular-dynamics studies of the vibrational propertiesand relaxation effects of a-Si:H. We will report on thesecalculations in a future publication.

ACKNOWLEDGMENTS

We thank Frangois Drolet for help in the initial stagesof this work, and A. Menelle for providing us with digi-talized listings of his experimental data. The calculationsreported here were performed on the Cray Research, Inc.X-MP supercomputer at the Ontario Center for Large-Scale Computation, thanks to a supercomputer accessgrant from the Natural Science and EngineeringResearch Council (NSERC) of Canada. Additional sup-port from NSERC as well as from the FCAR (le Fondspour la Formation de Chercheurs et 1'Aide a la Re-cherche) and Actions Structurantes programs of theGovernment of Quebec is also acknowledged. One of us(N.M. ) is grateful to NSERC for partial support.

'R. Zallen, The Physics ofAmorphous Solids (Wiley, New York,1983).

2S. R. Elliott, Physics of Amorphous Materials (Longtnan, Lon-

don, 1983).N. Mousseau and L. J. Lewis, in Proceedings of the Thirteenth

International Conference on Amorphous and Liquid Semi-conductors, edited by M. A. Paesler and R. Zallen (Elsevier,Amsterdam, in press).

4See, for instance, numerous articles in Proceedings of the FirstInternational Conference on Amorphous SemiconductorTechnology, edited by M. A. Paesler and R. Zallen (Elsevier,Amsterdam, in press).

5See, for instance, numerous articles in Proceedings of the Thir-teenth International Conference on Amorphous and LiquidSemiconductors, edited by M. A. Paesler and R. Zallen (El-sevier, Amsterdam, in press).

L. Guttman, Phys. Rev. B 23, 1866 (1981);L. Guttman and C.Y. Fong, ibid. 26, 6756 (1982).

7F. Wooten, K. Winer, and D. Weaire, Phys. Rev. Lett. 54, 1392(1985).

F. Wooten and D. Weaire, in Ref. 5.F. Wooten and D. Weaire, in Solid State Physics, edited by H.

Ehrenreich, F. Seitz, and D. Turnbull (Academic, New York,1987), Vol. 40, p. 1, and references therein.

Cx. Lucovsky, J. Non-Cryst. Solids 76, 173 (1985}.' F. Wooten and D. Weaire, J. Non-Cryst. Solids 97498, 349

(1987).' F. H. Stillinger and T. A. Weber, Phys. Rev. B 31, 5262

(1985).E. R. Cowley, Phys. Rev. Lett. 60, 2379 (1988).P. N. Keating, Phys. Rev. 145, 637 (1966).

' L. J. Lewis, N. Mousseau, and F. Drolet, in Atomic-Scale Cal-culations in Materials Science, Vol. 141 of Materials ResearchSociety Symposium Proceedings, edited by J. Tersoff, D. Van-derbilt, and V. Vitek, (MRS, Pittsburgh, 1989).A. Selmani, D. R. Salahub, and A. Yelon, Surf. Sci. 202, 269(1988).

' R. Biswas and D. R. Hamann, Phys. Rev. Lett. 55, 2001(1985).

' J. Berger, A. Selmani, C. Tannous, and G. Spronken, Surf. Sci.202, 255 (1988); F. Ladouceur, A. Selmani, C. Tannous, andG. Spronken, J. Phys. Cond. Matter 1, 4129 (1989); F. La-douceur, these de maitrise, Ecole Polytechnique de Montreal,1987.

R. W. Bickes, B. Lantzsch, J. P. Toennies, and K.Walaschewski, Faraday Discuss. Chem. Soc. 55, 167 (1973).A. Menelle, these de doctorat, Universite Pierre et Marie Cu-rie (Paris VI), 1987; see also R. Bellissent, A. Menelle, W. S.Howells, A. C. Wright, T. M. Brunier, R. N. Sinclair, and F.Jansen, preprint.

'See, for instance, Y. Cai and M. F. Thorpe, Phys. Rev. B 40,10 535 (1989).

W. D. Luedtke and U. Landman, Phys. Rev. B 40, 1164(1989).M. D. Kluge, J. R. Ray, and A. Rahman, Phys. Rev. B 36,4234 {1989).R. Biswas, G. S. Grest, and C. M. Soukoulis, Phys. Rev. B 36,7437 (1987).

~5J. H. Stathis, Phys. Rev. B 40, 1232 (1989).D. Beernan and B.L. Bobbs, Phys. Rev. B 12, 1399 (1975).Our models are available: requests can be sent via e-mail (toL.J.L.) at Bitnet address 1533~UMTLVR.