PHYSICAL REVIEW B VOLUME 47, NUMBER 7 15 FEBRUARY 1993-I

Model of hydrogenated amorphous silicon and its electronic structure

J. M. Holender* and G. J. MorganDepartment of Physics, Leeds University, Leeds LSg gJT, United Kingdom

R. JonesDepartment of Physics, Exeter University, Exeter EX/ $QI, United Kingdom

(Received 9 October 1992)

Using previously generated large models of pure amorphous silicon [J. M. Holender and G. J.Morgan, J. Phys. Condens. Matter 3, 7241 (1991)j we have now constructed structural models ofhydrogenated silicon. They are obtained by "hydrogenation" of our models of amorphous silicon,i.e. , by addition of the hydrogen atoms into a model of a-Si containing undercoordinated and over-coordinated atoms followed by relaxation using molecular dynamics. The electronic structure of themodels is calculated using the phenomenological tight-binding model. It is shown that addition ofhydrogen reduces drastically the density of electronic states associated with defects, producing aclearly defined gap.

In our previous paper we have built large models ofamorphous silicon and we have later calculated propertiessuch as the electronic density of states and the electricalconductivity. We have also successfully demonstratedthe experimentally observed, double sign anomaly of theHall coefficient.

In this report we expand our work to hydrogenatedamorphous silicon, a material of great practical impor-tance. The experimental and theoretical progress inunderstanding phenomena occurring in a-Si:H has beenreviewed in many papers. To generate structures ofamorphous silicon we used classical molecular dynam-ics with a phenomenological many body potential andthe electronic properties were calculated using the equa-tion of motion method combined with the empirical tight-binding model of Chadi. To apply the same approach tohydrogenated silicon we encounter two basic problems,namely the description of the Si-H interaction and theneed for parameters to model the electronic structure ofthe Si-H system. The latter problem can be solved by anextension of the phenomenological tight-binding modelof Chadi. There are papers with the parameters of suchmodels. We were also able to derive them by fittingthe energy levels of SiH4 and SiH3 to values obtainedusing the first principles method. Results obtained us-ing those models are very similar, so for consistency wepresent here results using the parameters from Ref. 7.

As far as the problem of Si-H interaction is concernedthe situation is much more complicated. There has beensome progress in obtaining empirical potentials for the Si-H system. The potentials based on the Stillinger-Webermodel of interaction in silicon are published in the recentpapers of Mousseau and Lewis. These potentials leadto a-Si:H structures which are in a good agreement withthe structural data (the radial distribution functions de-rived from experimental data) and vibrational densitiesof states. Their models give, however, a high density ofelectronic states in the gap region due to a high concen-tration of structural defects in their models. However,their potentials seems to describe basic features of the

H-Si interactions correctly, so a similar approach is usedin our study for structural relaxations. For Si-H, Si-Si-H,and H-Si-H interactions we used scaled Stillinger-Weberpotentials, while for H-H interactions we use the modelfrom Ref. 11. We apply the standard N-P-T moleculardynamics for the structure relaxation.

The full details of the electronic structure calcula-tions are given elsewhere. We use the equation of mo-tion method combined with the nearest-neighbor tight-binding model. 7 Harrison's r 2 scaling of the hoppingintegrals is used.

The electronic density of states, g(E), for our modelsof pure silicon has a "gap" where g(E) is very low. If wefilter states (in way described in Refs. 2 and 3) in order toobtain states corresponding to energies in the gap regionand calculate the partial density of states, it is clear thatthe undercoordinated atoms give a large contribution tothe gap states. It is worth noticing, however, that aboutM% of the gap states are associated with fully bonded(fourfold coordinated atoms) with slightly distorted ge-ometry.

We have recently published details of various modelsof amorphous silicon where a common feature is thepresence of threefold coordinated atoms. We use thesemodels as a starting point for the hydrogenation. Wetook our model of 1728 atoms containing about 6% un-dercoordinated atoms and 1' overcoordinated atoms asthe initial configuration. We need a defected (not a fullybonded) model because otherwise addition of hydrogenrequires large rearrangements in the network. In this waywe can clearly observe the effect of hydrogenation on theelectronic properties and structure. In Ref. 2 we dis-cuss some problems associated with the nearest-neighbor

~

~

NN) definition. In this study we use values rs; s; = 2.77and rs; H = 1.74 A. for the NN distance. The same cut-

offs are used in the tight-binding model for the hoppingintegrals.

If one is trying to add hydrogen to the amorphous net-work the most likely positions of hydrogen are those sat-urating the dangling bonds. We adopted two methods of

47 3991 1993 The American Physical Society

3992 BRIEF REPORTS

hydrogenation.As a first attempt we tried to add hydrogen to under-

coordinated atoms. Hydrogen is added to the threefoldcoordinated atoms in directions opposite to the existingbonds. If nq, nq, and n3 denote unit vectors in the di-rection of the bonds, a H atom is added in the directionn = —(nq+nq+ ns). The Si-H distance Rs;H is assumedto be the same as in SiH4, i.e. , Rs;H=1.48 A. . We check,however, if this new H atom is not too close to the ex-isting Si or H atoms. If the distance of a H atom to anyother H or Si atom is too small (less than 50% of RsiH)then we do not accept this addition. After attempts toadd H to all undercoordinated atoms the structure is re-laxed using the molecular dynamics technique. Usuallyafter the relaxation new defects are created so the pro-cedure of hydrogen addition and structure relaxation isrepeated.

In Fig. 1 the electron density of states g(E) for thestructure is shown. In all plots of the density of the elec-tronic states for the hydrogenated models the density ofstates of the initial (unhydrogenated) model is presentedfor comparison (dashed thin line). The concentration ofhydrogen is about 7% (1728 Si atoms and 126 H atoms).It is rather a low value in comparison with real materialswhich may contain up to 30% of H, but we are unableto incorporate more hydrogen using this scheme. Thereis a marked reduction in the density of gap states butthe value of g(E) in the gap region is definitively still toohigh in comparison with the experimental value (about5 x 10 s [(atom eV spin) ]. The density of the gap statesis, however, much lower than that for the Mousseau andLewis models.

We have found, however, a diferent algorithm for hy-drogenation which turns out to be much better. It worksas follow. The initial structure is chosen as before. Allthreefold coordinated atoms (and twofold coordinated ifsuch exist) are replaced by 3 (or 2) H atoms. If we de-note the undercoordinated atom by no and its neighborsby nr, nq, and ns, atom no is removed and three H atoms

C, a)

(b)

FIG. 2. A schematic illustration of the algorithm for hy-drogenation. The central threefold coordinated Si atom (a) isreplaced by three hydrogen atoms (b).

are added to the Si atoms nq, n2, and n3 in the directionof the previous bond joining with no. The length of theSi-H bond Rs;H is again set equal to 1.48 A.. The proce-dure is sketched in Fig. 2. If it happens that one of nr, nq,or n3 is also undercoordinated the H atom is not addedto this atom because the atom is going to be removed(Fig. 3). When all undercoordinated atoms are removedthe structure is relaxed using molecular dynamics. Af-ter this relaxation all H atoms which are unbonded orbonded to more than one Si atom are removed as well asthose bonded to already fourfold coordinated Si atoms.

(a)

0.5

(b)

0.2—

0.1—

0.0 !—10

I

—5

E LeV~

FIG. 1. The electronic density of states (spin eV atom)for the model of hydrogenated silicon (1728 Si atoms and 111H atoms) when H atoms are added to undercoordinated Siatoms. The dashed curve denotes the electronic density ofstates for the initial structure without hydrogen.

FIG. 3. The same as in Fig. 2 for two adjacent three-fold coordinated atoms which are replaced by four hydrogenatoms.

47 BRIEF REPORTS 3993

0.5

0.4—

0.2—

0.1—

0.0 I

—10I

—5

E (eV)

IlIIIIIIII

II II I

II II I

I I

I II

I I

I I

II

I I

IIIIIIIIII

III

I I

II

II

II

II

I I

2I

3

r (

SiSi---- SiH

HH

FIG. 4. The electronic density of states (spin eV atom)for the structure with no undercoordinated atoms (1568 Siatoms and 399 H atoms).

FIG. 6. The partial radial distribution functions (PRDF)for the final structure. The solid line denotes PRDF for Si-Si,the dashed one for Si-H, and the dotted one for H-H.

The procedure of removing the undercoordinated Siatoms (if one of the neighbors of removed atom is Hthis H is also removed), relaxation, and H removing isrepeated until there are no undercoordinated Si atoms.The electronic density of states for this structure is shownin Fig. 4.

The hydrogen concentration is much higher than us-ing the previous algorithm. For the structure shown inFig. 4 the H concentration is about 20% (there are 1568Si atoms and 399 H atoms). A great improvement is at-tained but there are still states in the gap. It should bestressed that in order to obtain such a reduction in thenumber of gap states we have to repeat the procedureof removing atoms and structure relaxation many times.If one simply removes undercoordinated atoms there isa small reduction in the number of the gap states. Thisstructure contains, however, fivefold coordinated atoms.

We proceed further and remove all overcoordinatedatoms in a similar fashion. We also repeat the cyclesof removing and relaxation. The final structure is com-posed of 450 H atoms and 1545 Si atoms. The H con-

0.5

centration is about 23%. The density of the electronicstates is shown in Fig. 5 and in Fig. 6 we show the par-tial radial distribution functions for the final structure.Some structural data are also presented in Table I. Theappearance of a clear gap (about 1.2 eV) is the mostimportant result.

Before discussing our models we should recall some ex-perirnental data. The density of defects (probably dan-gling bonds, but there is still controversy) is thought tobe about 10 cm for a-Si and 10 —10 cm fora-Si:H with 20% H. The first value corresponds to oneatom in 500, the second to 1 in 107. So unless one isdealing with an enormous model there should be no dan-gling bonds in the model of hydrogenated silicon. Thesenumbers also emphasize the importance of large mod-els when dealing with amorphous silicon, especially withdefects. The experimental density of states in the gapregion is equal to about 5 x 10 s cm s(eV) [10 s(spineV atom) ij for hydrogenated silicon. If the gap re-gion spans about 1.5 eV there should be about 10states jcms in a gap corresponding to one state in 5 x 10satoms. So again for any reasonable model there should

TABLE I. Structural properties of the initial and the finalstructure.

0.2—

0.1—

0.0I

—10I

—5

E (eV)

FIG. 5. The electronic density of states (spin eV atom)for the final structure (1545 Si atoms and 450 H atoms).

number of Si atomsnumber of H atomsH concentrationdensity/density of c-Sinumber of Si atoms withnumber of Si atoms withnumber of Si atoms withnumber of Si atoms withnumber of H atoms withnumber of Si atoms withnumber of Si atoms withnumber of Si atoms withnumber of Si atoms with

2NN3NN4NN5NN1NN4Si as NN3Si and 1H as NN2Si and 2H as NN1Si and 3H as NN

Initial Final

1728 15450 450

0% 23'Fo

0.939 0.8276 0

104 01598 154520 00 450

1598 11460 3500 470 2

3994 BRIEF REPORTS

be a very low probability of a gap state. As far as ourmodel is concerned all atoms are fully bonded and thereis a clear gap. A single state would be easily detectable.All Si atoms are fourfold coordinated and all H atomsare bonded to one Si atom. Our partial radial distri-bution functions (PRDF) are very reasonable and verysimilar both to those obtained from the ab initio calcu-lations of Ref. 13 as well as experimental ones. Theexperimental positions (for a-Si:H with 12%%uo of H) of thefirst two peaks in the PRDF for Si-Si are 2.34 and 3.84A. , while we obtain 2.36 and 3.76 A. (the values 2.38 and3.69 A, respectively, were given by Graczyk~s for a-Si:Hwith 25% of H). For Si-H the experimental values are1.48 and 3.13 A. , while the values for our model are 1.48and 3.14 A. Experimental data for the H-H PRDF arenot very accurate; however, a wide peak centered at 2.4A. is visible and we obtain a peak at 2.39 A. The densityderived from constant pressure relaxation is about 83%of the crystalline one. It is close to values for a-Si:H withhigh concentrations of H. The structural relaxationseems to be the very important part of our scheme whenthe stress and the distortion in the network are reduced.

When analyzing the results one should bear in mindthe methods for a-Si:H production. What we are tryingto do is to obtain a-Si:H by hydrogenation of a-Si whilein the real material hydrogen is added during production.Our algorithm leads to formation of microcavities withclusters of H atoms (Figs. 2 and 3). This is also consistentwith the results of ab initio calculations. It should be

emphasized, however, that these ab initio models are verysmall and usually no gap results using these methodsbecause there are too many defects.

We are convinced that our models are the best avail-able at present and it is feasible to increase the size of themodel when circumstances demand. We can also changethe H concentration by modifying the algorithm. Themodel described here is quite suKcient for studies of saytransport and optical properties. The main objective ofour work is to demonstrate that the addition of hydrogento existing models of a-Si produces a clear gap in the elec-tronic density of states. Further progress requires a bet-ter description of the interatomic interaction in the sys-tem. One can apply first principle methods, but thenthere is a very strong limitation on the size of the system,which is a very serious drawback for amorphous models,but such methods may be very important in derivingor verifying new phenomenological methods. Moleculardynamics based on the phenornenological tight-bindingapproach seems to be very promising as an intermediatesolution. s

All structures used in this study and those discussedin Refs. 1—3 are available on request on a floppy disk orvia electronic mail (phy6jangps. leeds. ac.uk).

We gratefully acknowledge the support of SERC. Weare grateful to Dr. Lewis for supplying us with his modelsof a-Si:H.

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Ho, Phys. Rev. B 45, 6839 (1992).B. Jones, J. Phys. C 21, 5735 (1988).' F. H. Stillinger and T. A. Weber, Phys. Rev. B 31, 5262(1985).N. Mousseau and L. J. Lewis, Phys. Rev. B 43, 9810 (1991).R. Alben, M. Blume, H. Krakauer, and L. Schwartz, Phys,Rev. B 12, 4090 (1975).F. Buda, G. L. Chiarotti, R. Car, and M. Parrinello, Phys.Rev. B 44, 5908 (1992).R. Bellissent, A. Chenevas-Paule, P. Chieux, and A.Menelle, J. Non-Cryst. Solids 778@78, 213 (1985); R. Bel-lissent, A. Menelle, A. Chenevas-Paule, and P. Chieux, J.Phys. (Paris) Colloq. 46, C8-93 (1985).J. F. Graczyk, Phys. Status Solidi A 55, 231 (1979).

'sW. Schiilke, Philos. Mag. B 43, 451 (1980).' D. A. Drabold, P. A. Fedders, O. F. Sandey, and J. D. Dow,Phys. Rev. B 42, 5135 (1990).