Amorphous clusters I. Electronic structure of Si clusters with N, P and As dopants

JOURNA

Journal of Non-Crystalline Solids 143 (1992) 232-240 North-Holland

L OF

NON-CRYSTALLIN SOLIDS E

Amorphous clusters I. Electronic structure of Si clusters with N, P and As dopants L.E. Sansores, R.M. Valladares,

J.A. Cogordan

and A.A. Valladares

lnstituto de Inuestigaciones en Materiales, UNAM, Apartado Postal 70-360, Mdxico D.F. 04510, Mexico

Received 25 October 1991

Amorphous impurity clusters of the type XSie0H28 with X = N, P and As have been studied using the well-known pseudopotential SCF Hartree-Fock method (and the HONDO Program). The local electronic density of states and charge density contours have been obtained. It is found that the covalent nature of the bonding in undoped silicon is altered by the presence of the dopants and both an ionic component and a shielding effect appear when N, P and As are substituted in the center of the amorphous cluster. Also, the local density of states in the neighborhood of a Si atom, nearest neighbor to the center of the cluster, indicates the presence of a new p-state in the band gap. There are quantitative differences in the electronic structure of the clusters as a function of the dopants. These results are analyzed in the light of the local changes and their relevance to the amorphous solid state properties.

1. Introduction T h e e l e c t r o n i c p r o p e r t i e s of a m o r p h o u s solids c o n t i n u e to b e a f a s c i n a t i n g c h a l l e n g e in spite of t h e p r o g r e s s t h a t t h e last two d e c a d e s have witnessed. A t t h e b e g i n n i n g o f t h e 1970s, t h e u n d e r s t a n d i n g of t h e e l e c t r o n i c s t r u c t u r e o f a m o r p h o u s m a t e r i a l s was g u i d e d by t h e c o n c e p t s a n d p r o p e r ties o f crystalline solids, by t h e n well u n d e r s t o o d . P r o g r e s s was slow, b u t n e v e r t h e l e s s by t h e e n d of t h e d e c a d e a b e t t e r u n d e r s t a n d i n g was a c h i e v e d t h r o u g h t h e use o f m o d e l h a m i l t o n i a n s o r t h r o u g h t h e study o f specific s a m p l e s with d e f i n i t e struct u r e s [1]. It was t h e n c l e a r t h a t e l e c t r o n i c b a n d s existed in spite o f t h e a b s e n c e o f p e r i o d i c i t y in the structure of the materials, a feature that r e v e a l e d t h a t s h o r t r a n g e o r d e r was m o r e decisive t h a n t h e strict t r a n s l a t i o n a l p e r i o d i c i t y r e q u i r e m e n t usually a s s u m e d n e c e s s a r y for b a n d f o r m a tion. T h e 1970s w i t n e s s e d t h e b r e a k t h r o u g h of S p e a r a n d L e C o m b e r [2] t h a t r e v o l u t i o n i z e d t h e field, since t h e h y d r o g e n a t i o n o f a m o r p h o u s silicon made possible the doping of amorphous semiconductors, a n d all t h a t s u b s e q u e n t l y f o l l o w e d for

device a p p l i c a t i o n s [3]. Since t h e n , m u c h r e s e a r c h has b e e n d e v o t e d to t h e u n d e r s t a n d i n g of t h e doping mechanism and the nature of the energy levels in a m o r p h o u s s e m i c o n d u c t o r s , b o t h experim e n t a l l y [4-10, a n d r e f e r e n c e s c o n t a i n e d therein] a n d t h e o r e t i c a l l y [11-15, a n d r e f e r e n c e s cont a i n e d therein]. P h o s p h o r u s a n d b o r o n have b e e n the m o s t widely s t u d i e d d o p a n t s since t h e y a r e c o m m o n l y u s e d in devices for t h e n a n d p layers. It is q u i t e difficult to d e a l with the effect of local i m p u r i t i e s in a m a t r i x which is n o t well d e f i n e d . F o r t u n a t e l y , m a n y of t h e e l e c t r o n i c p r o p e r t i e s w h i c h a r e e x h i b i t e d by t h e solid a r e of a local n a t u r e . In a p r e v i o u s p a p e r [13], it was shown t h a t c a l c u l a t i o n s of t h e e l e c t r o n i c p r o p e r ties o f a m o r p h o u s clusters with s u b s t i t u t i o n a l imp u r i t i e s t h a t b e l o n g e d to t h e s a m e c o l u m n in t h e p e r i o d i c t a b l e a r e very useful to d e t e r m i n e t h e p r o p e r t i e s of a m o r p h o u s solids with r e s p e c t to a s p e c t s such as t h e effect of d o p i n g or t h e p r o d u c t i o n of e l e c t r o n traps. I n p a r t i c u l a r it was d e m o n s t r a t e d t h a t no i m p u r i t y states a p p e a r in t h e f o r b i d d e n g a p o f Si a n d G e with g r o u p I V impurities, as was e x p e c t e d . M o r e o v e r , it b e c a m e e v i d e n t t h a t t h e local d e n s i t y of states at a site

0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

L.E. Sansores et al. / Amorphous clusters. I

next to the second and higher neighbors of the impurity does not change greatly with the impurity and that all these local densities are very similar, indicating that the perturbation due to the impurities is highly localized around the impurity and the nearest neighbor, and that, therefore, cluster calculation are quite useful as a simulation of the solid state properties. Cluster calculations have also been carried out [14] in order to understand the electronic density of states of a-Si, a - S i : H and a-Si:F, and the picture that emerges indicates that even though the dangling bonds give states around the center of the gap, hydrogenation and fluorination remove them allowing the doping mechanism to take place. Also, more recently, cluster complexes with periodic boundary conditions of a - S i : H doped with phosphorus and boron have been studied [15], and the conclusions seem to indicate that cluster calculations are a very useful tool for understanding some experimental results. Results such as those reported in refs. [13-15] and others [16,17] indicate that the electron states that are modified by the perturbations introduced by point impurities remain localized around them, and that a calculation in a small cluster will suffice to model the actual physical behavior of the solid, especially if the cluster has a structure that resembles, locally, that of the solid. There have been other cluster calculations [18,19] in which the randomness is included directly in the position of the donors. These calculations are performed for clusters which contain six donors located randomly in space, with elect r o n - e l e c t r o n interactions included. The calculation for each donor concentration was carried out for an ensemble of samples with different configurations of donors [20]. The authors argue that even though the cluster size is quite small, this does not cause serious limitations because, for the concentrations considered, the localization length is of the order of the average donor separation and does not exceed the sample diameter. In this paper, we present the first part of an extensive study of the electronic properties of groups III and V impurities in group IV amorphous clusters. This part considers a N, P or As impurity in the center of a tetrahedrally coordi-

233

Fig. 1. Tetrahedrally symmetric cluster used for these calculations with one central impurity, 20 silicon atoms and 28 hydrogen saturators. All rings are boat type.

nated silicon cluster that contains six-fold boat rings which are n o t present in the crystalline diamond structure, indicating that the clusters used in these calculations do not resemble the crystalline environment but are indeed of the amorphous type. The method used is an ab initio SCF H a r t r e e - F o c k and the program is a H O N D O that allows qualitative tendencies of the behavior of the gap to be studied, as well as the charge distribution and the local density of states in the cluster.

2. The clusters

The clusters used in ref. [13] and in the present work (fig. 1) are of the type XSi20H28 , with X = N, P or As, and the central impurity X is surrounded by four silicon nearest neighbors, twelve silicon second neighbors, four silicon third neighbors and 28 hydrogen saturators bonded to the outermost unsaturated Si atom bonds. These clusters have the following characteristics. (i) The impurity nearest neighbors are correctly bonded as in the solid, thereby better simulating the impurity surroundings as present in the solid. (ii) The cluster contains six-fold boat rings that are statistically present in amorphous materials

234

L.E. Sansores et a L / Amorphous clusters. I

but are n e v e r present in crystalline materials. Rings of the boat type appear due to the way the cluster was constructed and this is described next. The cage configuration of the outermost H atoms of the cluster, in the corners of a tetrahedron, arises due to the fact that it is necessary to rotate some of the bonds of the cluster to avoid overlapping of the hydrogen saturators located in the outermost shells. This twisting of the bonds gives as a consecuence the appearance of the six-fold boat rings. It should be noted that it is not possible to construct a three-dimensional crystal out of the tetrahedral cage described above. The fact that the cluster has only six-fold boat atom rings indicates that both eclipsed and staggered dihedral angles exist, and this is why it is claimed that the results reported herein can adequately describe the properties of amorphous semiconductors, both doped and undoped. In ref. [14], the clusters studied also had topological characteristics that took into account several types of silicon rings that are statistically found in the amorphous solid.

3. Method of calculation

Since the substituted impurities have an atomic structure of the type ns2np 3, the electronic structure is not a closed shell and therefore care should be exercised when carrying out the SCF calculations. No d-orbital polarization was included. As mentioned before, the program used was a H O N D O [21] in the open-shell H a r t r e e Fock approximation, and in order to take into account the core potential, pseudopotentials were used as reported in the literature [22]. It has been known for a long time that the H a r t r e e - F o c k approximation is a reliable method in determining total energies as well as qualitative tendencies in the energy scale. Also, the method is such that it allows for a clear understanding of the physical limitations of the calculations, and that is why it has become a method frequently employed in molecular studies. In performing these calculations the basis set must be chosen carefully. Formerly, the optimization of the sets was done on atoms, and then used

in a LCAO expansion. Better results however can now be attained by optimizing the basis set with respect to the total energy in small molecules [23]. The actual basis sets used for Si and H have been optimized in molecules, whereas those for the impurities have been optimized in atoms. All of them are of the minimal size, constructed from Gaussians contracted to one function and taken from ref. [13] for Si and H, and from ref. [22] for N, P and As. All basis sets used in the present calculations are reproduced in table 1. It is well known that the treatment of the lattice relaxation is one of the difficulties in dealing with amorphous materials since the ions are in metastable positions [15]. Therefore in order to avoid spurious geometrical strains, once the basis set has been chosen, care must be exercised in the selection of the optimized geometries because it is this equilibrium geometry which determines the electronic representation of the 'molecular' cluster. The relaxation procedure used in the present work is the one described in ref. [13]. Also, since it has been known for a long time that experimental results indicate that the impurities do not produce a large expansion of the lattice [24], the atoms beyond first neighbors were fixed to simulate the impedance of the rest of the lattice. The results given in the next section corroborate the validity of such an assumption. T h e relaxed distances found between the nearest-neighbor silicons and the impurities in these clusters are: N - S i = 2.177 A, P - S i = 2.450 ,~ and As-Si = 2.523 A as compared with the 2.40 found for Si-Si in ref. [13]. These equilibrium distances, together with those reported in ref. [13] for the other atoms, were those utilized in our calculations.

4. Results

In the H a r t r e e - F o c k calculations, a 'molecular' eigenvector, ~i(?), is written in terms of the contracted 'atomic' basis functions, ~b~(~), as a linear combination, q~i(~) = E c ~ 4 ~ ( ~ ) , ~z

(1)

L.E. Sansores et al. / Amorphous clusters. I Table 1 Basis set used in the present calculations Atom

Symmetry

Exponent

Coefficient

Si

3s

3.127896 1.816917 0.199650 0.134337 1.659000 0.383149 0.161900 0.044942

0.200717 -0.427960 0.672073 0.480353 -0.023819 0.390573 0.545449 0.192034

3.694107 1.271054 0.467733 0.173069 9.749368 2.269663 0.678060 0.202257

-0.171520 0.221727 0.587996 0.352701 0.062622 0.259393 0.496782 0.412617

3.292806 2.056402 0.295489 0.108451 2.245995 0.423865 0.150390 0.055516

0.216184 -0.452262 0.728296 0.429446 -0.024518 0.455759 0.539160 0.129835

3.452899 1.784671 0.290657 0.110228 1.578945 0.499472 0.194294 0.073401

0.127721 - 0.435329 0.685633 0.493845 -0.111633 0.283250 0.576780 0.301399

8.019570 1.292911 0.237862 0.396959

0.056752 0.260141 0.532846 0.291626

3p

235

from which contours of p(?) can be plotted in real space. To obtain the local density of states (LDOS), the following procedure is utilized. First, a given eigenvalue, ei, is chosen with the eigenvector ~i(~); since this eigenvector can be expressed as in eq. (1), the overlap product

W~(Ei) = E (@~(r) I~/"(~) )

(4)

becomes N

2s

2p

P

3s

3p

As

4s

4p

H

ls (basis parameters in Si-Hbond)

Wl,z(Ei) = E Sl.~vC?Izc;,

with S ~ = (~bul~b~) the overlap matrix. Due to the fact that this is a molecular calculation, the eigenvalues form a discrete set and in order to simulate the results for a solid, this S-like spectrum has to be smoothed out using Gaussian functions centered on the eigenvalues, {E/}. It should be clear that the width of the Gaussians has to be chosen adequately so as to smooth the curves while maintaining as much information as possible. It is then clear that Wu(e i) is the contribution of the ~b~ to the weight of the S-function. To obtain the contribution of the atomic orbitals with the origin at a given site A, the following sum has to be carried out:

WA(•i) = E W/z(Ei),

and finally, the total weight associated with the given state is

W('i) = E WA('i).

(7)

A

(2)

i=1 where the factor 2 in the equation is due to spin degeneracy and the sum is over occupied states. The charge density at point ? is then given by =

(6)

/z~A

where the coefficients c~ are all-essential to obtain Pu,~, the charge density matrix:

occ P~,,~ = 2 Y'~ c*Ucy,

(5)

(3)

These weights are used in the construction of the smooth spectrum mentioned above. Consideration of eq. (4) clearly shows that one can select certain orbitals in the sum, p or s orbitals for example, and eqs. (6) and (7) will give the corresponding contributions to the local and total density of states, respectively. In fig. 2 the charge contours are shown for the silicon cluster without impurities. It can be seen that in undoped amorphous silicon the contours correspond to a bonding that is strongly covalent, as expected.

236

L.E. Sansores et al. / Amorphous clusters. I

trons in n-doped bulk amorphous semiconductors, [25]. The LDOS curves at the central sites are shown in fig. 4 for undoped silicon and in fig. 5 for clusters doped with nitrogen, phosphorus and arsenic. The s- and p-contributions are indicated for the silicon atom and for each impurity at the center of the cluster. In undoped silicon, one finds a small s-contribution at the bottom of the conduction band which disappears for the three impurities. In all four cases the bottom of the valence band is s-like and the top is p-like; further, the bottom of the conduction band is p-like.

5. Discussion

Fig. 2. Charge contours for the Si2IH28 cluster around the central silicon atom and nearest-neighborsilicons.

When dopants are introduced, both an ionic contribution to the bonding and a shielding effect appear, although the covalent nature is not completely lost. Figures 3(a) refer to the total charge of the fully occupied states (56 full orbitals), whereas figs. 3(b) depict the charge due to the half-filled states corresponding to the 57th orbital. We believe that the degree of ionicity is better illustrated by the representation of the charge contours due to the 56 levels, whereas the shielding effect due to the lone electron is well represented in the charge distribution of the 57th orbital. The strongest ionicity and shielding appear in the case of nitrogen where the covalent character is almost lost, and these effects decrease as the atomic number of the impurities increases. Comparison of figs. 3(a) with figs. 3(b) shows the very interesting behavior of the half-filled level since the position probability of this electron has a spherical-like maximum around the impurities (slike states), and also an oriented secondary maximum around the silicon atoms (p-like states). These effects are the onset of the experimentally found hydrogen-like behavior of the excess elec-

It has been reported that for c-Si the three peaks observed experimentally, and identified theoretically as s-like, s,p-like and p-like, are affected in a-Si in such a way that the less energetic peaks, s-like and s,p-like, loose their identity and become one, as determined by XPS [26]. The present results seem to give evidence that, indeed, there is a tendency to smooth out the two lower peaks, as depicted in figs. 4 and 6. The states that appear due to s-orbitals in a covalent system are sensitive to the bonding topology, (i.e., ring statistics), and the smoothing out may be due to the fact that in an amorphous solid there are five and seven atom rings in addition to the six-atom boat-type rings typical of the cluster used in these calculations. For the concentrations used in the present calculations, phosphorus and arsenic behave in a similar manner since each one reduces the gap by shifting the conduction band downward and essentially leaving the top of the valence band unmodified. The gap reduction is largest for arsenic. Nitrogen, on the other hand, does shift the top of the valence band and the bottom of the conduction band also moves downward, but not as much as for P and As. There is an overall diminution of the gap for the three impurities, as found experimentally [2712 Nitrogen shows a decrease of the gap size for low concentrations and an increase of the gap as the concentration in-

237

L.E. Sansores et al. / Amorphous clusters. I

(a)

[b)

Fig. 3. Charge contours for the XSiz0H2s cluster around the central impurity atom and nearest-neighbor silicons. (a) The 56 fully occupied orbitals. (b) The 57th half occupied orbital.

creases [28,29]; our results unequivocally indicate a decrease for the concentrations studied. All three impurities shift the bottom of the valence band, but for nitrogen the effect is more dramatic, since an s-like peak appears in this region. Figure 5 shows that the size of the valence band increases with decreasing mass of the impurities, in agreement with experimental results [27,28]. An s-like state appears within the gap which is lower in energy for nitrogen than for phosphorus or arsenic. It can be observed that the depth decreases with increasing mass [5]. This is an effect that can be qualitatively related to the information obtained from the charge contours, since, as can be seen in fig. 3(b), the distance

from the center of the cluster at which the shielding effect is a maximum increases with increasing mass of the impurity, and therefore the energy of the electron in this state, the gap state, increases with increasing mass. This should be compared with the experimental results reported in ref. [25], where it is found that the size of the hydrogen-like states for P-doped Si is (10 _+ 1) A, whereas for As-doped Si it is (9 + 1) A.. In fig. 6 the LDOS is shown for the silicon atom nearest neighbor to the center of the undoped Si cluster, whereas figs. 7(a)-(c) depict the LDOS of the nearest neighbors for the impuritydoped clusters and shows the degree of hybridization of the silicon-impurity bonds. These figures

238

L.E. Sansores et al. / Amorphous clusters. [

also show that the band gap decreases from the value for undoped silicon as the mass of the impurity increases. Also, the size of the valence band decreases as the mass of th e impurity increases. Both the top of the valence band and the bottom of the conduction band shift to lower energies, but the top of the valence band is located at the same energy for all three impurities. In addition, a state whose symmetry is principally p-like is introduced in the gap by the impurity; the energy of this state decreases as the mass of the impurity decreases. The fact that the gap state is s-like for the LDOS near the center of the cluster and p-like for the nearest neighbor is related to the fact that as the lone electron approaches these Si atoms around the center of the cluster the minimun energy state has to be p-like to contribute to the minimization of the repulsion between impurities and the host atoms. These results agree with those reported by Yang et al. [15], for a - S i : H clusters doped with B and P with periodic boundary conditions. Figure 8 gives evidence of the validity of the supposition set forth above in regard to the impedance of the lattice. It can be seen that, in fact, the perturbation due to the impurities does not manifest itself beyond the nearest neighbors, thereby justifying the assumption that holding

16 14

a)

N Si2oH2e

12

o4

-I

'

-o.e

-o6

-o4

, A . . . .o.2

-~6.-2-r-0 ~

16 b)

P Si2oH2e

14 12

u) o o J

tO

8 6 4 2 0 -I

-0.8

-0.6

-0.4

-0.2

0.2

0.4

e) I4

As Si2oH2e

12

-I

Siz, Hz8

0

16

6

0.4

-0.8

.L. . . . .

-0.6

-0.4

-0.2

ENERGY

% ~

0

0.2

"

0.4

(o.u.)

5

Fig. 5. Local density of states at the impurity site for the (a) NSi20H2s cluster, (b) PSi20H2s cluster and (c) AsSi20H28 cluster. - - , total LDOS; , s-contribution; and . . . . . . , p-contribution. E i is the energy of the gap state.

2

/,,, O_I

YJ

-08

:; ;.,:... .... -

-0.6

t,,"'" .....

\v, 7-

-04

-0.2

ENERGY

......

, 0

0.2

- - -

....

0.4

rigid the second-neighbor silicons is adequate for the problem at hand.

(o.u.)

Fig. 4. Local density of states at the central site for the Si21H2s cluster. , total LDOS; - - - - - - , s-contribution; and . . . . . . , p-contribution. Vb is the bottom of the valence band, Vt i s the top of the valence band, E i is the energy of the gap state and C b is the bottom of the conduction band.

6. Summary and conclusions Amorphous cluster models have been used for some time to obtain information about the struc-

L.E. Sansores et al. / Amorphous clusters. I

ture and properties of amorphous solids. In particular, the electronic structure of amorphous materials has been the subject of systematic efforts by various research groups. To summarize, we have found that the charge density contours give evidence of the existence of both an ionic behavior of the bonding and a shielding effect. The ionicity and the shielding decrease as the atomic mass of the impurities increase. This behavior indicates the possible onset of the hydrogen-like characteristic of excess electrons in donor-doped semiconductors. Also, there is a qualitative systematic behavior for the L D O S near the impurities that depends on the atomic mass of the impurities. For example, both the energy gap, (C b - Vt), and the size of the valence band, (Vt - V b ) , decrease as the atomic mass is increased. All the impurities introduce a gap state whose energy E i moves upward towards the bottom of the conduction band as the impurity mass increases; this state is s-like near the impurity and p-like near the surrounding Si atoms, due to the onset of the hydrogen-like behavior described above. The position of the bottom of the valence band, Vb, systematically increases in energy as the mass of the impurity is increased while the bottom of the conduction band, Cb, decreases in energy with increasing mass. The fact that the silicon-nitrogen bonding is strongly ionic in character and that the impurity state in the gap is deep agrees with the experimental result that nitrogen introduces recombina-

239

I0 a)

N Si2oH 28

03 0 -I

-I

-0.8

-0.6

-0.4

-0.2

b)

u~ o _J

6

4

0

J-°'~ --]

It • v -.,, ,. .

--0,8 --0,6

.

.

--0.4 --0.2

. 0

0.2

0.4

IO c)

As SizoHze

8 (n 0 I

6

.J

4

%1 -0.8

-0.6

~°° -0.4

-0.2

0

0.2

0.4

(a.u.)

Fig. 7. L o c a l d e n s i t y o f s t a t e s at t h e silicon a t o m s n e a r e s t

neighbors to the impurity for the (a) NSi20H2s cluster, (b) PSi20H2s cluster and (c) AsSizoH28 cluster. , total LDOS; , s-contribution; and . . . . . . , p-contribution.

5

2 ,

o

0.4

P Si20H 28

ENERGY SiZl HZ8

0.2

I0

-I

6

0

.

-I

-0,8

-016

-0.4

-O.Z

......

0

0.2

014

ENERGY (a.u.) Fig. 6. L o c a l density o f states at t h e silicon atoms nearest-

n e i g h b o r s to t h e c e n t r a l silicon f o r t h e S i 2 1 H z s c l u s t e r . - - , total LDOS; , s-contribution; and ...... , p-contribution.

tion states in amorphous and crystalline silicon. Nitrogen introduces the biggest changes in the valence band while phosphorus and arsenic behave more or less the same. As experimentally observed, both P and As reduce the gap. The authors like to thank P. G r a n t and S. Muhl for some useful comments on this work,

240

L.E. Sansores et aL / Amorphous clusters. I ~6

12

10

As

•

N

/

I - I

-0.8

-0.6

-0.4 ENERGY

-0.2

0

0.2

"- .... 0.4

(a.u.)

Fig. 8. Local density of states (LDOS) for second silicon neighbors both for doped and undoped clusters. It is clear that the presence of the impurities barely have an. influence in the LDOS for these silicon atoms, thereby showing that the influence of these impurities is highly localized.

a n d J. C a m a c h o f o r d o i n g s o m e o f t h e f i g u r e s . T h e y w o u l d also like to t h a n k H . R i v e r o s f o r b r i n g i n g ref. [24] to t h e i r a t t e n t i o n . T h i s w o r k was p a r t i a l l y s u p p o r t e d by t h e O A S w i t h i n t h e P r o grama Multinacional de Materiales, and carried out on a SUN4/280 c o m p u t e r p r o v i d e d by DGSCA, UNAM.

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