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THE ELECTRONIC STRUCTURE OF A MODEL DEFECT IN HYDROGENATED AMORPHOUS
SILICON
D. Divincenzo, J. Bernholc, M. Brodsky
To cite this version:
D. Divincenzo, J. Bernholc, M. Brodsky. THE ELECTRONIC STRUCTURE OF A MODEL DE-
FECT IN HYDROGENATED AMORPHOUS SILICON. Journal de Physique Colloques, 1981, 42
(C4), pp.C4-137-C4-140. �10.1051/jphyscol:1981426�. �jpa-00220882�
JOURNAL DE PHYSIQUE
CoZZoque C4, suppl&ment au nO1O, Tome 42, octobre 1981 page C4-137
THE ELECTRONIC STRUCTURE OF A MODEL DEFECT I N HYDROGENATED AMORPHOUS SILICON+
D. P. DiVincenzo, J . ~ e r n h o l c * and M.H. Brodsky**
Moore School of E l e c t r i c a l Engineering, University of Pennsylvania, P h i k d e l p h i a , PA 19104, U.S.A.
*Enon Research and Engineering, Linden, NJ 07036, U. S. A.
**TBM Thomas J . Vatson Research Center, Y o r k t o m Heights, NY 10598, U.S.A.
Abstract.- We calculate the electronic properties of a model defect for hydrogen in hydrogen- ated amorphous Si. Our model is a vacancy in crystal Si with four H's satisfying the dangling bonds. Using a Green's function technique, we find the change in the density of states caused by the defect, as well as the local density of states for the Si-H bond and surrounding bonds. From several approaches, we extract information on band edge localiza- tion. Each approach gives a mobility edge of order tenths of an eV, therefore we conclude that compositional disorder has an effect comparable to that previously estimated for topological disorder. Conduction band effects are calculated to be similar but smaller.
I. Introduction.- In an attempt to understand the mechanism for band edge localization in hydrogen- ated amorphous Si (a-Si-H), several different types of microscopic disorder have been studied.
Within the assumptions of the ideal continuous random network, model calculations1 have deter- mined that bond length and bond angle variations are not as effective in causing band edge localization as dihedral angle distortions and topological disorder (e.g., 5-fold rings). In this paper we examine the spatially resolved electronic structure of a model Si-H defect in order to test the idea that the compositional disorder caused by the presence of H also makes an important contrib- ution to band edge localization. The results show a net depletion of band edge states for both the valence and conduction bands. For H induced localization alone, reasonable mobility edge values are obtained. Our results relate to the quantum well model2 of localization based on extended range percolation theory and the ~ n d e r s o n - ~ o t t ~ picture of disorder induced localization.
We choose to study the fully hydrogenated monovacancy in a crystal Si (x-Si) lattice.
Choosing the lattice to bc crystalline isolates the effect of H from other sources of disorder, e.g., bond distortion and topological disorder. The hydrogenated vacancy is also the simplest Si-H defect that is structurally compatible with the underlying x-Si lattice. Using a Green's function t e ~ h n i ~ u c ~ . ~ we directly obtain the change AN(E) in the host density of states NO(E) due to a single hydrogenated vacancy in an otherwise perfect infinite crystal. The technique also gives the energy and spatially resolved valence charge densities in the vicinity of the defect. The decay length for charge changes was previously used to give4 an estimate of E from extended range percolation arguments. In this work the local densities of states around thklhydrogenated vacancy gives both the shift in the mean energy and the band edge which we use as measures of the strength of the disorder. We then estimate E for reasonable H concentrations.
P
11. Calculations.- We compute the electronic structure of a model hydrogenated defect in x-Si. This defect is formed by removing one Si atom from the lattice and placing four H's at the end o: each of the resulting broken bonds. The Si-H bond length is fixed at its silane value of 1.48A; the distance between H atoms is 1.42& about twice the H-H molecular bond length. We have not studied lattice relaxation around the defect, but believe it to be be negligible because of bond restoration by H. The missing Si atom is replaced by 4 H's with a total of 4 tetrahedrally arranged valence electrons. The connectivity of the lattice is restored and, therefore, in terms of range effects, the local disturbance from the 4 Si-H bonds is mild compared to an isolated Si-H bond.
Therefore we consider the hydrogenated vacancy as a good estimate of clustered H on submicro- scopic voids, but an underestimate of the disturbance from an isolated Si-H bond in a - ~ i : ~ . ~ In terms of the total number of states disturbed, each model defect truly represents 4 Si-H bonds.
'This work is supported in part by the NSF on grant number DMR 76-80994, and by the Solar Energy Research Institute under Subcontract No. ZZ-0-9319.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981426
JOURNAL DE PHYSIQUE
The method of computation is a pseudopotential Green's function technique adapted4 from previous studies o n the bare mono-vacancy in x - ~ i . ~ This method is capable of giving electronic energies of defect states to the same accuracy as pseudopotential band structure calculations for perfect crystals. We use a host Green's function CO(r,E) derived from a self consistent plane wave pseudopotential calculation for x-Si which includes 4 valence and the 16 lowest conduction bands.
Solving Dyson's equation gives the Green's function G(r,E) for the host plus defect system. From this we extract the desired quantities: the total change in N0(E),
AN(E) =
- 2 s
I ~ [ G ( ~ , E ) - G 0 ( r , ~ ) ] d r"
All Space (1)the valence charge density p(r):
~ ( r ) =
- n J
2 Im G(r,E) d E Valence Bandand Nloc(E). the local density of states in some region in the vicinity of the defect:
Fig.1 The host density of states N'(E) for pure x-Si and the change AN(E) due to one hydrogenated vacancy. Left: The conduction band up to 4eV and the entire valence band;
Right: Expansion of thc band gap region.
111. Results.- Figure 1 shows NO(E) and A N ( E ) . ~ The points to note are that, in contrast to the bare Si vacancy,5 the defect has no bound states in the band gap or below the valence band. Nor are there sharp resonances within the valence band or within the first 3eV of the conduction band.
The structure in AN(E) results from small downward shifts of the original peaks in the NO(E) in the vicinity of the defect. There are also states states removed from both band edges. As Fig.1 shows, about 1.5 states have been removed per defect from the top 1.5eV of the valence band and about 0.1 stateddefect from the bottom 0.5eV of the conduction band.
Local densities of states, shown in Fig.2, provide morc detailed information about the spatial dependence of the valence band edge. These densities are averaged over a sphere (of diameter equal to the Si-Si bond length) centered on each bond. As we have pointed out earliep, the Si-H
bond charge is actually centered very close to the H atom. Note that there are some states left a t energies :ll the way up to the band edge. Even as far away as on the twelve next neighbor Si-Si bonds, 4A from the vacancy site, NlOc is still less than its bulk value.
The local densities of states Nloc(E) for four bond regions:.
1) S1-H bond; 2)Nearest Si-Si bond; 3)Next nearest Si-Si bond; 4) Si-Si bond in pure x- Si.
ENERGY [eV]
IV. Discussion.- Using the results for the three quantities p(r), AN(E) and N1,,(E), we now discuss three different approaches to the question of localization at the top of the valence band.
A. Spatial Variation of p(r)
-
Our earlier work4 used percolation theory to give one crude estimate of the width of the localization region. Briefly, we observed a depletion in p(r) for the top 0.25eV of the valence band in the vicinity of the hydrogenated defect. Assuming that this depletion healed exponentially into the bulk Si, the decay length, 2.7A, was used as an estimate of the radius from which band edge carriers are effectively excluded. From a classical percolation analysis for a collection of such excluded spheres, we concluded that in a sample with 15% H concentration, a large fraction of the carriers in the top 0.25eV of the valence band are localized, i.e., cannot percolate through the sample. Figure 2 is a confirmation of the extended range of excluded carriers near the top of the valence band.B. Anderson-Mott Localization due to AN(E) - From our present local density of states calculations, we now use the Anderson theory8 to make a simple estimate of the position of the mobility edge. Within a tight binding theory, E is approximately specified by8:
P
z exp
(J
Pn L P ( E ) ~ E ) = 1 E -EP
where V is the off-diagonal matrix element (assumed constant throughout the system), z is the coordination number, E is the on-site diagonal matrix element, and P(E) is the probability distribu- tion for E on all the sites of the system. In the present case, there are just two on-site energies:
ESi-H at the Si-H bond, ESi-si for the Si-Si bond. Therefore P(E) = a G(E-ESi-H)
+
(1-01) where a is the defect concentration. A reasonable value for a is 15%.
ESi-H=-6.2eV and ESi-Si=-5.4eV were determined from the first moment of the local density of states on the Si-H bond and on the Si-Si bond. Eq.4 now gives E VB = 0.1. From eq.4 we can also obtain an approximate but physically meaningful expression for P E Expanding in E /zV gives
P. P
E = aAE. From this expression and AE = ESi-H-ESi-Si = 0.8eV, we find E VB =0.1 This value is a lower limit on the mobility edge because of several simplifications whict ignore effects that P enhance localization. For example, the off-diagonal elements V are not constant and are smaller for the Si-H case. The narrower Si-H valence band, therefore, should in a way that is not as easily ,
calculable localize more states at the top of the valence band. The next section gives an ad hoc method for estimated the magnitude of the band edge localization in terms of the number of depleted states per defect.
C. Band Edge Depletion of NIoc(E)
-
Recall from Fig.1 that each hydrogenated vacancy removes states from both band edges. If the defect concentration f is so small that the defects do not interact with each other, then the total density of states is given by Nf(E)=N0!E)+fAN(E).However, for defect concentrations of order f=0.04 we find that Nf(E) calculated in this way is
JOURNAL DE PHYSIQUE
Lower limits of values of va- lence and conduction band mo- bility edges as a function of the defect concentration.
negative for both band edges. Such an unphysical result indicates that neighboring defects are strongly interacting. On the other hand, for energies deeper in the bands NO(E)>-fAN(E).
Conductivity can not take place in the absence of states. Assuming that the overlap effects tend to preferentially remove states from energies closest to the band edges, we use the condition
I
Nf(E )dE = 0 for estimates of lower limits on the valence and conduction band mobility edges.Fig. 3 Mows E > ~ and E CB as a function of f. For f30.04, E VB = 0.15 and E~~~ = 0.06eV for the valence and conckction bands, respectively. In reality and within the context of the P quantum well model, states in real space are depleted selective1 near the defects, creating barriers and further pushing the E 's into the bands. The ratio of E tB to E CB is consistent with the
assumptions of the quantum well model. P P P
Conclusions- Each of three distinct approaches agrees that the mobility edge lies more than O.leV into the top of the valence band. We obtain the physical picture that the localization follows from long range charge density exclusion from the Si-H bond. Because of the connectivity of the Si-H orbitals across the chosen model defect, the computed E 's are underestimates of the actual values.
We point out that even so the effect of compositional diforder is predicted to be as great as that of topological disorder; both give E on the order of tenths of an eV. Therefore, neither effect can be
neglected in a-Si:H. P
References-
1. F. Yonezawa and M. H. Cohen, in Fundamental Physics of Amorphous Semiconductors, Ed. by F. Yonezawa, (Springer, Berlin, 1981), p. 119.
2. M. H. Brodsky, Solid State Comm.
36,
(1980) 55.3. N. F. Mott and E. A. Davis, in ~ G t r o n i c Processes in Non-Crystalline Materials, (Claredon Press, Oxford, 1979), Ch. 2.
4. D.P. DiVincenzo, J. Bernholc, M.H. Brodsky, N.O. Lipari, and S.T. Pantelides, Tetrahedrally Bonded Amorphous Semiconductors, Ed. by R.A. Street, D.K. Biegelsen and J.C. Knights, AIP Conf. Proc.
73,
(1981) 156.5 . J. Bernholc, N.O. Lipari and S.T. Pantelides, Phys. Rev. B
21,
(1980) 3545.6. E.g., clustered and isolated Si-H concentrations are measurable by NMR, see J.A. Reimer, these proceedings.
7. We have presented similar results earlier (Ref. 4), but the present work uses a Si band structure which is more accurate in the valence band but which gives a band gap of 0.8 eV. We have also extended,our results to higher energy in the conduction band.
8. J. Ziman, Models of Disorder, (Cambridge University, Cambridge, 1979), Ch. 9.
