Microstructural analysis of nanostructured amorphous silicon–germanium alloys: Numerical modeling

Journal of Non-Crystalline Solids 357 (2011) 2620–2625

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Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l

Microstructural analysis of nanostructured amorphous silicon–germanium alloys: Numerical modeling R. Ben Brahim, A. Chehaidar ⁎ Research Unit in Mathematical Physics, University of Sfax, Faculty of Sciences, Department of Physics, P.O. Box. 1171, 3000 Sfax, Tunisia

a r t i c l e

i n f o

Article history: Received 18 September 2010 Received in revised form 28 February 2011 Available online 27 April 2011 Keywords: Amorphous silicon–germanium alloys Structure Numerical modeling

a b s t r a c t A detailed microstructural analysis of amorphous silicon–germanium alloys with germanium fraction ranging from 0.1 to 0.5 is performed by means of a numerical modeling technique. By substituting Ge atoms for Si atoms in nanoporous paracrystalline network of amorphous silicon, several amorphous silicon–germanium structures have been generated then relaxed. The main aim of our work is to study the effect of compositional heterogeneities on the structural properties of amorphous silicon–germanium alloys in comparison with the standard case, that of a homogeneous random distribution of the atoms. In the present work we envisage the two-phase amorphous silicon–germanium model proposed by Goerigk and Williamson to interpret their anomalous small-angle X-ray scattering measurements; it consists on a mixture of Ge-rich and Ge-poor domains at the nanoscale. The microstructure of our structural models is analyzed by examining the macroscopic mass density, the X-ray diffraction intensity, the radial distribution functions, the bond lengths and the coordination numbers within the first coordination shell of Si and Ge atoms. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The amorphous silicon–germanium (a-Si1 − xGex) alloys have known a technological interest essentially in the photovoltaic domain. Indeed, the addition of germanium in amorphous silicon leads to a smaller gap allowing more absorption in the solar spectrum and ensures more efficiency. They also present a fundamental interest which lies in their optoelectronic properties. Or the microstructure of these compounds controls their optical and electronic properties. From this fact we understand the growing interest in studying the microstructure of a-Si1 − xGex alloys. Many experimental studies have been reported in the literature in order to explore the structure of these alloys locally and at the nanoscale. Extended X-ray absorption fine-structure (EXAFS) has been used to investigate the local structure of a-Si1 − xGex alloys [1–5]. Some studies showed that the local environment of Ge atoms is completely random whereas others suggested an inhomogeneous distribution of Ge atoms in Ge-rich films. To examine the structure of a-Si1 − xGex alloys at the nanoscale, the small-angle X-ray scattering (SAXS) technique was used. Several SAXS data carried out for hydrogenated a-Si1 − xGex films [6–9] argue in favor of a heterogeneous microstructure of a-Si1 − xGex alloys at the nanoscale. This heterogeneity was interpreted as due partly to the presence of voids and/or hydrogen-rich clusters in hydrogenated amorphous films. More detailed nanoscale structural analysis has

⁎ Corresponding author. Tel.: + 216 98 488 247; fax: + 216 74274437. E-mail address: [email protected] (A. Chehaidar). 0022-3093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2011.03.008

been reported by Goerigk and Williamson [10–12] for a series of hydrogenated a-Si1 − xGex films using the anomalous small-angle Xray scattering (ASAXS) technique. The contribution of structural heterogeneities such as voids has been successfully separated from the total diffracted intensity. The remaining ASAXS has been successfully interpreted in the context of a two-phase model which consists on a mixture of Ge-rich and Ge-poor domains with correlation lengths in the nanometer range. On the other hand, some theoretical studies have been reported in the literature to aid at the interpretation of the EXAFS data for a-Si1 − xGex alloys [13–17]. These efforts have been concentrated on the analysis of the microstructure of these alloys at the local scale. Some works opted for a random mixture of the two types of atoms in the alloy whereas others showed the opposite. Moreover, some results disagree with others regarding the evolution of the bond lengths with the alloy composition. From this bibliography one might conclude that the microstructure of a-Si1 − xGex alloy is not so simple, contrary to what one might think, and it is not yet fully resolved. So, other efforts, particularly of theoretical aspect, are asked to provide more aid to the interpretation of experimental data and consequently a better understanding of the organization of matter in these alloys both locally and at the nanoscale. In this framework fit our theoretical works. We are interested particularly in the effect of compositional heterogeneities on the structural properties of a-Si1 − xGex alloy in comparison with the ideal case, that of a homogeneous random distribution of the atoms. In our previous work [19] we have envisaged atomistic models of a-Si1 − xGex alloy with nanoscale Ge-segregation. We have

R.B. Brahim, A. Chehaidar / Journal of Non-Crystalline Solids 357 (2011) 2620–2625

concluded that such phase-segregation must be discarded, in agreement with Tzoumanekas and Kelires [16] finding. In the present work we envisage a more reliable atomistic model for a-Si1 − xGex alloy, proposed by Goerigk and Williamson [10–12], which consists on a mixture of Ge-rich and Ge-poor areas with correlation lengths in the nanometer range. 2. Calculation method The starting point of our modeling of the structure of a-Si1 − xGex alloy is the structural model recently developed in our research team for the amorphous silicon, called the nanoporous paracrystalline model [18]. It consists on a distribution of paracrystallites and nanopores in a continuous random matrix of amorphous silicon. Our model is contained in a parallelepiped supercell with periodic boundary conditions. By substituting Ge atoms for Si atoms in the nanoporous paracrystalline network of a-Si, structural models for a-Si1 − xGex alloys are constructed. To generate the two-phase model, the substitution procedure is carried out in two stages. First, the Ge atoms are substituted for Si atoms within non-overlapping spherical volumes of predefined radius with a predefined Ge fraction. Then, the Ge atoms are substituted for Si atoms outside these spherical volumes with a different predefined Ge fraction. For comparison, we have prepared a-Si1 − xGex structures with homogeneous random distribution of Ge atoms. In this case, Ge atoms are substituted for Si atoms chosen randomly among the N atoms in the supercell. Once the structural model of a-Si1 − xGex alloy is obtained, we proceed to its relaxation. The structural relaxation is accomplished by the minimization of the total strain energy of the network. As we are concerned with covalent tetrahedrally-bonded solids, the short-range interatomic interactions prevail. The phenomenological model currently used to describe short-range valence forces in these systems is the Keating model [20]. As we deal with disordered solids, anharmonic effects become important and, consequently, must be taken into account. In the present work, the anharmonic Keating model proposed by Rücker and Methfessel [21] was used. Within the framework of this model, the strain energy of the system is given by: 


2 1 0 0 → → → → 0 V = ∑ αij r ij ⋅ r ij −rij 2 + ∑ βijk r ij ⋅ r ik + rij rik 3 i;j i;j;k

2

:

ð1Þ

The first sum in this expression is on all atoms i in the supercell and their four nearest neighbors specified by j. The second sum is on all → → atoms i and pairs of distinct neighbors. r ij and r ik are the vectors connecting atom i with its first-neighbors j and k, respectively. r0ij is the unstrained i–j bond length with r0ij = 2.35 , 2.40 and 2.45 for Si\Si, Si\Ge and Ge\Ge bonds, respectively. The force constants α and β essentially describe the bond-stretching and bond-bending restoring forces, respectively; their dependence laws are given by [21]: αij =

0 αij

rij0 rij

!4 ð2Þ

and 0 0

βjik =

0 βjik

rij rik rij rik

!7 = 2 :

ð3Þ

The total strain energy given by Eq. (1) is minimized by an iterative conjugate gradient method starting with initial configuration in which all the atoms in the supercell are randomly displaced from their crystalline positions. In this relaxation procedure, variations of the supercell sizes are allowed. Now, given the equilibrium coordinates of all the atoms of the relaxed network, its structural characteristics can be easily computed such as the pair correlation functions, the average bond lengths, the

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coordination numbers, the macroscopic mass density and the static structure factors. The partial pair correlation function, gij(r), is defined by the relationship: 2

dNij ðr Þ = 4πr n0 cj gij ðr Þdr

ð4Þ

where dNij(r) is the average number of atoms of j-type confined in the shell of radii r and r + dr centered at an atom of i-type taken as the origin, n0 is the macroscopic number density of the model, and cj is the fraction of j-type atoms. Given the finite sizes of our structural models, their computed g(r)s are corrected for the supercell size effects. From  Eq. (4) we deduce the average partial coordination numbers, Z ij , the  average total coordination number, Z total , the average partial bond   lengths, r ij , and the average total bond length, r total , defined by: rc

rc

0

0

 2 Z ij = ∫ dNij ðr Þ = 4πn0 cj ∫r gij ðr Þdr

ð5Þ

rc

 2 Z total = 4πn0 ∫ r ∑ ∑ cj gij ðr Þdr 0

i

r

j

ð6Þ

r

4πn0 cj c 3 1 c  r ij =  ∫rdNij ðrÞ = ∫ r gij ðr Þdr Zij Z ij 0 0

ð7Þ

and r

4πn c 3  r total =  0 ∫ r ∑ ∑ cj gij ðr Þdr i j Z 0

ð8Þ

where rc is the average separation between two first-neighbor atoms of i- and j-types. The partial static structure factors of the structural model are computed according to the relationship: Sij ðkÞ = 1 +

i 4πn0 ∞ h ∫ r gij ðrÞ−1 W ðr Þ sinðkr Þdr k 0

ð9Þ

in which W(r) represents the Lorch function given by [22]: W ðr Þ =

sinðπr = rmax Þ πr = rmax

ð10Þ

where rmax is the extent of the interatomic separation range of g(r). 3. Results The starting point of our calculations is the determination of the values of the force constants α0ij and β0jik (see Eqs. (2) and (3)) as empirical parameters in the strain potential model given by Eq. (1). They have been fixed at their crystalline values; they have been chosen in order to reproduce in a satisfactory manner the densities of vibrational states of crystalline compounds such as silicon, silicon–germanium solid solution and germanium [23,24]. The obtained values are α0SiSi = 49.23 N/m, β0SiSiSi = 6.90 N/m, α0GeGe = 42.09 N/m, β0GeGeGe = 5.72 N/m, α0SiGe = 43.32 N/m, β0SiGeSi =β0GeGeGe and β0GeSiGe =β0SiSiSi [19]. A nanoporous paracrystalline model for a-Si was first constructed from a 980-atom diamond structure with four nanopores of ~1 nm in diameter and one paracrystallite of ~1.6 nm in diameter. By substituting Ge atoms for Si atoms in the preceding a-Si network, many structures for a-Si1 − xGex alloys, with x ≈ 0.1, 0.2, 0.3, 0.4 and 0.5, have been generated then relaxed. In the case of two-phase models, the ratio between the Ge fractions inside and outside the spheres is fixed at ~0.1. For each network the structural properties have been computed numerically and averaged over nine different configurations for each alloy composition. The macroscopic mass density of each structural model, defined by the total mass of the atoms divided by the volume of the relaxed

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supercell, has been computed numerically. The results are depicted in Fig. 1. For comparison, we have reported in the same figure the experimentally derived densities for ion-implanted a-Si1 − xGex films by Laaziri et al. [25], and the predicted values using the Vegard's law. In the framework of this law, the mass density of the a-Si1 − xGex alloy is given by the relation: ρ = ð1−xÞρSi + xρGe

ð11Þ



 2 2 2 2 IðkÞ = fSi cSi cGe + cSi SSiSi ðkÞ + fGe cSi cGe + cGe SGeGe ðkÞ

0.5

0,5 0.4 0.3 0.2

1

1,5

2

0.1

²²

I (k)

in which ρSi and ρGe are the mass densities of a-Si and a-Ge, respectively. Assuming a nanoporous paracrystalline structure for a-Si and a-Ge, we have obtained ρSi = 2.257 g/m3 and ρGe = 5.158 g/m3. The analysis of the microstructure of our structural models at the intermediate range has been accomplished by simulating the X-ray diffracted intensity, I(k). The later is expressed in terms of the partial structure factors as follows [26]:

0

ð12Þ

+ 2fSi fGe cSi cGe ðSSiGe ðkÞ−1Þ where fSi and fGe are the atomic scattering factors of Si and Ge, respectively. In our present calculations, we have used a simple phenomenological model describing the dependence of the atomic scattering factor on the scattering vector, viz.:

0 0.5

−K2 k

f ðkÞ = K1 e

ð13Þ

+ K3

where Ki are empirical constants adjusted to experimental data for pure a-Si [25] and a-Ge [27]. A best fit has been obtained with K1 = 13.49, K2 = 0.15 and K3 = 0.73 for Si atom, and K1 = 31.53, K2 = 0.13 and K3 = 2.10 for Ge atom. The computed X-ray diffraction intensities are reported in Figs. 2 and 3. The partial pair correlation functions, gSi\Si(r), gSi\Ge(r) and gGe\Ge(r) computed for our structural models are reported in Fig. 4. More insights

4,00 experimental data two-phase model Vegard's law

0

2

4

6

k

0,5 0.4

1

0.30.2 0.1

8

10

12

(Å-1)

Fig. 2. The X-ray diffracted intensities computed for the two-phase a-Si1 − xGex networks (upper panel) and for the homogeneous one-phase models (lower panel). The arrows indicate the Ge fraction (x). The inset in the lower panel shows a direct comparison of the SAXS computed for two-phase structure and one-phase network with x = 0.5. The experimental data, shown by empty circles, are from Shevchik et al. [29].

into the short-range order of our structural models are inferred from the total and partial average bond lengths, rtotal ,  r Si−Si ,  r Si−Ge and  r Ge−Ge , and  also from the total and partial average coordination numbers, Z total ,     Z Si−Si , Z Si−Ge , Z Ge−Si and Z Ge−Ge . The calculation has been performed in conformity to the Eqs. (6) and (7) with the cutoff distance, rc, fixed at the first minimum of the pair correlation function (rc =2.8 Å). The results relative to the bond lengths are reported in Fig. 5, and those relative to the coordination numbers in Fig. 6. For comparison, we have reported in the insets of these figures experimental data derived by EXAFS measurements on Si-rich a-Si1 − xGex:H films by Chapman et al. [3].

Density (g.cm-3)

3,50

3,00 4. Discussion

2,50

0

0,1

0,2

0,3

0,4

0,5

0,6

alloy composition, x Fig. 1. Variation of macroscopic mass density of a-Si1 − xGex alloys with the alloy composition. Circles denote mass densities computed for our relaxed two-phase structural models, squares those predicted by Vegard's law, and triangles the experimentally derived values on ion-implanted silicon–germanium alloys films (from Laaziri et al. [25]). The lines are drawn as guides to the eyes.

The fundamental property critical for testing microscopic models for covalent tetrahedrally-coordinated amorphous solids is the macroscopic density. Density measurements for a-Si1 − xGex alloys have led to values smaller than their crystalline counterparts [5,10,25,28], depending on the alloy composition and preparation method. As can be noted from Fig. 1, the two-phase a-Si1 − xGex structures reproduce the experimentally derived mass densities [25] for all the examined alloy compositions. On the other hand, the Vegard's law underestimates the amorphous alloy density, in agreement with experimental findings [5,10,25]. We retrieve here practically the same results as those obtained for structural models with homogeneous random distribution of Ge atoms as well as with Ge-atoms segregation [19]. The macroscopic mass

R.B. Brahim, A. Chehaidar / Journal of Non-Crystalline Solids 357 (2011) 2620–2625

0,5

1

I (k)

0

0

2

4

6

8

10

12

k (Å-1) Fig. 3. The X-ray diffracted intensities computed for three two-phase networks with four (bottom), two (middle) and one (top) compositional nanoheterogeneities of ~1.7, ~1.3 and ~1 nm in diameter, respectively. The inset shows a direct comparison of the computed SAXS for these networks.

density is thus insensitive to the compositional heterogeneity in amorphous silicon–germanium alloy. Now we turn our attention to the local structural characteristics of our computer-generated models for a-Si1 − xGex alloys. Direct experimental data about atomic structure in amorphous materials, at shortrange as well as medium-range, were usually provided by X-ray or neutron diffraction experiments. Unfortunately, no diffraction data were reported for a-Si1 − xGex alloys, except those reported by Shevchik et al. [29] for a-Si0.5Ge0.5 film prepared by sputtering. In our present work, we have paid a particular attention to this structural characteristic

45 40 Ge-Ge

Pair correlation function

35 30 25 Si-Ge

20 15 Si-Si

10 5 0

to show if compositional nanoheterogeneities in the alloy is reflected in the diffracted intensity. As shown in Fig. 2, our two-phase structural model with x = 0.5 reproduces reasonably the overall aspect of the experimental data of Schevchik et al. So, It accounts for the intense small-angle scattering (k b 1 Å− 1), and the four features in the largeangle scattering (k N 1 Å− 1) range which decrease in intensity with increasing k. This figure shows also that this aspect is maintained for all the Ge concentrations. However, the diffracted intensity enhances when the Ge fraction increases. This fact results indeed, from the importance of the atomic scattering factor of Ge atom relative to that of Si atom. As can be noted from this figure also, the large-angle scattering in the twophase a-Si1 − xGex network is indistinguishable from its counterpart in the structure with homogeneous random distribution whatever the alloy composition. The small-angle scattering, however, shows a noticeable enhancement in the presence of compositional nanoheterogeneities, as can be noted from the Inset in the bottom of this figure. From these results one might conclude that the large-angle scattering is insensitive to the details of the Ge arrangement in the amorphous silicon–germanium alloy. This is not the case, however, for the SAXS. The size effect of the compositional heterogeneity on the SAXS has been examined by computing the diffraction intensity for different networks which differ by the ‘sphere’ size; the concentration and the size of the nanopores are the same for all the examined networks; even more, the volume fraction of each phase [defined by the ratio of the total number of atoms in each phase to the total number of atoms in the supercell] as well as its alloy composition are maintained fix. The results obtained for x = 0.5 are depicted in Fig. 3. As expected, the SAXS of a-Si1 − xGex alloys is affected by the compositional-heterogeneity size. Indeed, the SAXS enhances with increasing compositional-heterogeneity size. We expect more pronounced effect with larger heterogeneities. The real-space information regarding the topological and chemical disorder in our structural models for a-Si1 − xGex alloys is provided by the radial pair correlation function. Fig. 4 shows the results of our calculations for the alloy composition x = 0.5 only, but the same behavior holds for the other alloy compositions examined in this work. As can be noted from this figure, the three partial pair correlation functions show the same overall aspect which reminds us of that of pure a-Si [18]. It shows that only the two first-neighbor shells are well defined (first- and second-nearest neighbors, respectively). The first two corresponding peaks are separated by a null minimum. The thirdneighbor shell peak, at ~4.5 , appears as a shoulder at the right hand side of the second-neighbor shell peak. Beyond 5 , the correlation between pairs of atoms becomes relatively weaker and the medium seems to be quasi-continuous. The two-phase a-Si1 − xGex network leads to practically the same pair correlation functions as those obtained for structures with homogeneous random distribution of Ge-atoms. We arrive now to the analysis of the short-range topological and chemical disorders in our two-phase models for a-Si1 − xGex alloys via two structural parameters such as the average bond length and the average coordination number. The computed values are shown graphically as functions of the alloy composition in Figs. 5 and 6, respectively. For comparison, the same figures show the results obtained for a-Si1 − xGex networks with homogeneous random distribution of Ge atoms. An overall observation of Fig. 5 does not show noticeable differences between the two examined configurations. In both configurations the bond lengths depend on the alloy composition. The total bond length increases linearly with the alloy composition and follows, in both models, the Vegard's law given by [30]:    r = ð1−xÞ rSi−Si + x rGe−Ge

0

4

8

12

r(A) Fig. 4. Partial pair correlation functions, gSi\Si, gSi\Ge and gGe\Ge, computed for the twophase model (continuous line) and the homogeneous model (dashed line) for a-Si1 − xGex alloy with x ≈ 0.5.

2623

ð14Þ

  where r Si−Si = 2:38 Å and r Ge−Ge = 2:47 Å are the computed average bond lengths in nanoporous paracrystalline models for pure a-Si and aGe, respectively. The behaviors of the Si\Si, Si\Ge and Ge\Ge bond lengths, however, cannot be described by linear laws. A detailed observation of Fig. 5 shows that the presence of compositional

2624

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2,49

2,47

average bond length (A)

2,45

2,43

2,41

2,39

2,37

2,35

0

0,1

0,2

0,3

0,4

0,5

0,6 0

0,1

0,2

0,3

0,4

0,5

0,6

alloy composition, x

alloy composition, x

Fig. 5. Bond lengths within the first coordination shell computed for the homogeneous one-phase a-Si1 − xGex structures (left panel) and the heterogeneous two-phase a-Si1 − xGex networks (right panel). Circles, squares, triangles and diamonds indicate the total, Si\Si, Si\Ge and Ge\Ge bond lengths, respectively. The lines are drawn as guides to the eyes.

4,0 total

3,5

total

Ge-Si

Ge-Si

coordination number

3,0 Si-Si

2,5

2,5 Si-Si

2,0 1,5

2,0

1,0 Si-Ge

1,5

0,5 0,0

0

0,2 0,4 0,6

Si-Ge

1,0

0,5

Ge-Ge

Ge-Ge

0,0 0

0,1

0,2

0,3

0,4

alloy composition, x

0,5

0,6

0

0,1

0,2

0,3

0,4

0,5

0,6

alloy composition, x

Fig. 6. Coordination numbers within the first coordination shell computed for homogeneous one-phase a-Si1 − xGex structures (left panel) and heterogeneous two-phase a-Si1 − xGex networks (right panel). Insets: Comparison between calculated Ge\Ge coordination numbers (filled diamonds) and experimental data (empty diamonds) from Chapman et al. [3]. The lines are drawn as guides to the eyes.

R.B. Brahim, A. Chehaidar / Journal of Non-Crystalline Solids 357 (2011) 2620–2625

heterogeneities in a-Si1 − xGex alloys manifests itself by a less important variation of the Ge\Ge bond length with the alloy composition. The comparison between our theoretical results and the experimental data of Chapman et al. [3] shows that the two-phase model reproduces better the measured Ge\Ge bond lengths in a-Si1 − xGex alloys than the homogeneous one-phase model. As can be noted from Fig. 6, the distinction between the heterogeneous two-phase model and the homogeneous one-phase structure for a-Si1 − xGex alloys through the coordination number is not so obvious since the behavior of this structural parameter shows a small difference between the two examined configurations. The comparison between our theoretical results and the experimental data of Chapman et al. [3], shown in the insets of this figure, shows that the two-phase structural model reproduces in the same way as the homogeneous model and even slightly better the measured coordination numbers ZGe\Ge. 5. Conclusion We have examined the structural properties of several atomistic models for a-Si1 − xGex alloys with alloy composition ranging from 0.1 to 0.5. These structures have been generated by substituting Ge atoms for Si atoms in nanoporous paracrystalline model of a-Si. Two types of models have been constructed: homogeneous one-phase model, and heterogeneous two-phase model. The short-range topological and chemical disorders in the a-Si1 − xGex alloys are found to be practically insensitive to the presence of compositional nanoheterogeneities in these alloys. This is not the case however for the SAXS which is found to enhance with increasing the compositional-nanoheterogeneity size. From our results one might conclude that the two-phase model is the best representation of the structure of a-Si1 − xGex alloy. The simulation of the ASAXS data for this model should be considered in the near future to make comparison with experiment.

2625

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