On the origin and the structure of the first sharp diffraction peak of amorphous silicon
OOn the origin and structure of the first sharp diffraction peak of amorphous silicon
Devilal Dahal, Hiroka Warren, and Parthapratim Biswas ∗ Department of Physics and Astronomy, The University of Southern Mississippi, Hattiesburg, Mississippi 39406, USA
The structure of the first sharp diffraction peak (FSDP) of amorphous silicon ( a -Si) near 2 ˚A − is addressedwith particular emphasis on the position, intensity, and width of the diffraction curve. By studying a number ofcontinuous random network (CRN) models of a -Si, it is shown that the position and the intensity of the FSDPare primarily determined by radial atomic correlations in the amorphous network on the length scale of 15 ˚A. Ashell-by-shell analysis of the contribution from different radial shells reveals that key contributions to the FSDPoriginate from the second and fourth radial shells in the network, which are accompanied by a backgroundcontribution from the first shell and small residual corrections from the distant radial shells. The results fromnumerical calculations are complemented by a phenomenological discussion of the relationship between thepeaks in the structure factor in the wavevector space and the reduced pair-correlation function in the real space.An approximate functional relation between the position of the FSDP and the average radial distance of Si atomsin the second radial shell in the network is derived, which is corroborated by numerical calculations. Keywords: Amorphous silicon, Pair-correlation function, Static structure factor, First sharp diffraction peak
I. INTRODUCTION
Professor David Drabold has contributed significantly inthe field of amorphous materials. It is therefore an opportunemoment to contribute to his Festschrift on a topic which isvery close to his heart. The first sharp diffraction peak (FSDP)is a distinct feature of many noncrystalline solids, which arecharacterized by the presence of a peak in the low wavevectorregion (1–2 ˚A − ) of the structure factor of the solids. Al-though the origin of the FSDP in many multinary glasses isnot yet fully understood from an atomistic point of view, ithas been shown that the FSDP is primarily associated with thepresence of the short-range and medium-range order, whichentail voids, chemical ordering, large ring structures, localtopology, and atomic correlations between constituent atomsin the amorphous environment of the solids [1–5].The FSDP is ubiquitous in many disordered condensed-phase systems. Numerous experimental [1, 2, 6–8] and the-oretical [3, 9, 10] studies have reported the (near) universalpresence of the FSDP in glasses and liquids/melts. In glasses,the origin of the FSDP can be largely attributed to the pres-ence of layered structures [11], interstitial voids [3–5], chem-ical disorder [4], and large ring structures [8] in the networks,which constitute a real-space description of atomic correla-tions on the nanometer length scale. Elliott [3] has shown thatthe FSDP in binary glasses can be interpreted as a prepeakin the concentration-concentration structure factor, which iscaused by the presence of the chemical ordering between con-stituent atoms in the networks. Likewise, the interstitial voidshave been found to play an important role in the formation ofthe FSDP in tetrahedral amorphous semiconductors [4], e.g., a -Si. On the other hand, Susman et al. [8] have reported that inbinary AX glasses, the A–A and A–X correlations within theextended ring structures can give rise to the FSDP. Busse andNagel [11] have suggested that the existence of the FSDP in g -As Se can be ascribed to the inter-layer atomic correlations ∗ To whom correspondence should be addressed: [email protected] in the glassy network. Experimental studies on GeSe andGeSe glasses by Armand et al. [12] have indicated that theGe–Ge atomic correlation on the length scale of 6-7 ˚A is theprimary cause of the FSDP, which is supported by molecular-dynamics studies by Vashishta et al. [9].The behavior of the FSDP in covalent glasses often showsan anomalous dependence with respect to temperature [2, 13],pressure [13, 14], and composition [6, 15]. Following thewell-known Debye-Waller [16] behavior, one may assumethat the peaks in the structure factor should decrease withthe increase of the temperature of the system. However, thefirst (sharp) diffraction peak of many glassy systems has beenfound to remain either invariant or become more intense andnarrower at high temperature [2, 17]. A notable exception isvitreous silica ( v -SiO ), which does not follow the behaviorstated above. The intensity of the FSDP of v -SiO has beenobserved to decrease with increasing temperature, due to thethermally induced motion of the atoms and the associated dif-fused scattering [18], leading to the broadening of the firstpeak [13]. Likewise, the position and the width of the FSDPhave been observed to vary with the pressure or density ofthe glasses [14, 17]. Neutron diffraction [19] and molecular-dynamics studies of densified v -SiO [20] have indicated thatthe intensity and the width of the FSDP can change with thedensity of the samples/models. These changes can be at-tributed to the frustration induced by the reduction of Si–O–Si bond angles and the changes in the Si–Si and O–O atomiccorrelations on the length scale of 4–10 ˚A when the system isdensified. A similar conclusion can be made for GeO glass,when the glass is densified [21]. The addition of extrinsicatoms in glassy networks has been also found to affect thefirst sharp diffraction peak. Lee and Elliott [15] have notedthat the inclusion of extrinsic atoms in v -SiO can change thechemical ordering of the interstitial voids in the glassy net-work, which can alter the shape/width of the FSDP.While the great majority of earlier studies mostly examinedthe origin and the behavior of the FSDP in borate, chalco-genide, oxide, and silicate glasses [5, 22–24], there exist onlya few studies [3, 4, 25] that address the structure of the FSDPin tetrahedral amorphous semiconductors, such as a -Si and a r X i v : . [ c ond - m a t . d i s - nn ] J a n a -Ge. Elliott and coworkers [4, 25] have addressed the prob-lem at length, but their studies are primarily focused on theorigin of the extended-range oscillations (ERO) in a -Si. Theresults from their studies, which are based on the (Fourier)inversion of experimental structure-factor data of Fortner andLannin [26] and highly defective a -Si models of Holender andMorgan [27], suggest that the ERO arise from the preferentialpropagation of second-neighbor correlations in the network,which in turn can significantly affect the intensity of the FSDPup to a radial length scale of 20 ˚A. However, no systematicanalysis of the results with respect to the size of models isprovided and, thus, in the absence of direct numerical evi-dence, it is not clear to what extent the intensity of the FSDPis truly affected by atomic correlations originating from radialdistances beyond 15 ˚A.The key purpose of this paper is to provide a systematicstudy of the structure of the FSDP, with an emphasis on theposition, intensity, and width of the peak, with the size ofthe models. In addition, the origin of the FSDP in a -Si isaddressed by obtaining a quantitative estimate of the contri-bution of atomic pair correlations from different radial shellsand their effect on the intensity and position of the FSDP. Therelationship between the peaks in the structure factor and itsreal-space counterpart, the reduced pair-correlation function,is addressed, and an approximate functional relation betweenthe position of the FSDP in a -Si and the radial distance of theatoms in the second radial shell of the amorphous network isobtained. Throughout this paper, we shall use the term FSDPto refer to the first peak of the structure factor of a -Si at Q =1.9–2 ˚A − in discussing our results. Likewise, the term prin-cipal peak will be used to indicate the second peak at Q = 3.6˚A − . For amorphous silicon, this terminology has been usedpreviously by others [4, 25], and it is consistent with the factthat the peak at Q is indeed the first peak of S ( Q ) and thatit is reasonably sharp and strong with a value of the intensity S ( Q ) , which is about 67% of the intensity of the principalpeak. A further justification of the use of the terminology willbe evident later from our discussion of the results in sectionIIIA.The rest of the paper is planned as follows. Section II pro-vides a brief description of the simulation method for produc-ing atomistic models of a -Si via the modified Wooten-Winer-Weaire (WWW) [28, 29] method, the calculation of the ra-dial pair-correlation function, and the structure factor for thesemodels. This is followed by results and discussion in sectionIII, with an emphasis on the origin and the structure of theFSDP. The conclusions of this study are presented in sectionIV. II. MODELS AND METHODS
For the purpose of generating atomistic models of a -Si,we have employed the well-known WWW method. The de-tails of the method can be found in Refs. [28, 29]. Here, wehave used the modified version of the method, developed byBarkema and Mousseau (BM) [29]. In the modified WWWapproach, one starts with a random configuration that consists of N atoms in a cubic supercell of length L . The volume ofthe supercell is chosen in such a way that the mass density ofthe model corresponds to about 2.28 g.cm − , as observed in a -Si samples produced in laboratories [30, 31]. Initially, fol-lowing the BM ansatz, the nearest neighbors of each atom areso assigned that a tetravalent network is formed [32]. This isachieved by choosing a suitable nearest-neighbor cutoff dis-tance, up to 3 ˚A, between Si atoms. The resulting tetravalentnetwork is then used as the starting point of the WWW bond-switching algorithm. New configurations are generated byintroducing a series of WWW bond switches, which largelypreserve the tetravalent coordination of the network and theenergy of the system is minimized using Monte Carlo (MC)simulations. The acceptance or rejection of a proposed MCmove is determined via the Metropolis algorithm [33] at agiven temperature. Here, the energy difference between twoconfigurations is calculated locally by using the Keating po-tential [34], which employs an atomic-index-based nearest-neighbor list of the tetravalent network during MC simula-tions. In addition, the total energy of the entire system isrelaxed from time to time using the Stillinger-Weber poten-tial [35]. Finally, the configurations obtained from the mod-ified WWW method were relaxed using the first-principlesdensity-functional code S IESTA [36]. For the models with 216atoms to 3000 atoms, a full self-consistent-field calculation,using the generalized-gradient approximation (GGA) [37] anda set of double-zeta basis functions, was carried out. Theremaining models of size from 4096 atoms to 6000 atomswere treated using the non-self-consistent Harris-functionalapproach [38] with a single-zeta basis set in the local den-sity approximation (LDA) [39]. To conduct configurationalaveraging of simulated data, we have generated 10 models foreach size starting with different random configurations usingindependent runs.Once the atomistic models are generated, the calculationof the structure factor proceeds by computing the reducedpair-correlation function. The latter is defined as G ( r ) =4 πrn [ g ( r ) − , where g ( r ) and n are the pair-correlationfunction and the average number density of a model, respec-tively. Assuming that the distribution of the atoms in a dis-ordered network is isotropic and homogeneous, the structurefactor, S ( Q ) , can be written as, S ( Q ) = 1 + 1 Q (cid:90) ∞ G ( r ) sin( Qr ) dr ≈ Q (cid:90) R c G ( r ) sin( Qr ) dr, (1)where R c is the length of the half of the cubic simulation cell.The conventional periodic boundary conditions are used tominimize surface effects and to calculate the pair-correlationfunction in Eq. (1). III. RESULTS AND DISCUSSION
Equation (1) suggests that the shape of the FSDP can befully determined via the Fourier (sine) transformation of thereduced pair-correlation function G ( r ) , provided that G ( r ) → as r → R c . Since the shape of the FSDP is primarily de-termined by the structure factor in the vicinity of Q ≈ − , it is apparent that one requires sufficiently large mod-els of a -Si, in order to satisfy the condition above, for an ac-curate determination of the FSDP. To this end, we first val-idate the structural models of a -Si, obtained from the mod-ified WWW method. Since the latter is a well-establishedmethod, we restrict ourselves to the pair-correlation function(PCF), the bond-angle distribution (BD), and the coordina-tion number (CN) of Si atoms in the network. It has beenshown elsewhere [40] that the knowledge of the PCF, the BD,and the CN of the atoms are sufficient to establish whethera structural model can produce the correct electronic and vi-brational properties of a -Si or not. The full structure factorand the normalized bond-angle distribution, obtained from aset of 3000-atom models of a -Si, are plotted in Figs. 1 and2, respectively. For the purpose of configurational averag-ing, the results were averaged over 10 independent modelsof an identical size. The simulated values of S ( Q ) in Fig. 1can be seen to agree well with the corresponding experimen-tal data reported in Ref. [31]. Likewise, the full width at halfmaximum (FWHM) of the bond-angle distribution in Fig. 2,about 21.4 ◦ , matches with the observed value of 18 ◦ –24 ◦ ob-tained from the Raman “optic peak” measurements [41]. TheFWHM of the bond-angle distribution for the WWW modelsis also found to be consistent with those obtained from high-quality molecular-dynamics simulations [42, 43], and data-driven information-based approaches [40, 44], developed inrecent years. A further characterization of the models is pos-sible by examining the statistics of the CN of Si atoms, thedihedral-angle distribution, and the presence of various irre-ducible rings in the amorphous structures. However, sincethe WWW models have been extensively studied and vali-dated in the literature, we will not linger over the validationissue and get back to the central topic of this paper by listingthe coordination-number statistics of the atoms and some keystructural properties of the WWW models in Table I. The cor-responding results for the DFT-relaxed models are providedin Table II. A. Characterization, origin, and the structure of the FSDP inamorphous silicon
Figure 3 shows the structure factor of a -Si obtained fromfour different models, of size from 216 atoms to 3000 atoms,and experiments [30]. As before, the simulation data are ob-tained by averaging over 10 independent models for each size,whereas the experimental data refer to as-implanted samplesof a -Si in Ref. [30]. An examination of Fig. 3 leads to the fol-lowing observations. Firstly, it is apparent that the 216-atommodel shows a marked deviation from the experimental datanear the FSDP, indicating noticeable finite-size effects orig-inated from small models of linear size of about 16 ˚A. Bycontrast, the larger models, consisting of 1000 to 3000 atoms,have produced the peak intensity more accurately. Secondly,all the models consistently underestimate the position of the Q (Å -1 ) S ( Q ) FIG. 1. The structure factor of a -Si obtained from simulations andexperiments. The simulated data are from 3000-atom WWW modelsof density 2.28 g.cm − , whereas the experimental data correspond tothose from Ref. [31]. The simulated data are obtained by averagingover 10 models from independent runs.
70 80 90 100 110 120 130 140 150
Bond angle ( θ) N o r m a li ze d bond - a ng l e d i s t r i bu ti on FIG. 2. The normalized bond-angle distribution for a -Si, obtainedfor 3000-atom WWW models. The full width at half maximum(FWHM) corresponds to a value of 21.4 ◦ . The distribution is ob-tained by averaging over 10 independent models.TABLE I. Structural properties of a -Si models before DFT relax-ation. The number of i -fold-coordinated atoms (in percent) in thenetwork is indicated as n i . Bond lengths and bond angles/widthsare expressed in ˚A and degree, respectively. The results are obtainedby averaging over 10 configurations using a nearest-neighbor cutoffvalue of 2.8 ˚A.Size Bond angle & width Coordination number Bond length N (cid:104) θ (cid:105) ∆ θ n n n (cid:104) r (cid:105)
216 109.25 9.11 100 0 0 2.35300 109.25 9.32 100 0 0 2.35512 109.26 9.41 100 0 0 2.351000 109.27 9.16 100 0 0 2.352000 109.27 9.31 99.95 0 0.05 2.353000 109.26 9.39 99.94 0 0.06 2.354096 109.26 9.26 99.95 0 0.05 2.355000 109.27 9.31 99.97 0 0.03 2.356000 109.26 9.39 99.96 0 0.04 2.35 experimental FSDP [30] at Q =1.99 ˚A − , by an amount of TABLE II. Structural properties of DFT-relaxed models of a -Si. Thetotal number of i -fold-coordinated atoms (in percent) present in therelaxed networks is indicated as n i . Average bong lengths and bondangles/widths are expressed in ˚A and degree, respectively. Aster-isks indicate the use of single-zeta basis functions and the non-self-consistent Harris-functional approximation for relaxation of largemodels.Model size Bond angle & width Coordination number Bond length N (cid:104) θ (cid:105) ∆ θ n n n (cid:104) r (cid:105)
216 109.11 10.14 100 0 0 2.36300 109.15 10.22 100 0 0 2.36512 109.13 10.45 100 0 0 2.361000 109.15 10.14 100 0 0 2.362000 109.14 10.3 99.95 0 0.05 2.363000 109.13 10.4 99.96 0.01 0.03 2.364096 ∗ ∗ ∗ − . One can surmise a number of possible reasons forthis discrepancy. These include the inadequacy of the clas-sical potentials, the uncertainty of the actual density of the a -Si sample(s) used in experiments, and a possible sample-to-sample dependence of the experimental results. The last pointcan be appreciated by noting that the experimental value of Q , for as-implanted samples of a -Si, reported in Refs. [26],[30], and [31] differ from each other by about 0.07 ˚A − (seeFig. 9). Finally, a first-principles total-energy relaxation ofthe models, using the density-functional code S IESTA [36],somewhat remedies this issue at the expense of the reductionof the peak intensity. This is illustrated in Fig. 4, where wehave plotted both the reduced pair-correlation function (see in-set) and the configurational-average structure factor from ten3000-atom models before and after total-energy relaxation.The increase of the peak height of G ( r ) upon relaxation isnot surprising in view of the fact that first-principles relax-ations minimized the total energy of the system by reducingthe bond-length disorder at the expense of a minor increaseof the bond-angle disorder. The latter is reflected in the root-mean-square (RMS) width, ∆ θ , of the bond-angle distributionbefore and after relaxation in Tables I and II, respectively.By contrast, the shape of the structure factor near the FSDPremains more or less the same after relaxation, except for asmall shift of the FSDP toward the higher values of Q .Having addressed the overall shape of the structure factorand the FSDP for a number of models of varying sizes, wenow examine the origin of the FSDP in terms of the real-space structure of a -Si networks. While it is well-understoodthat the FSDP in a -Si arises from the medium-range order inthe network, which entails a length scale of a few to severalangstroms, a quantitative characterization of the contributionfrom different radial shells is still missing in the literature. Weaddress this aspect of the problem by examining the role of ra-dial atomic correlations in forming the FSDP, via the Fouriertransform of the reduced PCF, and provide a quantitative mea-sure of the contributions that originate from the increasingly Q (Å -1 ) S ( Q ) Exp.N = 216N = 1000N = 2000N = 3000
FIG. 3. The structure factor of a -Si in the vicinity of the FSDPfrom simulations and experiments. Experimental data (•) correspondto as-implanted samples from Ref. [30], whereas simulated data referto 216-atom ( (cid:7) ), 1000-atom ( (cid:4) ), 2000-atom ( (cid:78) ), and 3000-atom ( (cid:4) )unrelaxed WWW models. Q (Å -1 ) S ( Q ) UnrelaxedDFT-relaxed r (Å) G (r) FIG. 4. Effects of approximate first-principles relaxations on the po-sition and the intensity of the FSDP of a -Si for a 3000-atom modelbefore ( (cid:4) ) and after ( • ) relaxation. A small shift of the diffractionpeak toward higher values of Q is accompanied by a slight reductionof the peak intensity in the relaxed model. The corresponding re-duced pair-correlation functions near the first peak are shown in theinset. distant radial shells in the amorphous environment of silicon.This can be achieved by writing, S ( Q ) = 1 + F ( Q ) = 1 + n (cid:88) i =1 F i ( Q ; R (cid:48) i , R (cid:48) i +1 ) , where, F i ( Q ; R (cid:48) i , R (cid:48) i +1 ) = 1 Q (cid:90) R (cid:48) i +1 R (cid:48) i G ( r ) sin( Qr ) dr. (2)In Eq. (2), F i ( Q ; R (cid:48) i , R (cid:48) i +1 ) is the contribution to F ( Q ) fromthe reduced PCF, G ( r ) , at distances between R (cid:48) i and R (cid:48) i +1 .The contribution from a given radial shell can be obtained bya suitable choice of R (cid:48) i and R (cid:48) i +1 , where R (cid:48) i +1 > R (cid:48) i , and anappropriate set { R (cid:48) , . . . , R (cid:48) n } covers the entire radial (integra-tion) range to obtain the full F ( Q ) . For example, a choice of −2−1 0 1 2 3 4 5 6 2 3 4 5 6 7 8 9 10 11 12 13 14 15 G (r) r (Å)3000−atom model FIG. 5. The reduced pair-correlation function, G ( r ) , of a -Si ob-tained from configurational averaging of ten 3000-atom WWW mod-els. The presence of the first six radial shells, which extend up to adistance of ≈
12 ˚A, is highlighted in different colors. R (cid:48) = 0 ˚A and R (cid:48) = 2.8 ˚A yields F ( Q ; R (cid:48) , R (cid:48) ) , and R (cid:48) =2.8 ˚A and R (cid:48) = 4.9 ˚A provides F ( Q ; R (cid:48) , R (cid:48) ) . The origin ofthe FSDP and the principal peak can be studied by computingvarious F i ( Q ) in the vicinity of 2 ˚A − and 3.6 ˚A − , respec-tively. The appropriate values of R (cid:48) i for different radial shellscan be obtained by inspecting the reduced PCF of a -Si. Thisis illustrated in Fig. 5, by plotting the configurational-average G ( r ) obtained from a set of ten 3000-atom models. We shouldemphasize that the radial shells correspond to the radial re-gions between two neighboring minima in the reduced PCF.Except for the first radial shell, the radial regions, defined by apair of consecutive minima in G ( r ) , are not necessarily identi-cal to the corresponding atomic-coordination shells due to theoverlap of the atomic distribution from different coordinationshells.Figure 6 shows the contribution to F ( Q ) in the vicinity of2 ˚A − from the first six radial shells. The plots for differentradial shells are indicated by the corresponding shell color asdepicted in Fig. 5. It is evident that the chief contribution tothe FSDP comes from F ( Q ) , which is followed by F ( Q ) and F ( Q ) in the descending order of magnitude. F ( Q ) and F ( Q ) play a crucial role in determining both the intensityand the position of the FSDP, while F ( Q ) and F ( Q ) con-tribute very little to none. By contrast, F ( Q ) monotonicallychanges in the vicinity of the FSDP and thus contributes tothe intensity (and the shape) of the FSDP near Q to somedegree but does not play any noticeable role in determiningthe position of Q . It is therefore apparent that the positionof the FSDP in a -Si is primarily determined by the informa-tion from the second radial shell, followed by the fourth andsixth radial shells, whereas the rest of the distant radial shellsprovide small perturbative corrections. The enumeration ofthe radial shell-by-shell contribution to F ( Q ) is a significantresult to our knowledge, which cannot be quantified from aphenomenological understanding of the Fourier transform of G ( r ) in Eq. (1). A similar analysis reveals that the contri-bution to the principal peak at 3.6 ˚A − mostly arises from F ( Q ) , F ( Q ) , F ( Q ) , and F ( Q ) , in the decreasing order of Q (Å -1 ) -1-0.500.511.5 F i ( Q ) F (Q)F (Q)F (Q)F (Q)F (Q)F (Q)F(Q) F i (Q) = F(Q; R i , R i +1 ) FIG. 6. The contribution to the FSDP, F i ( Q ) , near Q , originatingfrom the first six radial shells and the total F ( Q ) (blue). The resultscorrespond to 3000-atom WWW models, and are averaged over tenconfigurations. The color of the plots corresponds to the color of theradial shells in Fig. 5. magnitude. Once again, F ( Q ) is found to provide a positivebut monotonically decreasing contribution with increasing Q in the vicinity of the principal peak. Thus, the peak at 3.6˚A − is principally contributed by the first four radial shells inthe reduced PCF. This observation amply justifies the use ofthe term ‘principal peak’ to describe the peak at 3.6 ˚A − inthe structure factor of a -Si. Figure 7 shows the results for theprincipal peak using the same color code as in Fig. 5. B. Relation between peaks in
S(Q) and
G(r)
The results presented in the preceding section on the basisof the partitioning of F ( Q ) reveal that the information fromthe second and fourth radial shells largely determine the struc-ture, i.e., the position, intensity, and width, of the FSDP in a -Si. We now provide a physical interpretation of the numer-ical results and demonstrate that the emergence of the firsttwo peaks in S ( Q ) , near 2 ˚A − and 3.6 ˚A − , respectively, canbe deduced simply from the knowledge of the reduced PCFand the behavior of the integral, involving the sinc(x) (i.e., sin x/x ) function, which defines the structure factor. Notingthat the structure factor can be written as, S ( Q ) = 1 + F ( Q ) = 1 + 1 Q (cid:90) R c G ( r ) sin( Q r ) dr = 1 + (cid:90) R c r G ( r ) (cid:20) sin( Q r ) Qr (cid:21) dr, (3)it is elementary that the peaks in F ( Q ) (and hence S ( Q ) )should appear approximately for those values of Qr for whichboth sin( Qr ) /Qr and rG ( r ) are maximum. Here, the r val-ues in Qr are given by the maxima of rG ( r ) . Since themaxima of sin( Qr ) /Qr and sin( Qr ) are very close to eachother [45] for Qr > , and the maxima of G ( r ) and rG ( r ) practically coincide, one may use the maxima of sin( Qr ) and G ( r ) in determining the approximate location of the first twopeaks in S ( Q ) . This implies Qr must satisfy, sin( Qr ) = 1, Q (Å -1 ) -0.500.51 F i ( Q ) F i (Q) = F(Q; R i , R i +1 ) F (Q)F (Q)F (Q)F (Q)F (Q)F (Q)F(Q) FIG. 7. The contribution, F i ( Q ) , to the principal peak at Q = 3.6˚A − from the first six radial shells of the reduced pair-correlationfunction. The total F ( Q ) is shown in blue color for comparison.The results are obtained via configurational averaging of data fromten 3000-atom models. or Qr = (4 m + 1) π/ , where m =0, 1, 2, . . . etc. Since thefirst two maxima of G ( r ) are given by r ≈ r ≈ m = 0 does not admit a physical solu-tion [45], the first major contribution to the F ( Q ) comes fromthe second radial shell for r = 3.8 ˚A and m = 1 . This gives, Q = 5 π/ (2 × r ) = 2.07 ˚A − . Likewise, the next contri-bution comes from, for m = 2 , the fourth radial shell with apeak at r ≈ G ( r ) . This yields, Q = 9 π/ (2 × r ) = 1.95 ˚A − . A similar analysis shows that the principal peak( Q ) gets its share from the first radial shell, for m = 1 , at Q = 5 π/ (2 × r ) = 3.34 ˚A − , which is followed by the sec-ond radial shell, for m = 2 , at Q = 9 π/ (2 × r ) = 3.72 ˚A − ,the fourth radial shell, for m = 4 , at Q = 17 π/ (2 × r ) =3.69 ˚A − , and the third radial shell, for m = 3 and r = 5 . ,at Q = 13 π/ (2 × r ) = 3.57 ˚A − . The exact position ofa peak in S ( Q ) is determined by the sum of the contributionfrom the relevant radial shells, which introduce a minor devi-ation from the individual estimate above due to the approxi-mate nature of our calculations. Table III presents a summaryof the results obtained from the reasoning above. The esti-mated position of the peaks in F i ( Q ) , for i =1 to 6, is listedin the Table. The first column, shown in light gray shading,corresponds to the maxima ( r i ) of G ( r ) up to a radial distanceof 11 ˚A, whereas the second row, indicated by dark gray cells,lists the values of Q r = (4 m + 1) π/ for m =1 to 6. The re-maining six rows, between columns 1 and 8, indicate the peakpositions in F i ( Q ) that are obtained by dividing the Qr val-ues by the corresponding r i value from the first column. Theestimated positions of the FSDP and the principal peak for anumber of combination of ( r i , m ) are shown in Table III bygreen and red colors, respectively.Conversely, assuming that the FSDP is located at Q ≈ − , one arrives at the conclusion, by dint of our logic,that the contribution to the FSDP should come from r =(4 m + 1) π/ (2 × Q ) = π/ , π/ , π/ , and π/ , etc.,for m = 0 to 3. The first value of r , for m = 0, does not pro-vide a physical solution but the remaining values at 3.93 ˚A, TABLE III. Estimated values of the peak positions, Q , in F i ( Q ) ,obtained from Q r = (4 m + 1) π/ (dark gray cells in the secondrow for m =1 to 6) and the maxima of G ( r ) (first column) in ˚A. Thepositions of the FSDP and the principal peak (PP) in F i s are indicatedby green and red colors, respectively. The radial shells that contributeto the FSDP and the PP can be directly read off the first column.Maxima 1 2 3 4 5 6 ← m of G ( r ) ← Qr F F F F F F G ( r ) (cf. Fig. 5).A similar analysis can be done for the principal peak. Theargument presented here suffices to explain why the informa-tion from the distant radial shells, for r ≥ ˚A, cannot con-tribute significantly in the formation of the FSDP. At large ra-dial distances, when the reduced PCF rapidly vanishes and theconcomitant numerical noises in G ( r ) become increasinglystronger, sin( Qr ) /Qr cannot find, or sample, suitable valuesof r with a large G ( r ) , for Q values near the FSDP, to satisfythe condition above. This leads to small F i ( Q ) for the distantradial shells. We have verified that the analysis presented hereis consistent with the results from numerical calculations of F i ( Q ) . R c (Å) S ( Q )
216 300 512 1000 2000 3000 4096 5000 6000S(Q ) = 1.52 S(Q ) = 1.37S(Q ) = 1.55 DFT-relaxedUnrelaxedExpt 1Expt 2Expt 3
FIG. 8. The dependence of the intensity of the FSDP, S ( Q ) , withthe radial cutoff distance, R c , for a number of models of differentsizes, as indicated in the plot. The experimental values of S ( Q ) reported in the literature are shown as horizontal dashed lines: 1) S ( Q ) =1.52 from Ref. [30]; 2) S ( Q ) =1.37 from Ref. [26]; and 3) S ( Q ) =1.55 from Ref. [31]. The results and discussion presented so far indicate that theradial information from the reduced PCF of up to a lengthscale of 15 ˚A plays a significant role in the formation of theFSDP. To further establish this point, we now conduct a sys-tematic study of the structure of the FSDP in terms of the R c (Å) F W H M ( Å - )
216 300 512 1000 2000 3000 4096 5000 6000
Expt 1 = 0.54Expt 2 = 0.57
DFT-relaxedUnrelaxedExpt 1Expt 2Expt 3
Expt 3 = 0.77
FIG. 9. The full width at half maximum (FWHM) of the FSDP at Q for a number of models before ( (cid:4) ) and after ( • ) DFT relaxations. Thehorizontal dashed lines indicate the experimental values of 0.54 ˚A − (green), 0.57 ˚A − (black), and 0.77 ˚A − (indigo) for as-implantedsamples of a -Si from Refs. [30], [31], and [26], respectively. intensity and the width of the peak. The variation of thepeak intensity with the size of the models is studied by plot-ting the value of S ( Q ) against R c for a number of DFT-relaxed/unrelaxed models, consisting of 216 atoms to 6000atoms. Since R c is given by the half of the linear size of themodels, Fig. 8 essentially shows the dependence of S ( Q ) onthe radial pair correlations up to a distance of R c , through theFourier transform of G ( r ) . It is clear from the plots (in Fig. 8)that the intensity of the FSDP for both the relaxed and unre-laxed models varies considerably until R c increases to a valueof the order of 14 ˚A. This roughly translates into a modelof size about 1000 atoms. For even larger values of R c , thepeak intensity is more or less converged to 1.48 for the unre-laxed models but considerable deviations exist for the valueof DFT-relaxed models from the experimental value of S ( Q ) of 1.52 in Ref. [30]. The deviation of the peak intensity fromthe experimental value for small models of a -Si can be readilyunderstood. Since G ( r ) carries considerable real-space infor-mation up to a radial distance of 15 ˚A, possibly 20 ˚A for verylarge models, small models with R c values less than 15 ˚Acannot accurately produce the peak position using Eq. (1). Onthe other hand, the peak intensity for the DFT-relaxed mod-els deviates noticeably (about 0.2–12%) from their unrelaxedcounterpart and the experimental value for as-implanted sam-ples in Refs. [30], [26], and [31]. This apparent deviation forthe bigger models is not particularly unusual and it can beattributed, at least partly, to: 1) the use of approximate total-energy calculations in the relaxation of large models, via thenon-self-consistent Harris-functional approach using minimalsingle-zeta basis functions; 2) the intrinsic difficulties asso-ciated with quantum-mechanical relaxations of large models;and 3) the sample dependence of experimental results, show-ing a considerable difference in the value of S ( Q ) for as-implanted samples in Fig. 8, which is as high as 0.18 fromone experiment to another. Thus, the results obtained in thisstudy are well within the range of the experimental values re-ported in the literature [26, 30, 31].The full width at half maximum, or FWHM, of the FSDP )0.40.50.60.70.8 F W H M ( Å - ) FWHM = 0.54 S ( Q ) = . S ( Q ) = . r XY = -0.9 N = 216N = 300N = 512N = 1000N = 2000N = 3000N = 4096N = 5000N = 6000LS fitFWHM = 0.77
FIG. 10. A scattered plot showing the presence a clear correlationbetween the FWHM and S ( Q ) of the FSDP for a number of modelsof varying system sizes. The solid (red) line corresponds to the linearleast-square (LS) fit of the data, whereas r XY = − . indicates thePearson correlation coefficient for the data sets. The horizontal andvertical dotted lines indicate the experimental values of FWHM and S ( Q ) , respectively, obtained for as-implanted samples of a -Si. for different models is plotted against R c in Fig. 9. A some-what high value of the FWHM for the large DFT-relaxed mod-els is a consequence of the reduction of the peak intensity.As the intensity of the peak reduces, the FWHM increasesslightly due to the widening of the diffraction plot away fromthe peak. An inspection of Figs. 8 and 9 appears to suggestthat the values of FWHM and S ( Q ) are somewhat correlatedwith each other. In particular, amorphous-silicon models ex-hibiting smaller values of S ( Q ) (in Fig. 8) tend to producesomewhat larger values of FWHM in Fig 9, irrespective of thesize of the models and DFT relaxation. This is apparent inFig. 10, where FWHM and S ( Q ) values for all configura-tions and sizes are presented in the form of a scattered plot.A simple analysis of FWHM and S ( Q ) data by computingthe Pearson correlation coefficient, r XY , confirms the sugges-tion that FWHM and S ( Q ) values are indeed linearly cor-related with each other and have a correlation coefficient of r XY = − . . The linear least-square (LS) fit of the dataare also shown in Fig. 10 by a solid (red) line. The greatmajority of the FWHM and S ( Q ) values in Fig. 10 can beseen to have clustered along the straight line within a rectan-gular region bounded by the experimental values of FWHMand S ( Q ) , from 0.54 to 0.77 ˚A − and 1.37 to 1.55, respec-tively. Likewise, the dependence of the position of the FSDPwith R c for the unrelaxed and DFT-relaxed models is illus-trated in Fig.11. For the unrelaxed models, Q is observedto converge near 1.96 ˚A − , whereas the corresponding valuefor the DFT-relaxed models hovers around 1.97 ˚A. In both thecases, Q is within the range of the experimental values, from1.95 ˚A − to 2.02 ˚A − , shown in Fig. 11.In summary, a systematic study of a -Si models, consistingof 216 to 6000 atoms, firmly establishes that the structure ofthe FSDP in a -Si is mostly determined by radial pair correla-tions up to a distance of 15 ˚A, as far as the size of the largestmodels employed in this study is concerned. Further, the ma-jor contribution to the FSDP arises from the second and fourth R c (Å) Q
216 300 512 1000 2000 3000 4096 5000 6000Q = 1.99 Q = 1.95Q = 2.02 DFT-relaxedUnrelaxedExpt 1Expt 2Expt 3
FIG. 11. The dependence of the position of the FSDP, Q , withthe size of the models before ( (cid:4) ) and after (•) DFT relaxations. Thehorizontal lines correspond to the experimental value of Q for as-implanted samples of a -Si from Refs. [30] (green), [31] (black), and[26] (indigo). radial shells, along with small residual contributions from thedistant radial shells at a distance of up to 15 ˚A. C. The FSDP and the radial shell structures of a -Si Earlier, in sections IIIB and IIIC, we have demonstratedthat the position of the FSDP, Q , is primarily determined by F ( Q ) and, to a lesser extent, F ( Q ) . This leads to a possibil-ity of the existence of a simple functional relationship between Q and a suitable length scale in the real space involving theradial atomic correlations in the network. In this section, wewill show that an approximate relationship between Q andthe average radial distance, (cid:104) R (cid:105) , of the atoms in the second(radial) shell does exist. Below, we first provide a rationalebehind the origin of this relationship, which is subsequentlycorroborated by results from direct numerical calculations.The first hint that an approximate relationship may exist fol-lows from the behavior of Q with the (mass) density, ρ , of themodels. In Fig. 12, we have plotted the variation of Q against ρ for a -Si. For this purpose, the density of a set of 3000-atommodels is varied, within the range from 2.12 g.cm − to 2.32g.cm − , by scaling the length of the cubic simulation cell andthe position of the atoms therein. This involves a tacit as-sumption that for a small variation of the density, by about ± a -Si do not include any extended defects and voidsin the network, the scaling assumption is reasonably correctand suitable to produce models with a small variation of thedensity. Figure 12 presents the results from our calculations,which show a linear relationship between Q and the density, ρ , of the models. This linear variation of Q with ρ is not par-ticularly unique to a -Si; a similar behavior has been observedexperimentally by Inamura et al. [14, 17] for densified silica.The results from Fig. 12 and the experimental data from Density (g.cm -3 ) Q ( Å - ) Q = 2.02Q = 1.99Q = 1.95 UnrelaxedDFT-relaxedExpt 1Expt 2Expt 3
FIG. 12. The variation of the peak position ( Q ) for 3000-atommodels of a -Si with its mass density before ( (cid:4) ) and after (•) DFTrelaxations. The value of Q has been observed to vary linearly withthe density of the model. The experimental values of Q (horizon-tal dashed lines) correspond to as-implanted samples of density 2.28g.cm − from Refs. [31] (black), [30] (green), and [26] (indigo). Refs. [14] and [17] suggest that Q can vary approximatelylinearly with the average density, ρ , of the models/samples.Since ρ is inversely proportional to the cubic power of thesimulation cell size ( L ) for a given number of atoms, Q alsovaries as /L when the density is varied by rescaling thevolume. Thus, for homogeneous and isotropic models withno significant variation of the local density, which the WWWmodels satisfy in the absence of extended defects and voids, itis reasonable to assume that Q ∝ /r ij , where r ij ( ρ ) is thedistance between any two atoms in the network, at sites i and j , of average density ρ . In view of our earlier observation thatthe position of the FSDP is largely determined by F ( Q ) (seeFig. 6), one may posit that r ij values between R and R in G ( r ) mostly affect the peak position at Q . These consid-erations lead to the suggestion that by substituting r ij by itsaverage value of (cid:104) r ij (cid:105) = (cid:104) R (cid:105) for the atoms in the second ra-dial shell, Q (cid:104) R (cid:105) should remain constant, on average, upondensity variations via volume rescaling. Likewise, one can in-voke the same reasoning and may expect Q (cid:104) R (cid:105) should bealso constant but only approximately, due to the limited roleand contribution of the atoms in the fourth radial shell in de-termining the position of Q .The efficacy of our argument can be verified by resultsfrom direct numerical calculations. A plot of Q (cid:104) R (cid:105) (and Q (cid:104) R (cid:105) ) versus the average density ρ in Fig. 13 (and Fig. 14)indeed confirms our prediction. It may be noted that theobserved (absolute) deviation, ∆ , of Q (cid:104) R (cid:105) values in thedensity range 2.15–2.3 g.cm − in Fig. 13 is of the order of ± . σ , where σ is the (largest) standard deviation obtainedby averaging results from 10 independent models for eachdensity. By contrast, the corresponding deviation for Q (cid:104) R (cid:105) in Fig. 14 is found to be more than two standard deviation, asindicated in the plot. The large deviation of Q (cid:104) R (cid:105) values isnot unexpected in view of the small contribution of F ( Q ) (tothe FSDP) that originates from the fourth radial shell. Thus,the results from Fig. 13 lead to the conclusion that Q is ap- Density (g.cm -3 ) Q 〈 R 〉 FIG. 13. The relation between Q and (cid:104) R (cid:105) of the atoms in thesecond radial shell. A constant value of Q (cid:104) R (cid:105) with respect to thedensity of a -Si models indicates that Q is approximately propor-tional to the inverse of (cid:104) R (cid:105) . The horizontal black line indicates theaverage value of Q (cid:104) R (cid:105) within the density range shown in the plot. Density (g.cm -3 ) Q 〈 R 〉 ∆ ≈ 2.5σ ∆ FIG. 14. The dependence of Q and (cid:104) R (cid:105) for the atoms in the fourthradial shell of a -Si in the density range from 2.15 to 2.3 g.cm − . Thelarge deviation of Q (cid:104) R (cid:105) values, indicated by ∆ ≈ . σ , from aconstant value suggests that no simple relationship between Q and (cid:104) R (cid:105) exists. proximately proportional to the inverse of the average cubicpower of the radial distance, (cid:104) R (cid:105) , of the atoms in the sec-ond radial shell in a -Si. It goes without saying that the useof (cid:104) R (cid:105) , instead of (cid:104) R (cid:105) , does not change the conclusion ofour work, as the difference between these two values is foundto be about 1.92–2.1 ˚A , for the mass density in the range of2.15 to 2.3 g.cm − , which simply shifts the plot (in Fig. 13)vertically downward by a constant amount.We end this section by making a comment on the possiblerole of distant radial atomic correlations, or extended-rangeoscillations (ERO), in G ( r ) on the FSDP, based on our prelim-inary results from 6000-atom models. Although the presenceof ERO in ultra-large models of a -Si beyond 15 ˚A is an undis-puted fact [25], a direct determination of the effect of the EROon the FSDP in a -Si is highly nontrivial due to the presenceof intrinsic noises in G ( r ) at large radial distances. Numeri-cal calculations using 6000-atom models of a -Si indicate that Q (Å -1 ) S ( Q ) r i = 15 År i = 16 År i = 17 År i = 18 ÅR C = L/2 FIG. 15. The structure factor, S ( Q ) , in the vicinity of the FSDPfrom 6000-atom a -Si models. The change in S ( Q ) due to varyingradial cutoff distances, r i , is found to be less than one standard devi-ation ( ∆ S ( Q ) ≈ . σ for r i = 15 ˚A, and 0.46 σ for r i = 18 ˚A).The standard deviation, σ , is obtained from using the maximal radialcutoff L/ , which is given by 24.85 ˚A. The results correspond to theaverage values of S ( Q ) obtained from 10 configurations. only a minute fraction of the total intensity of the FSDP resultsfrom the radial region beyond 15 ˚A. These calculations do notinclude any possible artifacts that may arise from the noises in G ( r ) at large distances. The observed deviation in the peak in-tensity, due to the truncation of the radial distance at 15 ˚A andat higher values, is found to be about 1–2%, which is less thanone standard deviation ( σ ) associated with S ( Q ) , obtainedfrom using the maximal radial cutoff distance R c (= L/ , asfar as the results from 6000-atom models are concerned. Fig-ure 15 shows the variation of the intensity near the FSDP forfive different cutoff values from 15 ˚A to 18 ˚A and 24.85 ˚A. Itis apparent that the changes in S ( Q ) near Q are very smallas the radial cutoff value increases from 15 ˚A to 18 ˚A. Thesesmall changes in the intensity values are readily reflected inFig. 16, where the fractional errors, with respect to S ( Q, R c ) ,associated with the calculation of S ( Q, r i ) are plotted against Q for r i = 15 ˚A to 18 ˚A. Thus, as far as the present studyand the maximum size of the models are concerned, the EROdo not appear to contribute much to the FSDP. However, anaccurate study of the ERO in a -Si would require high-qualityultra-large models, consisting of several tens of thousands ofatoms, and a suitable prescription to handle noises in G ( r ) atlarge distances. These and some related issues concerning theorigin of the ERO in a -Si and their possible role in S ( Q ) willbe addressed in a future communication. IV. CONCLUSIONS
In this paper, we have studied the origin and structure of theFSDP of a -Si with an emphasis on the position, intensity, andwidth of the diffraction peak. The study leads to the follow-ing results: 1) By partitioning the contribution of the reducedPCF to the FSDP, which originates from the Fourier trans-form of radial atomic correlations in the real space, a quanti-tative measure of the contribution to the FSDP from different0 Q (Å -1 ) -0.03-0.02-0.0100.010.020.03 ∆ S / S ( Q , R c ) r i = 15 År i = 16 År i = 17 År i = 18 Å FIG. 16. The fractional error associated with the calculation ofthe structure factor in the vicinity of the FSDP with a varying radialcutoff distance, r i , from 15 ˚A to 18 ˚A. ∆ S ( Q ) is the absolute errorand R c (=24.85 ˚A) is the half-length of the cubic simulation cellfor 6000-atom models. The error due to the truncation of the radialdistance at r i can be seen to be around 1–2%, which is well withinone standard deviation of S ( Q ) (see Fig. 15). radial shells is obtained. The results show that the positionof the FSDP in a -Si is principally determined by atomic paircorrelations in the second, fourth, and sixth radial shells, inthe descending order of importance, supplemented by smallresidual contributions from beyond the sixth radial shell; 2) Aconvergence study of the position, intensity, and width of theFSDP, using a set of models of size from 216 to 6000 atoms,suggests that the minimum size of the models must be at least1000 atoms or more in order for the results to be free fromfinite-size effects. This approximately translates into a radiallength of 14 ˚A, which is consistent with the results obtained from the radial-shell analysis of the reduced PCF; 3) A theo-retical basis for the results obtained from numerical calcula-tions is presented by examining the relationship between thepeaks in the structure factor and the reduced PCF. Contrary tothe common assumption that the peaks in the structure factorand the reduced PCF are not directly related to each other, wehave shown explicitly that the knowledge of the reduced PCFalone is sufficient not only to determine the approximate po-sition of the FSDP and the principal peak but also the relevantradial regions that are primarily responsible for the emergenceof these peaks in the structure factor, and vice versa; 4) Thestudy leads to an approximate relation between the positionof the FSDP and the average radial distance of the atoms inthe second radial shell of a -Si networks. For homogeneousand isotropic models of a -Si with no significant variation ofthe local density, it has been shown that the position of theFSDP is inversely proportional to the cubic power of the av-erage radial distance of the atoms in the second radial shell.The result is justified by providing a phenomenological ex-planation – based on experimental and computational studiesof the variation of the FSDP with the average density of a -Sisamples and models – which is subsequently confirmed by di-rect numerical calculations for a range of density from 2.15 to2.3 g.cm − . ACKNOWLEDGEMENTS