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1 This article was downloaded by:[ebscohost EJS Content Distribution] On: 19 March 2008 Access Details: [subscription number ] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Philosophical Magazine First published in 1798 Publication details, including instructions for authors and subscription information: Structural study of supercooled liquid silicon T. H. Kim a ; A. I. Goldman b ; K. F. Kelton a a Department of Physics, Washington University, St. Louis, Missouri 63130, USA b Ames Laboratory USDOE and Department of Physics and Astronomy, Iowa State University, Ames, Iowa, 50011, USA Online Publication Date: 01 January 2008 To cite this Article: Kim, T. H., Goldman, A. I. and Kelton, K. F. (2008) 'Structural study of supercooled liquid silicon', Philosophical Magazine, 88:2, To link to this article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Philosophical Magazine, Vol. 88, No. 2, 11 January 2008, Structural study of supercooled liquid silicon T. H. KIM*y, A. I. GOLDMANz and K. F. KELTONy ydepartment of Physics, Washington University, St. Louis, Missouri 63130, USA zames Laboratory USDOE and Department of Physics and Astronomy, Iowa State University, Ames, Iowa, 50011, USA (Received 3 June 2007; in final form 6 November 2007) For many years, theoretical studies using model and ab initio potentials have predicated the existence of a liquid/liquid phase transition in silicon, based on a continuous change of the liquid A5 structure to A4. In contrast, we report here a quantitative analysis of data from high-energy X-ray diffraction measurements of containerlessly-processed supercooled liquid silicon that demonstrates that the fractions of regions with A5 and A4 order instead remain essentially constant with supercooling, but that the coherence length of the A5 order increases. 1. Introduction In addition to the impressive practical applications of crystalline Si, the liquid phase is of significant fundamental interest, being a member of a small group of elemental liquids (Si, Ge, Sb, Bi and Ga) that share similar thermodynamics [1]. For several decades, the structure of supercooled liquid Si and its transition from a metallic dense-packed structure at high temperatures to a semiconducting open network at lower temperatures has remained a topic of intense interest. A first-order amorphous liquid (AL) phase transition near 1450 K, originally proposed by Spaepen and Turnbull [2], was supported by pulsed laser heating [3] and ion implantation [4] experiments and molecular dynamics (MD) simulations [5]. However, more recent MD simulations using the well-established Stillinger Weber (SW) potential [6 9] and ab initio potentials [10] predicted that this was actually a liquid liquid phase transition from a more-dense liquid to a less-dense liquid. Based on the MD simulations made using the SW potential, a decrease in the first-shell coordination number (CN) was expected during supercooling, which is in conflict with recent experiments showing a constant CN [11]. The evolution of the structure factor, S(q), and pair correlation function, g(r), derived from those scattering data, however, indicted that there were changes in the atomic structure beyond the first coordination shell. The nature of that ordering was unclear. *Corresponding author. thkim@physics.wustl.edu Philosophical Magazine ISSN print/issn online ß 2008 Taylor & Francis DOI: /

3 172 T. H. Kim et al. 2. Structural analyses Here, we examine the local structures of liquid Si obtained from a reverse Monte Carlo (RMC) [12 15] treatment of the experimentally determined g(r) in terms of their bond-angle distributions and by a combined Honeycutt Andersen (HA) index [16 18] and bond orientational order (BOO) parameter [19, 20] analysis. These studies demonstrate an increase in the coherence length of the white-tin (A5) structure that is characteristic of liquid silicon instead of the expected ordering of the liquid toward the diamond cubic (A4) structure that is characteristic of the crystal phase. No evidence for a liquid/liquid phase transition is therefore observed at the accessible temperatures. Further, these results indicate that liquid Si evolves towards a structure with a local order that is not characteristic of the crystal phase. This behaviour is reminiscent of the recently discovered development of increased local icosahedral short-range order in many simple elemental and alloy metallic liquids [21 23] Pair correlations and RMC fits The experimental data used as input for the RMC analysis were derived from time-resolved in-situ high-energy (125 kev) X-ray diffraction measurements on supercooled liquid silicon in a high-vacuum environment without a container using the beamline electrostatic levitation (BESL) technique. The BESL technique, the experimental procedures followed and the methods used to compute S(q) and g(r) from the experimental data have been discussed elsewhere [11, 24]. Structural models for the liquid as a function of temperature were constructed from g(r) by RMC simulations using 5000 atoms that were placed in a cube of side L ¼ A with periodic boundary conditions. The large number of atoms reduces the impact of truncation errors [13]. Experimental values for the number density, 0, were used in the simulation: 0 ¼ (atom/a ) at 1815 K, at 1584 K and at 1382 K [25]. For these three temperatures, the simulations were started with a variety of initial configurations to ensure reproducibility. Atom movements during the simulation were excluded if the resulting distances between atoms were less than 2.1 A in g(r). The final configurations were described in terms of their bond-angle distributions, Honeycutt Andersen indices [17, 18] and bond orientational order parameters [19, 20]. Figure 1a shows the X-ray structure factors, S(q), computed from the diffraction data for the electrostatically levitated liquid from above the liquidus temperature to approximately 316 K below the liquidus temperature. The evolution of a doublepeaked structure near 2.5 and 3.5 A 1 with increasing supercooling is clearly evident, primarily due to the growth of a shoulder on the high-q side of the first peak. The development of this double peak signals a change in the atomic structure of the liquid. Figure 1b shows g(r) computed from S(q) using gðrþ ¼1 þ 2 Z 1 q½sðqþ 1ŠsinðqrÞdq, ð1þ 4r 0 0

4 Structural study of supercooled liquid silicon 173 Figure 1. (a) The experimental structure factor, S(q), for supercooled liquid silicon during cooling (solid lines). (b) The pair correlation function for the equilibrium (1815 K) and supercooled (1584 K and 1382 K) liquids (solid lines). The hollow circles are the results of a reverse Monte Carlo fit to the g(r) data. The thick solid lines in (a) show the S(q) values calculated from the Fourier transform of the fit to g(r). A residual experimental background in the BESL experiment causes the agreement with the experimental S(q) to be worse than for g(r). where 0 is the average number density. Like the coordination number [11], the nearest-neighbour distance in the liquid, r 1, remains approximately constant with decreasing temperature. However, there are changes in the higher-order peaks, indicating that the observed evolution of S(q) arises from changes in the medium-range order. The solid lines in figure 1b are the results from a reverse Monte-Carlo fit to the g(r) calculated from equation (1). The second peak in g(r) is composed of two sub-peaks at approximately 3.45 and 4.0 A, in agreement with previous reports by Ansell et al. and Watanabe et al. [26, 27]. The fit to the stable liquid data is similar to that obtained by Petkov [14], and to the ab-initio results obtained by Jakse et al. [28]. The final atomic configurations obtained from the fits to the experimental data starting from different initial configurations were very similar, giving confidence that the results were robust and could be analyzed to obtain insight into the local structures that develop with supercooling.

5 174 T. H. Kim et al. r (A) Analysis of RMC results N th nearest neighbor atom Figure 2. Average distances from a central atom to its first eight neighbours for the RMC ensembles obtained for the equilibrium ¼ 1815 K (diamond) and supercooled ¼ 1584 K (circle) and 1382 K (square) liquids. The bar shows the typical standard deviation for the distances Average distances of first eight neighbours. Figure 2 shows the average distances measured from a central atom to its first eight neighbours for the RMC ensembles obtained for the equilibrium and supercooled liquids. The amorphous structures are characterized by a range of distances (see representative standard deviation in the figure) as expected from the broad peaks in g(r). The first six neighbours belong to the first shell of g(r) and the seventh and eighth neighbours belong to the second peak. The average distances decrease only slightly during supercooling, consistent with the observed small increases in the density [25] and indicating that the local structure of liquid Si remains essentially constant. That these data are identical to those obtained from a first principles MD calculation [10] demonstrate that the RMC ensembles obtained from the experimental data are good representations of the liquid structures and can be analyzed in more depth to better understand the evolution in short-range order with supercooling Bond angle distribution. The distributions of bond angles for Si, the angle between the bonds connecting a central atom to two neighbouring atoms within the nearest-neighbour distance, r 1, in the atomic configurations obtained by RMC, are shown for three temperatures in figure 3. All of the bond angle distributions have maxima at approximately 58 and 92. This is strikingly similar to distributions obtained from previous MD simulations [29] and supports the validity of our RMC structures. Viewing the structure in terms of dominant clusters, the bond angle with the highest density for a diamond cubic structure (A4) is 109 ; for a white-tin cluster (A5) the prominent angles are 74,94 and 149. No peaks near 109 are observed in the RMC structures, suggesting that the short-range order is not dominated by A4 order. However, little evidence from this method is given for A5 order either, except perhaps the very small increase with supercooling of the bond angle distribution near 92 (the arrow indicated in figure 3). Since the bond distribution analysis is not

6 Structural study of supercooled liquid silicon 175 Frequency Number of pairs Angle ( ) Figure 3. The bond angle distribution for liquid Si during cooling. The hollow squares represent the equilibrium liquid data (1815 K), the hollow circles represent the supercooled data (1584 K) and the filled triangle for the supercooled liquid (1382 K). A small increase in the intensity of the bond angle distribution near 92 is indicated by the arrow HA index Figure 4. An HA index analysis of the liquid configuration obtained from a reverse Monte Carlo fit to the g(r) data. The hollow bars indicate the 1815 K data, the hatched bars are for the 1584 K data and the cross bars are for the 1382 K data. Inset: HA index analysis; the hollow bars indicate the results for an A4 cluster and the filled bars for an A5 cluster (all clusters contained 19 atoms). particularly sensitive to the local structure in the liquid, however, other techniques such as the Honeycutt Andersen index and bond orientational order analyses were used to provide better measures of the local configurations Honeycutt Andersen index analysis. Figure 4 shows the results of an HA index analysis of the atomic configuration obtained from the simulation. In this technique all pairs of atoms in the cluster are taken, in turn, as a root pair and the local structure around this pair is described by four indices. These indices denote the

7 176 T. H. Kim et al. following: (1) to which peak in g(r) the pair belongs; (2) how many atoms are counted as nearest-neighbours of the root pair; (3) the number of pairs of these nearest-neighbours that are close enough to form a bond; (4) a classification for clusters with identical sets of the first three indices but with different bond arrangements. A more complete discussion is provided elsewhere [17, 18]. In the HA analysis, a leading index of 1 refers to 1st nearest neighbours and a leading index of 2 to 2nd nearest neighbours. In order to relate the indices to structural motifs we point out that some indices are more prominent for particular structures. For example, the 2211, 2101, 1421 and 2441 indices are characteristic of a bulk fcc structure; hcp structures contain these indices, but the 1422 and 2331 are also prominent. Pertinent to the liquid data discussed here, the 1101, 1201, 2101 and 2311 indices are prominent for an A5 cluster and only the 1001 and 2101 indices are prominent for an A4 cluster (see inset to figure 4). Note that, the 2101 pairs are present in A5 and A4 types of local order so this index is not an instructive indicator of the local structure. Based on these considerations, the analysis results shown in figure 4 indicate that the A5 local structure is dominant in the supercooled liquid, as expected. However, the 1001 and 2101 indices are also prominent, suggesting a significant presence of the A4 structure as well. The relative number of pairs for the 1001 and 1101 HA indices suggests that the local order of the liquid is composed of a mixture of approximately 60% A5-type and 40% A4-type clusters and remains constant during supercooling. The almost constant magnitude of all indices, except 1201, 2101 and 2211, suggests that changes in the nearest-neighbour configuration with supercooling are insignificant. However, a measurable change in the 2211 index with supercooling indicates that A5 type ordering beyond nearest neighbour distances is expected. In addition, the 1201 index is present when we consider the 19 atom A5 cluster including the 1st, 2nd and 3rd nearest neighbours. On supercooling, the count of atom pairs with a leading HA index of 2, corresponding to root pairs in the second peaks of g(r), increases significantly more than those with leading HA index of 1, indicating that changes in structure are occurring beyond the first coordinate shell. These simulation results are reproducible even with different initial configurations (note the small error bars). In summary, the HA analysis of structures from RMC simulation indicates that there is an extension of A5 type order beyond the first shell and no transformation from A5 to A4 type ordering of liquid Si with supercooling Bond orientational order parameter analysis. To test this proposal, a bond orientational order parameter analysis was applied, which provides additional information concerning the changes that occur with increased supercooling. A set of order parameters {Q lm ð! r Þ¼Y lm ðð! r Þ, ð! r ÞÞ}, expressed in terms of the spherical harmonics {Y lm (, )}, are associated with the orientation of atomic pairs. To account for equivalent structures that are oriented differently, the rotationally invariant combination is used: Q l ¼ 4 X l 1=2 Q 2l þ 1 m¼ l lm 2 : ð2þ

8 Structural study of supercooled liquid silicon 177 (a) Q l (b) Q l l l l Figure 5. BOO analysis of liquid configurations, considering various numbers of nearest neighbour shells: (a) 1st; (b) 1st and 2nd nearest neighbours. The hollow bars indicate the 1815 K data, the hatched bars are for the 1584 K data and the cross bars are for the 1382 K data. Insets: (a) 1st nearest neighbours of A5 cluster (6 atoms, filled bars) and diamond cubic cluster (4 atoms, hollow bars). Left inset of (b): various nearest shells in the A5 cluster. The hollow bars represent BOO indices for 1st neighbours of A5 cluster and the filled bars for 1st and 2nd neighbours. Right inset of (b): enlarged figure for l ¼ 7 as a function of supercooling. 7 Different structures are characterized by Q l s of different intensity [19, 20]. For example, cubic symmetry is identified by having a first non-zero Q 4 value, whereas for icosahedral structures Q 6 is the first non-zero value. However, for isotropic arrangement, Q l s are always zero. The BOO results for supercooled liquid Si are shown in figure 5. The BOO was computed from the 1st nearest neighbour distribution in the RMC structure. The cut-off distance was chosen to be the 1st minimum in the radial distribution function, RDF ¼ 4r 2 0 gr ðþ, the same distance used to compute the coordination number. The BOO parameters calculated from the RMC structures obtained from the experimental data at 1815 K, 1584 K and 1382 K, using only the nearest neighbours, are given in figure 5a. All of the Q l s are non-zero, indicating orientational ordering in the liquid. As shown by the inset of figure 5a the BOO parameters for an A5 cluster and an A4 cluster assuming nearest neighbours are significantly different. The almost constant intensity of the BOO for the experimental structures, then, indicates that the symmetry of the nearest neighbour structure does not change significantly with supercooling (figure 5a). This is in agreement with the HA index analysis discussed earlier and is consistent with our proposal related to a constant mixture of A4 and A5 types order. However, that analysis did indicate ordering in the next nearest neighbour environment

9 178 T. H. Kim et al. with supercooling. When 1st and 2nd nearest neighbours in the RMC structures are considered, the BOO results (figure 5b) agree with this, showing a change in the BOO intensity between the two temperatures, especially for Q 7 (the left inset of figure 5b, as indicated by the arrow), the parameter that is most different in A4 and A5 structure (inset to figure 5a). The cut-off distance was chosen to be the 2nd minimum in g(r). The fact that a decrease in l ¼ 7 is shown in figure 5b is consistent with A5 type order evolution in higher coordination shells. Taken together, then these observations are consistent with an increasing length scale (coherence length) of the A5 ordering in the liquid, as opposed to a transition towards an A4 order, in agreement with the experimentally observed evolution of the second peak in g(r). 3. Conclusion In conclusion, information about the evolution of the local structure of liquid silicon with supercooling has been determined from high-energy X-ray diffraction studies of electrostatically levitated samples. The atomic structures determined from the measured S(q) using a reverse Monte Carlo simulation were analyzed in terms of the Honeycutt Anderson and bond orientational order parameter techniques. All methods suggest that the supercooled liquid silicon maintains the A5 structure (with some A4), with the coherence length of the A5 order increasing with supercooling. The ordering of the A5 structure is consistent with previous experimental observations, such as a constant coordination number, constant nearest-neighbour distances and the evolution of the first peak in S(q) and of the second peak in g(r) with supercooling. These observations are in opposition to theoretical predictions of a liquid liquid structural phase transition toward a diamond cubic structure (characteristic of crystal Si) in supercooled liquid Si, Further, they indicate that barring the intervention of phase transitions such as crystallization, liquids will order towards a preferred structure that is characteristic of the liquid state and the nature of the atomic bonding. Acknowledgements The work at Washington University was partially supported by the NSF under grant DMR and by NASA under Contract No. NNM04AA016. MUCAT and the Ames Laboratory are supported by the U.S. Department of Energy, Office of Science under Contract No. W-7405-Eng-82. References [1] H. Tanaka, Phys. Rev. B (2002). [2] F. Spaepen and D. Turnbull, AIP Conf. Proc (1978).

10 Structural study of supercooled liquid silicon 179 [3] M.O. Thompson, G.J. Galvin and J.W. Mayer, Phys. Rev. Lett (1984). [4] E.P. Donovan, F. Spaepen, D. Turnbull, et al., J. Appl. Phys (1985). [5] W.D. Luedtke and U. Landman, Phys. Rev. B (1988). [6] C.A. Angell and S.S. Borick, J. Phys. Condensed Matter (1999). [7] S. Sastry and C.A. Angell, Nature Mater (2003). [8] S.S. Ashwin, U.V. Waghmare and S. Sastry, Phys. Rev. Lett (2004). [9] C.A. Angell, J. Shao, M. Grabow, Non-equilibrium Phenomena in Supercooled Fluids, Glasses and Amorphous Materials (World Scientific, Singapore, 1996), p. 50. [10] T. Morishita, Phys. Rev. Lett (2006). [11] T.H. Kim, G.W. Lee, B. Sieve, et al., Phys. Rev. Lett (2005). [12] R.L. McGreevy, J. Phys. Condensed Matter 3 F9 (1991). [13] R.L. McGreevy, M.A. Howe, D.A. Keen, et al., Inst. Phys. Conf. Ser (1990). [14] V. Petkov and G. Yunchov, J. Phys. Condensed Matter (1994). [15] D.A. Keen and R.L. McGreevy, Nature (1990). [16] H. Kimura, M. Watanabe, K. Izumi, et al., Appl. Phys. Lett (2001). [17] J.D. Honeycutt and H.C. Andersen, J. Phys. Chem (1987). [18] A.S. Clarke and H. Jonsson, Phys. Rev. E (1993). [19] P. Steinhardt, D.R. Nelson and M. Ronchetti, Phys. Rev. Lett (1981). [20] P.J. Steinhardt, D.R. Nelson and M. Ronchetti, Phys. Rev. B (1983). [21] K.F. Kelton, G.W. Lee, A.K. Gangopadhyay, et al., Phys. Rev. Lett (2003). [22] G.W. Lee, A.K. Gangopadhyay, K.F. Kelton, et al., Phys. Rev. Lett (2004). [23] T.H. Kim and K.F. Kelton, J. Chem. Phys (2007). [24] A.K. Gangopadhyay, G.W. Lee, K.F. Kelton, et al., Rev. Scient. Instrum (2005). [25] W. Rhim and K. Ohsaka, J. Cryst. Growth (2000). [26] S. Ansell, S. Krishnanz, J.J. Feltenz, et al., J. Phys. Condensed Matter 10 L73 (1998). [27] M. Watanabe, K. Higuchi, A. Mizuno, et al., J. Cryst. Growth (2006). [28] N. Jakse, L. Hennet, D.L. Price, et al., Appl. Phys. Lett (2003). [29] C.S. Liu, Z.G. Zhu, J. Xia, et al., Phys. Rev. B (1999).