LETTER pubs.acs.org/JPCL
A Single-Component Silicon Quasicrystal Jessica C. Johnston, Spencer Phippen, and Valeria Molinero* Department of Chemistry, University of Utah, Salt Lake City, Utah 84112-0850, United States ABSTRACT: Quasicrystals are structures with long-range order and no translational periodicity. Monatomic quasicrystals were predicted for model potentials, but no single-component atomic quasicrystal of an actual element has been reported to date. A dodecagonal quasicrystal was recently predicted to form in bilayer water. Water and silicon present striking similarities in their phase behavior, raising the question of whether quasicrystals may occur in silicon. Here, we show, using molecular simulations, that a confined silicon bilayer forms a quasicrystal upon compression between smooth surfaces. The quasicrystal is stable in a narrow region of the phase diagram and forms spontaneously upon cooling the liquid bilayer in a wide range of pressures. Cooling the liquid between atomically detailed plates incommensurate with the quasicrystal leads to its spontaneous formation at 1 atm of lateral pressure. This suggests that the silicon quasicrystal could be obtained in experiments at room pressure by tuning the structure and interactions of the surfaces. SECTION: Nanoparticles and Nanostructures
T
he potential applications of nanoscale silicon in technology have made the study of low-dimensionality silicon structures a growing area of research. Morishita et al. reported the formation of a bilayer silicon crystal tiled by flat hexagonal rings in registry between the two layers,1 analogous to the structure of bilayer ice initially reported by Koga et al.2 Electronic structure calculations of Bai et al. using DFT under the PBE approximation have suggested that the flat hexagonal bilayer (H) silicon crystal is a quasi-2D semimetal, similar to graphene, with an indirect zero band gap.3 Using DFT-GGA ab initio calculations, Morishita et al. concluded that planar H bilayer silicon (without dangling bonds) is a semiconductor and the H corrugated structure (with dangling bonds) is metallic, and suggested that the silicon bilayers are candidate materials for ultrathin fieldeffect transitions.4 The analogies in the phase behavior of water and silicon extend beyond the formation of the H bilayer crystal. Although water and silicon are chemically unrelated, they form similar bulk structures and have related bulk phase diagrams.5,6 Among the similarities, at room pressure, water and silicon form tetrahedral crystals that are less dense than the liquids to which they melt, leading to their characteristic decrease in melting temperature with pressure. Water and silicon also present polyamorphism.7-9 Low-density amorphous ice and amorphous silicon are random tetrahedral networks, and their high-density glasses also have related structures.10 The polyamorphic transition in silicon is accompanied by a radical change in electronic properties, from semiconductor in the low-density amorphous phase to metal in the high-density one.8 The common feature to water and silicon, responsible for their similarities in their phase diagrams and structures, is the tetrahedral topology of their intermolecular interactions. It is the locally tetrahedral structures that they favor, r 2011 American Chemical Society
not the nature of the interactions (hydrogen bonds or covalent), that determine the structures and phase behavior of these substances.5 Quasicrystals are structures with long-range order but no translational periodicity.11,12 Most quasicrystals occur in metallic alloys, usually with multiple components and rarely with just two elements.13 Electron diffraction patterns observed in selected areas of agglomerates of thread-like crystalline silicon nanoparticles vapor-deposited on mica were interpreted in ref 14 as arising from a local dodecagonal quasicrystalline arrangement of crystalline silicon nanoclusters. In another publication by the same authors, that diffraction pattern was assigned, however, to tubule-like crystalline structures of silicon.15 A one-dimensional arrangement of crystalline stripes of silver about 1.5 nm high and of two widths (12.5 and 17.6 Å16) deposited on cleaved GaAs(110) has been characterized through scanning tunneling spectroscopy and shown to follow the “silver mean” quasi-periodic distribution.17 To date, however, there has not been a conclusive report of a single-element quasicrystal that is itself not composed of crystalline particles. The theoretical plausibility of quasicrystals made of tetravalent elements has been discussed in the literature, particularly in the context of crystalline phases of silicon that display local icosahedral ordering18,19 and models for the amorphous phases of silicon and germanium.20 No such tetrahedrally coordinated quasicrystal has yet been realized. In a recent communication, we predicted a dodecagonal quasicrystal and a crystal fully tiled by pentagons in the phase diagram of confined bilayer water.21 The water Received: December 17, 2010 Accepted: January 20, 2011 Published: February 02, 2011 384
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Table 1. Probability of Obtaining Each Structure by Spontaneous Cooling of Bilayer Liquid Silicon between the Smooth Platesa pL (GPa)
stable phase
H
W
0
H
1
1.22
H
1
1.83
H
1
2.44
H
1
3.05 3.35
H W
1 1
3.66
S
0.8
5.49
S
7.31
S
C
T
S
0.2 0.6
0.4 0.6
0.4
a
The structures formed are hexagonal (H), coffins (C), tic-tac-toe (T), and square (S) crystals and wheels (W) quasicrystal.
quasicrystal and novel bilayer pentagon-based ice polymorphs are denser than hexagonal bilayer ice.21 The quasicrystal was formed through lateral compression of the water bilayer without any templating from the confining surfaces, which were smooth. The average coordination number of the water molecules in the quasicrystal was 4.4. On the basis of the strong similarities in phase diagrams and structures for water and silicon, a natural question is whether silicon also makes a quasicrystal and a bilayer crystal fully tiled by pentagons. Answering this question is the focus of this Letter. We use molecular dynamics (MD) simulations with the Stillinger-Weber (SW) potential to investigate the phase behavior of two layers of silicon confined between two parallel smooth Lennard-Jones 9-3 (L-J-93) walls. The SW silicon model is best among empirical potentials in reproducing the experimental melting temperature Tm of the bulk diamond cubic crystal at 1 atm; it adequately reproduces the melting line of that crystal up to pressures of about 7 GPa, and it predicts a melting temperature for the Si136 guest-free silicon clathrate crystal in excellent agreement with experiments.22 The fidelity of the model in reproducing the liquid-crystal equilibrium of Si136 is significant for this work because pentagonal silicon rings predominate in that crystal which is exclusively tiled with dodecahedra and hexakaidecahedra. We first investigated which structures were spontaneously obtained upon quenching the high-temperature bilayer liquid silicon (L) at constant lateral pressures pL up to 7.31 GPa. The results of 45 independent quenching simulations are summarized in Table 1. Five distinct phases are produced through first-order transitions from the liquid state; only one of these, the hexagonal bilayer crystal (H, shown in panel 3 of Figure 1), was previously reported for the silicon bilayer.1,3 The remaining four phases encompass a novel crystal that we name squares (S) that consists of two layers of silicon in registry, each tiled by square rings (panel 9 of Figure 1), and three phases based on pentagons, recently found for bilayer water,21 the wheels (W) dodecagonal quasicrystal and the coffins (C) and tic-tac-toe (T) crystals. C, T, and H are four-coordinated crystals; S is five-coordinated. The average coordination in the quasicrystal is 4.4. We first focus on the crystal phases of bilayer silicon and their stability; then, we discuss the quasicrystal and its formation from the liquid phase. Figure 2 presents the phase diagram of the silicon bilayer, displaying the equilibrium coexistence lines between liquid (L) and the low-temperature phases. H and S
Figure 1. Structural evolution of bilayer silicon upon cooling. Panels 1-9 show the bilayer structures confined between the smooth LJ-93 plates at the temperature T and lateral pressure pL indicated in the labels (these state points are indicated in Figure 2). The upper layer is shown in blue and the lower layer in red. Bonds connect atoms within 3 Å. Cooling the liquid at 0.1 MPa produces H bilayer crystal, while quenching the liquid at 3.35 GPa leads to a first-order transition to the W quasicrystal. The S crystal is obtained by cooling at the highest pressure, 7.31 GPa. The lower row presents the diffraction patterns of the H crystal (6-fold rotational symmetry), the W quasicrystal (12-fold), and the S crystal (4-fold) at their melting temperatures at the corresponding pL.
are the stable crystals of bilayer silicon at low and high pressure, respectively, and display an interesting structural analogy to the corresponding stable crystal phases of bulk silicon. The structure of H looks like a two-dimensional (2D) projection of the diamond crystal. S looks like a quasi-2D projection of the β-tin phase of bulk silicon. Different from H, that is four-coordinated like the bulk diamond crystal, this “projection” entails a decrease in the coordination number from six in β-tin to five in the bilayer S crystal. The slopes of the melting curves of the stable crystals (given by (dT/dpL)D = ΔV/ΔS) have identical signs as their bulk counterparts, producing the same characteristic V-shaped phase diagram. C is a bilayer crystal with four-fold rotational symmetry exclusively tiled by five-member rings organized in the Cairo pentagonal tiling, dual of the 32.4.3.4 tiling. 87% of the rings that tile the T crystal are pentagons, and the two layers are not fully in registry. We refer the reader to ref 21 for the description of the structures of C and T and their diffraction patterns. Table 1 385
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the endeavor of finding optimum conditions to produce this silicon analogue of bilayer graphene worth pursuing. We now turn our attention to the silicon quasicrystal and its formation from the liquid phase. Quenching bilayer liquid silicon at 1.83 e pL e 3.35 GPa exclusively yields wheels (W), a quasicrystal with the same structure as that formed in bilayer water.21 The liquid to quasicrystal transition is sharp and can be reversed upon heating, with little hysteresis. The structure of the wheels quasicrystal is shown in panel 6 of Figure 1. W is composed of two layers of silicon in registry; 86% of the rings tiling each layer are pentagons. The diffraction pattern, also shown in Figure 1, reveals the 12-fold rotational symmetry of the quasicrystal. Each layer of W has the structure of the dodecagonal quasicrystal formed by the two-dimensional Lennard-Jones-Gaussian model of refs 28 and 29 that, like water and silicon, has two distinct length scales. W is the first quasicrystal predicted for silicon. Although the W quasicrystal is the preferentially formed phase for a wide range of pressures, W is mostly metastable with respect to H and S. We note, however, that the quasicrystal is the stable phase of bilayer silicon at pL = 3.35 GPa with Tm = 1646 ( 10 K, while Tm of H and S are 1626 ( 10 K. The liquid to quasicrystal transition appears to be first-order and is accompanied by small changes in energy and entropy. The enthalpy difference between the liquid and quasicrystal at the phase transition at 3.35 GPa is just 4.88 ( 0.02 kJ mol-1, about 1/3 of the thermal energy RT at that temperature. The change in entropy is concomitantly modest, 3.0 ( 0.2 J K-1 mol-1, about 15% of the ΔS of melting of bulk silicon.23 The ΔS between the liquid and quasicrystal at the H-L-W triple point is about half of the ΔS between L and H at similar conditions. We attribute the small ΔH and ΔS of melting of the quasicrystal to its intrinsic disorder and the significant ordering of the liquid as it approaches the coexistence temperature. This is apparent in the central row of Figure 1. The ordering involves (i) a steady increase in the fraction of atoms in registry between the two layers of the liquid, from 53% at 2080 K to 82% at 1656 K and (ii) the appearance, in the stable liquid, of the wheel motif (a central pentagon surrounded by four pentagons and one hexagon21) and the edge-sharing pentagonal prism (a bilayer cluster with three edge-sharing pentagons, part of the main motif of the C). The ubiquity of W and C motifs in the structure of the liquid at around 3 GPa is responsible for the low hysteresis observed for the L-W and L-C transition. The hollow squares in Figure 2 indicate the average nonequilibrium transition temperatures Tf upon cooling the liquid. The Tf of the liquid to quasicrystal transition are very close to the corresponding equilibrium temperatures Tm. This is not the case for the crystals (with the notable exception of the L to C transition at 3.6 GPa) for which there is a significant gap between Tf and Tm, commensurate with the difference in structures between liquid and crystals. The maximum degree of supercooling is observed for the crystallization of H, consistent with the lack of sizable H nuclei in the supercooled liquid (Figure 1). The crystallization of the S at 7.3 GPa, on the other hand, is favored by the presence of large S patches already in the stable liquid phase. An analysis of the structural transformation of the liquid upon cooling and to which extent it reflects the structure of the underlying crystals will be presented in a separate communication. We have shown that the similarities between the phase diagrams of water and silicon, and their differences, extend to the confined bilayers. Molecular simulations reveal the rich phase
Figure 2. Phase diagram of bilayer silicon confined between smooth plates. The filled circles indicate equilibria between the bilayer liquid (L) and wheels (W) quasicrystal in green, hexagonal (H) crystal in maroon, coffins (C) crystal in red, tic-tac-toe (T) crystal in orange, and squares (S) crystal in black. The empty squares show the average freezing temperatures in the cooling ramps, using the same color coding as the melting temperatures. Blue numbers 1-9 signal the T and pL of the corresponding structures in Figure 1. The W quasicrystal is the stable phase of bilayer silicon at pL = 3.35 GPa. Note that the normal melting point of the H bilayer is about 420 K higher than that for the bulk diamond silicon crystal8,23 (white circle).
shows that C and T are spontaneously obtained by cooling liquid silicon under pressure. These four-coordinated crystals tiled by pentagons are, however, always metastable phases of bilayer silicon, at least for the confining interfaces used in this study. As discussed in ref 21, C looks like a two-dimensional projection of bulk ice XII. There is no ice XII analogue for bulk silicon, however, as ice XII is one of the densest four-coordinated water crystals and silicon is known to favor an increase in coordination upon compression. Of the many bulk silicon crystals, only Si I (diamond cubic), III (BC8), IV, VIII, IX, and XII (R8) are fourcoordinated; except for Si I, they are all metastable.24 The same situation is observed for the silicon bilayer; H is the only stable four-coordinated crystal. The effect of pressure on the coordination of the crystals is different in bulk water and silicon, and the bilayers mirror that difference. Compression of bulk and bilayer water produces only four-coordinated crystals.21 For silicon, instead, the increase in pressure is accompanied by an increase in coordination of the stable phases, from four in diamond cubic to six in β-tin in bulk24 and from four in H to five in S in the bilayer. We recently demonstrated that H bilayer ice has an anomalously high melting temperature, up to 45 °C higher than the melting temperature of bulk ice.21,25 This is also the case for bilayer H silicon; the melting temperature of H at 0.1 MPa, 2080 K, is about 420 K above the Tm of the bulk silicon diamond crystal. The origin of this exceptionally high melting temperature is discussed in ref 25. The implication is that if H bilayer silicon could be deposited from the vapor phase, as recently reported for H bilayer ice in experiments,26,27 it would be stable up to very high temperatures, well above ambient temperature. That consideration, along with its promising electronic properties,3 makes 386
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behavior of the silicon bilayer, with stable crystals that are quasi2D analogues of 3D bulk phases and the formation of a quasicrystal that is mostly tiled by pentagons and has 12-fold rotational symmetry. Monatomic quasicrystals have been reported before in simulations,28-32 but never for a model parametrized to represent an actual element. The prediction of a single-component silicon quasicrystal is a first step toward the realization of a monatomic quasicrystal under laboratory conditions. The main challenge that remains for the synthesis of the silicon quasicrystal is the selection of appropriate confining surfaces that would favor its formation at ambient pressures. In the final paragraph of this Letter, we demonstrate that the silicon quasicrystal could be synthesized without the application of lateral pressure to the bilayer. Modifications of the silicon-surface attraction with the smooth LJ-93 surfaces do not significantly affect the phase behavior of the silicon bilayer; to shift the H-L-W triple point to negative pressures, the surface must have atomic structure. A principle to stabilize the quasicrystal at pL = 1 atm should not be based on direct templating of the W by the atoms of the surface (as there may be no surface with the appropriate structure) but rather on the surface favoring the formation of denser silicon structures without compression. We found that cooling bilayer liquid silicon confined between two triangular lattice plates with atomic spacing incommensurate with that in silicon ordered structures (2.8 Å in the plates and ∼2.45 Å in silicon) at pL = 1 atm results in a transition to the W quasicrystal for interactions of silicon with the plate atoms between 7 and 14 kJ/mol, to the H crystal for weaker attractions, and to the S crystal for stronger ones. Figure 3 shows that the silicon atoms of the dodecagonal quasicrystal do not align or follow the triangular pattern of the plates. Calculations of the melting temperatures of W, H, and S indicate that under these conditions, the quasicrystal is metastable with respect to either H or S, and its spontaneous formation is favored by a low barrier of nucleation from the liquid. More studies are needed to elucidate the principle by which the surfaces control the stability of the silicon bilayer. The encouraging results obtained with the triangular lattice plates suggest that appropriately chosen interfaces could be used to stabilize the silicon quasicrystal under conditions amenable for its preparation and the study of its electronic, optical, and mechanical properties in experiments.
Figure 3. The dodecagonal quasicrystal spontaneously forms between triangular lattice plates (TL) upon cooling bilayer liquid silicon at pL= 1 atm. (A) View of the silicon quasicrystal obtained by quenching liquid silicon between the TL for silicon-plate attraction ε = 10.1 kJ/mol. Silicon atoms are located at the vertices of the blue and red lines, for the upper and lower layer, respectively. The diffraction pattern is identical to that of the dodecagonal quasicrystal formed between the smooth LJ-93 plates. (B) Detail of the quasicrystal structure, showing the atoms of the lower plate (white balls). The silicon atoms are not aligned with the atoms of the triangular lattice plates.
’ METHODS Silicon was modeled with the Stillinger-Weber (SW) potential.33 The bilayers contained 1377 or 5508 silicon atoms (cell dimensions of about 7.5 and 15 nm, respectively) and were confined between smooth Lennard-Jones 9-3 walls with σ = 2.63 Å and ε = 4.04 kcal/mol at a fixed distance D = 7.4 Å or, when expicitly indicated, flat triangular lattice plates (TL) in registry (as in ref 25) at fixed D = 7.4 Å. The interatomic distance in TL was 2.8 Å, and each surface atom interacted with silicon with the two-body term of the SW potential with σ = 3.1 Å and ε = 3.35, 6.7, 10.1, 11.8, 13.5, or 16.9 kJ/mol. The atoms in the plates were fixed. It was demonstrated in ref 34 that the liquid-vapor equilibrium of water confined between TL plates was insensitive to the vibration of the atoms in the plates; thus, we do not expect the liquid-crystal or liquid-quasicrystal to be significantly different if the atoms in the surface were allowed to vibrate. Simulations were performed in the isothermal isostress NpLT ensemble, where pL is the lateral pressure, using
LAMMPS.35,36 Temperature and pressure were controlled with a Nose-Hoover thermostat and barostat, with relaxation times of 0.96 and 4.81 ps, respectively. Equations of motion were integrated with the velocity Verlet algorithm with a time step of 1.9 fs. Quenching simulations were performed at constant pL and linearly varying T at a cooling rate of 21 K ns-1. The probabilities of Table 1 were computed as the fraction of events over five independent quenching simulations at each pL. Rings of connected silicon atoms (within 3 Å, the first minimum in the radial distribution functions) in each of the two layers of the structures were identified and counted. The equilibrium melting temperatures Tm were determined using the phase coexistence method37 following the protocols of ref 21. ΔH of melting was computed as the difference in enthalpy of the two phases, each 387
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averaged over a simulation at Tm; ΔS = ΔH/Tm. The diffraction patterns were computed as the intensity of the 2D structure factor of the systems averaged over 4000 independent configurations.
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’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We thank Michael Engel and C. Austen Angell for their comments on their manuscript and Sergey Buldyrev for the code to compute the diffraction patterns. This work has been supported by the Arnold and Mabel Beckman Foundation through a Young Investigator Award to V.M. ’ REFERENCES (1) Morishita, T.; Nishio, K.; Mikami, M. Formation of Single- and Double-Layer Silicon in Slit Pores. Phys. Rev. B 2008, 77, 081401. (2) Koga, K.; Zeng, X.; Tanaka, H. Freezing of Confined Water: A Bilayer Ice Phase in Hydrophobic Nanopores. Phys. Rev. Lett. 1997, 79, 5262–5265. (3) Bai, J.; Tanaka, H.; Zeng, X. C. Graphene-Like Bilayer Hexagonal Silicon Polymorph. Nano Res. 2010, 3, 694–700. (4) Morishita, T.; Russo, S. P.; Snook, I. K.; Spencer, M. J. S.; Nishio, K.; Mikami, M. First-Principles Study of Structural and Electronic Properties of Ultrathin Silicon Nanosheets. Phys. Rev. B 2010, 82, 045419. (5) Molinero, V.; Moore, E. B. Water Modeled As an Intermediate Element between Carbon and Silicon. J. Phys. Chem. B 2009, 113, 4008– 4016. (6) Angell, C. A.; Bressel, R. D.; Hemmati, M.; Sare, E. J.; Tucker, J. C. Water and Its Anomalies in Perspective: Tetrahedral Liquids with and without Liquid-Liquid Phase Transitions. Phys. Chem. Chem. Phys. 2000, 2, 1559–1566. (7) Mishima, O.; Stanley, H. E. The Relationship between Liquid, Supercooled and Glassy Water. Nature 1998, 396, 329–335. (8) Mcmillan, P. F.; Wilson, M.; Daisenberger, D.; Machon, D. A Density-Driven Phase Transition between Semiconducting and Metallic Polyamorphs of Silicon. Nat. Mater. 2005, 4, 680–684. (9) Mishima, O.; Calvert, L. D.; Whalley, E. An Apparently 1stOrder Transition between 2 Amorphous Phases of Ice Induced by Pressure. Nature 1985, 314, 76–78. (10) Benmore, C. J.; Hart, R. T.; Mei, Q.; Price, D. L.; Yarger, J.; Tulk, C. A.; Klug, D. D. Intermediate Range Chemical Ordering in Amorphous and Liquid Water, Si, And Ge. Phys. Rev. B 2005, 72, 133201. (11) Levine, D.; Steinhardt, P. J. Quasicrystals: A New Class of Ordered Structures. Phys. Rev. Lett. 1984, 53, 2477–2480. (12) Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J. W. Metallic Phase with Long-Range Orientational Order and No Translational Symmetry. Phys. Rev. Lett. 1984, 53, 1951–1953. (13) Tsai, A. P.; Guo, J. Q.; Abe, E.; Takakura, H.; Sato, T. J. A Stable Binary Quasicrystal. Nature 2000, 408, 537–538. (14) Kamalakaran, R.; Singh, A. K.; Srivastava, O. N. Quasicrystalline Decagonal Phase of Si Clusters Evaporated in Helium and Annealed. Phys. Rev. B 2000, 61, 12686–12688. (15) Kamalakaran, R.; Singh, A. K.; Srivastava, O. N. Formation and Characterization of Nanoparticle-Bearing Threads of Silicon, Germanium and Tin. J. Phys.: Condens. Matter 2000, 12, 2681–2689. (16) Eom, D.; Jiang, C.-S.; Yu, H.-B.; Shi, J.; Niu, Q.; Ebert, P.; Shih, C.-K. Scanning Tunneling Spectroscopy of Ag Films: The Effect of Periodic versus Quasiperiodic Modulation. Phys. Rev. Lett. 2006, 97. (17) Smith, A. R.; Chao, K.-J.; Niu, Q.; Shih, C.-K. Formation of Atomically Flat Silver Films on Gaas with a 'Silver Mean' Quasi Periodicity. Science 1996, 273, 226–228. 388
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