NANO LETTERS
Pressure-Driven Sphalerite to Rock Salt Transition in Ionic Nanocrystals: A Simulation Study
2004 Vol. 4, No. 9 1581-1585
Ben J. Morgan and Paul A. Madden* Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford OX1 3QZ, U.K. Received April 23, 2004; Revised Manuscript Received June 14, 2004
ABSTRACT The pressure-driven phase transformation from the sphalerite (B3) to rock salt (B1) crystal structures is studied in “constant stress” molecular dynamics simulations of ionic nanocrystals. Direct observation and calculated diffraction patterns confirm the transition. The mechanism is described and is shown to have consequences for the overall shape of the nanocrystals and in the formation of grain boundaries. The effects of pressurization rate and nanocrystal size on the transition pressure are discussed.
Many II-VI and III-V semiconductors adopt the fourcoordinate B3-sphalerite (zinc blende) or B4-wurtzite structures in their crystalline ground states and undergo pressuredriven phase transitions to the denser six-coordinate rock salt structure at pressures between 2 and 15 GPa.1 Nanoparticles of these materials also transform upon the application of pressure, and Alivisatos and co-workers2,3 have shown substantial particle size effects on the transition pressure and noted major changes in particle shape in the course of the transformation. They showed that the shape changes were consistent with the pattern of atomic displacements that occur in the bulk transformation mechanism. They explained an increase in transformation pressure with decreasing particle size by noting that, for a particle whose shape had been optimized to minimize the surface energy in the initial structure, a concerted rearrangement via the bulk transition mechanism would result in the creation of high energy surfaces of the final structure and therefore raise the energetic barrier for the transition relative to the bulk, where such surface effects are absent. This mechanism for the increased energy barrier would also apply to the reverse transformation, provided that the particles were annealed at high pressures for sufficiently long to allow the low energy surfaces of the product phase to be formed. This has been observed by Decremps et al.4 who trapped ZnO particles in the rock salt phase under ambient conditions, where it is metastable. They have noted how novel materials with desirable properties could be synthesized by these means. In this paper we describe an examination of this scenario in computer simulations of the B3fB1 transformations of nanocrystals in the size range 3-5 nm in diameter. Similar calculations have been performed previously by Molteni et * Corresponding author. E-mail:
[email protected]. 10.1021/nl049403d CCC: $27.50 Published on Web 08/05/2004
© 2004 American Chemical Society
al.5 on considerably smaller Si particles. The particles are made up of a generic ionic material, broadly similar in ion size ratio to CuCl or ZnO, for which the mechanisms of the B3fB16 and the B4fB17 have already been established, using simulation methods similar to those employed here. These mechanistic pathways appear very similar to those seen in crystallographic studies on real materials.1,8,9 The details of the interaction potentials for our ionic interactions are given in the cited references 6 and 7. The nanocrystals are rhombic dodecahedra, formed by cleaving the B3 structure along equivalent [110] planes, which form the low-energy surfaces for this structure.10 The crystals (comprising 1000-4000 ions) are surrounded by a pressurization medium consisting of a binary atomic mixture, where the size ratio (6:5) and concentration ratio (3:1) of the atoms are chosen to inhibit crystallization. The potential energy Uij due to the interaction between two atoms is described by a truncated Lennard-Jones 8-4 potential
Uij )
[
(( ) ( ) )
4ij 0
σij 8 σij rij rij
4
+ ij for rij e 21/4σij
(1)
for rij > 2 σij 1/4
where σij is the sum of the atomic radii and ij sets the interaction energy scale. Using a truncated potential allows the interations to be evaluated very efficiently, using celllisting methods.11 An interaction potential softer than the more normal 12-6 gives an increased atom mobility, again helping maintain fluidity at high pressures. This also increases the cutoff range of the potential, decreasing the efficiency of the cell-listing algorithm, giving rise to a greater computational overhead associated with modeling the Lennard-Jones fluid. The 8-4 potential was used since this
Figure 1. Molecular dynamics “snapshot” of a freshly cleaved, thermalized, nanocrystal embedded in a binary Lennard-Jones fluid.
Figure 3. Time evolution of the structure factor of the cluster throughout the simulation. The system transforms from a zinc blende structure to a rock salt structure at approximately 9.0 GPa.
Figure 2. Temperatures of the nanocrystal and the pressurizing Lennard-Jones fluid along with the internal pressure throughout a part of a simulation.
ensured the pressurization medium remained fully fluid at the pressures of interest, while still being relatively shortranged. The same interaction potential was used for the atom-ion interactions, with the anions and cations given the same radii as the large and small atoms, respectively, and the same parameter was used for all interactions (∼120 K). The system, comprising the nanocrystal and the surrounding fluid, of up to 30 000 atoms, was assembled in periodic boundary conditions; a snapshot of a starting configuration is shown in Figure 1. The molecular dynamics simulations used the “constantpressure” Andersen12 method, as implemented by Martyna et al.13 The ions and atoms were connected to Nose´-Hoover chain thermostats of length 5 at a temperature of 500 K. In the course of the simulation, the external pressure was increased in increments of 0.15 GPa with the interval, ∆t, between pressure increases varied between 3.6 and 121 ps. Although this rate of pressurization is large compared to the 1582
experimental (diamond anvil) conditions, as Figure 2 shows, for ∆t ) 14.4 ps, it was slow enough to allow the internal pressure to reequilibrate at each new pressure and for the temperature to be maintained close to 500 K. To sharpen the structural features that emerge from the simulation trajectory, we have calculated configurational properties from “thermally quenched” configurations, obtained by selecting configurations periodically along the MD run and performing a few steps of a conjugate-gradient potential energy minimization until an energy equal to the harmonic vibrational energy has been removed. Figure 3 shows calculated “powder” diffraction patterns for a 3246 ion cluster at a series of pressures along a run. The diffraction pattern has been calculated for a single particle and then averaged over all possible orientations. The patterns clearly show that there is a significant change in structure. Assignment of peaks confirms the initial B3 structure and indicates that there is a transition to B1 at a pressure just below 9.1 GPa. In the transition region the pattern is suggestive of a β-tin-like intermediate, as also seen in the bulk simulations.6 Figure 4 shows the ion-ion potential energy for the same 3246 ion cluster during the section of the simulation that contained the transition. This energy does not include contributions due to the interaction between the surface ions and the surrounding Lennard-Jones fluid. Also shown in this figure are the number of octahedrally coordinated ions, and a measure of the instantaneous cluster shape, obtained by diagonalizing the moment of inertia tensor for the cluster, IRβ )
∑i mi(r2i δRβ - ri,Rri,β)
(2)
Nano Lett., Vol. 4, No. 9, 2004
Figure 4. Time evolution of three system properties throughout the simulation. The upper curve is a plot of the ion-ion potential energy for the system. The middle plot shows the combination of the moments of inertia, M ) 2Izz - Ixx - Iyy, and the bottom graph gives the number of octahedrally coordinated cations in the simulation.
where the sum runs over all ions. The quantity plotted is M, M ) 2I2zz - I2xx - I2yy
(3)
where the z direction corresponds to the principal axis with the largest eigenvalue of the inertia tensor. A near-spherical cluster gives a small value for M, whereas a positive value corresponds to a prolate ellipsoidal shape. As the pressure is gradually ramped (with ∆t = 14.4 ps), the cluster is compressed along the z direction, and elongates in the x and y directions. This change in shape, shown by the increase in M, is accompanied by a gradual increase in the ion-ion potential energy as a result of the ions being forced together. After ≈260 ps (at which point the pressure is 8.8 GPa) there is a sudden sharp increase in the potential energy, with a corresponding further change in the shape of the cluster, before the energy plateaus. Accompanying these sudden changes in the shape and energy of the cluster, the number of cations in octahedral coordination environments also jumps suddenly from zero to around 450. After this first jump the number of octahedral cations continues to increase until it reaches nearly 700 at ≈325 ps. The relatively slow increase of the potential energy to the right of 275 ps shows how the crystal responds much more weakly to the subsequent increases in pressure, as is expected of the denser sixcoordinate structure formed during the transition. The mechanism of the B3fB1 transformation observed for this system, identified at the atomistic level, is the same as seen in the bulk. It was discussed in depth in references 6 and 7: as we will now explain it is directly manifested on a larger scale in the shape and morphology of the transformed nanocrystals. The transition occurs in two steps. There is an initial tetragonal distortion, in which the anion and cation planes are compressed in the [001]B3 direction and expanded in the [010]B3 and [100]B3 directions. This corresponds to a Nano Lett., Vol. 4, No. 9, 2004
Figure 5. Orthographic projections of a 3246 ion cluster showing views down the three rotational axes as defined by the moment of inertia, before and after the transition. The view down the z axis is in the upper left corner of each figure. Before the transition: Izz ) 2771, Ixx ) 2752, Iyy ) 2618. After the transition: Izz ) 3135, Ixx ) 2284, Iyy ) 2099.
flattening of the coordination tetrahedra and results in a diatomic β-tin-like structure, with 4+4 coordination. In the second stage the cation sublattice moves in the [001]B3 direction to form a local cation coordination environment of two nearest anions, with two slightly further away. At the same time as the cation motion there is a concerted shift of alternate layers of the anion sublattice in opposite [100]B3 and [1h00]B3 directions. This converts the 2+4 cation coordination site into an octahedral environment, with the initial [001]B3 direction becoming a [111]B1 direction of the product phase. The initial tetragonal distortion is responsible for the overall shape change of the nanocrystal, as indicated by the moments of inertia. The initial tetragonal distortion takes place by contracting the same crystallographic axis for all the tetrahedra in the cluster, so as to minimize the distortion of the crystal lattice. Figure 5 shows orthographic projections of the cluster before and after the transition, with each set containing the views along the three principal axes of the 1583
Figure 6. 3246 ion cluster after the B3fB1 transition, viewed along the z axis. The orange lines mark three Σ3 grain boundaries.
inertia tensor, with the view down the z axis in the upper left. In the first set, the cluster has a similar shape along all three axes. The second set of views, however, shows how the tetragonal distortion of the coordination tetrahedra leads to a flattening of the cluster as it undergoes a concerted deformation. This produces an asymmetry with Izz becoming much larger than Ixx and Iyy. The atomic arrangement seen in the z-projection after the transformation in Figure 5 suggests the formation of several grain boundaries, separating symmetrically tilted grains. These grain boundaries are highlighted in Figure 6. Their formation is a consequence of the two-step nature of the transition; a similar grain structure is seen in the bulk simulations when the pressure ramping is very rapid (see figure 12 of reference 6). After the initial tetragonal distortion, the cations are in symmetrical 4+4 coordinated sites and can move into the 2+4 distorted octahedron coordination environment either along [001]B3 or [001h]B3 directions (movement parallel to the shortest rotational axis of the distorted cluster). Since the movement of a cation between two neighboring anions forces them to move apart, in turn creating the octahedral hole and leading to the two anion layers sliding apart, the choice of direction of motion of the cations determines the direction of the subsequent motion of the anion layers. Viewed from the [001]B3 direction, this movement of adjacent anion layers is seen as a distortion of the anion lattice as the [111]B1 planes are formed, and the two possible relative motions of adjacent anion layers are seen as two possible different orientations of the final B1 lattice. If all the cations were to move in the same direction, a perfect B1 nanocrystal would form; however, in all the nanocrystal transformations observed thus far grain boundaries have been formed between regions of different lattice 1584
Figure 7. The same configuration as shown in Figure 6. Only cations that have moved from tetrahedral to octahedral coordination environments are shown. Ions that have moved in a positive z direction (out of the image) are colored red, while those that have moved in a negative z direction are blue.
orientation as shown in Figure 6. Figure 7 shows the same cluster from the same orientation with the cations color-coded according to which direction they have moved in the z direction. Motion in opposite directions produces regions with different lattice tilts and grain boundaries in between. In the bulk simulation, with periodic boundary conditions, at pressurization rates comparable to those used here, single crystals of the B1 phase are formed invariably. The reasons for this difference are not yet understood. The multigrain texture of the transformed particles is evident in the width of the Bragg peaks of the transformed rock salt phase in Figure 3. This is a clear difference to the observations of Tolbert and Alivisatos2 who specifically remark that the Bragg peak widths were the same before and after the transformation. However, their observations were for the wurtzite f rock salt transformation, and it remains to be seen whether this or the differing time scale between experiment and simulation is responsible. In the final Figure 8 we have collected the data on clusters of four different sizes at different pressurization rates and with different thermal histories (i.e., after being equilibrated for different amounts of time before entering the transition re´gime). The same phenomena we have described above were evident in all of these calculations. For particles containing less than about 1500 ions, the increased importance of the surface effects tends to mask the structural transformations and the particles tend to amorphize, rather than transform coherently. The results suggest an increase in the transition pressure with pressurization rate and with decreasing particle size. The latter effect was noted by Tolbert and Alivisatos,2 but care should be taken in comparing these observations. They Nano Lett., Vol. 4, No. 9, 2004
In conclusion, the B3 f B1 pressure driven phase transition was studied by computer simulation for nanocrystals of about 5 nm diameter. The transition was observed directly and by calculating “powder” diffraction patterns and occurs at a substantial overpressure compared with the thermodynamic transition pressure at T ) 0 K. The mechanism was observed to be the same as previously seen in simulations of bulk systems undergoing the same transition. This mechanism has direct consequences for changes in the overall shape of the cluster and in the formation of observed grain boundaries. Acknowledgment. This work was supported by EPSRC grant GR/R57584/01. References Figure 8. Transition pressures for 2484 and 3246 ion clusters with a range of compression rates.
were studying particles that were coated by a “surfactant” layer, which stabilizes the surface and which is believed to produce nicely facetted nanoparticles. As is evident from the figures, our surfaces are quite disordered, especially after the transformation. Surface energy effects could therefore be more pronounced in the experiments. The transition pressures we observe for the nanoparticles are lower than we find in the bulk simulations (∼14 GPa) at pressurization rates similar to those used here. However, since the bulk transition pressure is substantially higher than the thermodynamic transition pressure at T ) 0 K (5.5 GPa), as obtained from the common tangent to the internal energy against volume curves,6 it is likely that bulk pressure is affected by periodic boundary effects. Further work is underway to clarify this point.
Nano Lett., Vol. 4, No. 9, 2004
(1) Nelmes, R. J.; McMahon, M. I. Semicond. Semimet. 1998, 54, 145. (2) Tolbert, S. H.; Alivisatos, A. P. Annu. ReV. Phys. Chem. 1995, 46, 595. (3) Wickham, J. N.; Herhold, A. B.; Alivisatos, A. P. Phys. ReV. Lett. 2000, 84, 923. (4) Decremps, F.; Pellicer-Porres, J.; Datchi, F.; Itie´, J. P.; Polian, A.; Baudelet, F.; Jiang, J. Z. App. Phys. Lett. 2002, 81, 4820. (5) Molteni, C.; Martonˇa´k, R.; Colombo, L.; Parrinello, M. J. Chem Phys. 2002, 117, 24, 11329. (6) Wilson, M.; Hutchinson, F.; Madden, P. A. Phys. ReV. B 2002, 65, 094109. (7) Wilson, M.; Madden, P. A. J. Phys.: Condens. Matter. 2002, 14, 4629. (8) Sowa, H. Z. Kristallogr. 2000, 215, 335. (9) Sowa, H. Acta Crystallogr., Sect A: Found. Crystallogr. 2001, 57, 176. (10) Hamad, S.; Cristol, S.; Catlow, C. R. A. J. Phys. Chem. B 2002, 106, 11002. (11) Frenkel, D.; Smit, B. Understanding Molecular Simulation Academic Press: New York, 1996. (12) Andersen, H. C. J. Chem Phys. 1980, 72, 2384. (13) Martyna G. J.; Tobias, D. J.; Klein, M. L. J. Chem. Phys. 1994, 101, 4177.
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