Size Dependence of Structural Metastability in Semiconductor Nanocrystals

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Science  18 Apr 1997:
Vol. 276, Issue 5311, pp. 398-401
DOI: 10.1126/science.276.5311.398


The kinetics of a first-order, solid-solid phase transition were investigated in the prototypical nanocrystal system CdSe as a function of crystallite size. In contrast to extended solids, nanocrystals convert from one structure to another by single nucleation events, and the transformations obey simple unimolecular kinetics. Barrier heights were observed to increase with increasing nanocrystal size, although they also depend on the nature of the nanocrystal surface. These results are analogous to magnetic phase transitions in nanocrystals and suggest general rules that may be of use in the discovery of new metastable phases.

In order to expand the range of available solid-state materials, it is important to discover pathways that lead to metastable, high-energy structures. Sometimes, a high-energy form of a solid is observed to persist indefinitely at ambient conditions. For instance, diamond does not revert to graphite under ambient conditions, and high-pressure phases of AlN (1) and MgTe (2) do not revert to their lower energy phases upon release of pressure. There are many more cases, however, for which dense, high-energy phases of solids, created under conditions of high pressure or temperature, or both, spontaneously transform to a lower energy structure on a rapid time scale. A general understanding of what determines the energetic barriers between crystal structures does not currently exist but is essential for the rational synthesis of new materials. The development of routine syntheses of nanocrystals creates the opportunity to study metastability as a function of a new variable, the size of the crystal (3). Such experiments may prove analogous to well-known studies of supercooling in liquid droplets (4).

In extended solids, the transformation kinetics and microscopic pathways from one solid structure to another are difficult to determine. First-order, solid-solid phase transitions nucleate at defects, which are present at equilibrium even in the highest quality crystals. As a transformed region of the crystal grows larger, mechanical forces may generate new defects, which in turn act as new nucleation sites. These phase transitions then proceed by complex kinetics involving multiple nucleation and domain fracture (5,6). Thus, it is difficult to compare theoretical calculations of structural stability under pressure to experimental results. For example, theoretical stability calculations of defect- and strain-free bulk Si have shown that it remains metastable in the diamond structure (stable phase under ambient conditions) up to 64 GPa, whereas the thermodynamic transition to the β-tin structure (high-pressure phase) was calculated to be at 8 GPa (7). Experimentally, this transformation was observed at 11 GPa (8). These discrepancies are the result of differences in barrier heights between simulation and experiment. Thermodynamic calculations do not include a kinetic barrier, and, experimentally, crystalline defects lower the barrier for nucleation.

The kinetics of phase transitions in nanocrystals are simpler than in extended solids because it is possible to make nanocrystals that contain very few defects. In a high-quality sample, each nanocrystal is, on average, a faceted single crystal (9). Defects can be annealed out more easily in nanocrystals than in extended solids, because the distance a defect must travel to reach the surface in a nanocrystal is much smaller and the temperature required for annealing is lower (10). The effect of any residual defects are restricted to a specific nanocrystal; in contrast, in the extended solid nucleation at defects can propagate the phase transition through a large volume. Furthermore, nanocrystals undergo solid-solid phase transitions by single nucleation events, because the time required for propagation of a phase front across a distance of nanometers is less than the time separation between successive nucleation events in one crystallite (11-13).

In this study we investigated the kinetics of the four- to six-coordinate transformation from the wurtzite to rock-salt structure in CdSe and CdS nanocrystals. The transformations were investigated as a function of pressure (0 to 13 GPa) and temperature (300 to 500 K) by x-ray powder diffraction and optical absorption (14). We prepared CdSe nanocrystals 23 to 43 Å in diameter, with a narrow size distribution and high crystallinity, according to the methods given in (15). The resulting crystallites had wurtzite crystal structure and a faceted, hexagonal shape with an aspect ratio of 1.1:1 (16) and were coated with a monolayer of surfactant, tri- n -octyl phosphine oxide. The nanocrystals were dissolved in ethylcyclohexane, which was used as the pressure-transmitting medium in a diamond anvil cell (17). We also synthesized Cd32S14(SC6H5)36 · DMF4(DMF = N,N -dimethylformamide), which is a monodisperse Cd32S50 molecule 15 Å in diameter with zincblende structure, using the methods given in (18).

The transformation from wurtzite to rock salt in CdSe nanocrystals is evident in the x-ray powder diffraction patterns (Fig.1) and occurred at a pressure well above the bulk upstroke transition pressure of 2.8 GPa (19, 20). This elevation in transition pressure results from the effects of cluster size on both the kinetic barrier, which scales with the width of the hysteresis of the phase transition, and the thermodynamic transition pressure, which is taken as the center point of the hysteresis loop. Earlier work has demonstrated that the only change in thermodynamic transition point as a function of size arises as a result of differences in the energy at the interface between nanocrystal and pressure medium between the four- and six-coordinate phases (11). This causes the entire hysteresis curve to shift to higher pressure in smaller sizes.

Figure 1

(A) X-ray diffraction patterns with increasing pressure obtained for CdSe nanocrystals 43 Å in diameter at 383 K. Q = 2π/d, whered is distance in angstroms. The data show an upstroke transformation from wurtzite to rock salt between 6.2 and 6.7 GPa. (B) The diffraction linewidths of the nanocrystals, which are inversely proportional to the domain size as a result of Debye-Scherrer broadening, do not change as a result of the phase transition, consistent with single nucleation. The rock-salt diffraction lines at 7.2 GPa (dots) are the same width as the wurtzite lines [solid line, 3.5 GPa, x-ray data; dashed line, atmospheric pressure simulation based on the size determined by transmission electron microscopy (TEM)]. All lines have been shifted to have coincident centers at ΔQ = QQ line center. The proposed transition pathway (12) is shown on the right.

To separate kinetic from thermodynamic size effects on the phase transition, we investigated the hysteresis as a function of temperature (Fig. 2). With increasing pressure, the wurtzite unit cell volume decreased smoothly up to a critical pressure, at which point there was an abrupt decrease to the rock-salt unit cell volume as a result of the phase transition. Upon release of pressure, the system recovered to the four-coordinate structure [a mixture of wurtzite and zincblende (12)], but at a much lower pressure than the upstroke transformation. The width of the hysteresis is related to the observation time and to the ratio of the barrier height to the thermal energy, kBT, where kB is Boltzmann’s constant and T is temperature. As the temperature increased from 383 to 433 K, the hysteresis narrowed observably (Fig. 2).

Figure 2

Hysteresis curves for the four- to six-coordinate transformation in CdSe nanocrystals 43 Å in diameter at 383 K and 433 K. Data were obtained from optical absorption and x-ray diffraction measurements. Arrows indicate directions of increasing and decreasing pressure. We derived the unit cell volumes by integrating the optical absorption features to obtain the fractions of wurtzite (WZ) and rock salt (RS) present and calculating the volume from the data in (12). The schematic representation at the top depicts changes in the potential energy curves of the two structures with pressure.

At even higher temperatures the rate of transformation from wurtzite to rock salt became comparable to the observation time, and the kinetics of the structural phase transition were observed directly. In time-dependent measurements (Fig. 3), the pressure was increased abruptly (within a few seconds), and the phase of the nanocrystals was monitored as a function of time. The transformation was observed to proceed by a single exponential decay, indicating that there is one rate-determining step.

Figure 3

Transformation from wurtzite to rock salt with time for CdSe nanocrystals 34 Å in diameter at a constant pressure of 4.9 GPa and two temperatures. The differential of optical density (OD) with respect to time (t) was calculated by subtraction of the first absorption feature in the wurtzite phase in two consecutive spectra. The linear fits shown were used to calculate the rate constant at each temperature.

The barrier height at a particular pressure, which can be determined from the temperature dependence of the rate constant by assuming simple Arrhenius kinetics, was found to vary as a function of nanocrystal size, from 0.5 to 2.4 eV per nanocrystal at 4.9 GPa, as the size increased from 23 to 43 Å in diameter (Fig. 4). The barrier at the thermodynamic transition point will be larger than the barrier at 4.9 GPa but should follow the same trend with size (21). For small sizes, it appears that the barrier height per CdSe unit is constant, that is, it increases linearly with the number of unit cells. This result is consistent with the expectation that the barrier height is an extensive property of the system and scales with volume (3).

Figure 4

Changes in activation energy (E a) versus size (number of unit cells) for CdSe nanocrystals at 4.9 GPa. We calculated the number of unit cells using the sizes obtained from optical absorption measurements and TEM. The volume of a unit cell is 1.124 × 10−28m3 obtained from (12).

The interface between the nanocrystal and the pressure-transmitting medium also influences the barrier height. This contribution is relatively small in the CdSe nanocrystals but dominates in extremely small crystallites. The room-temperature hysteresis of the structural transition in Cd32S50 crystallites (Fig.5) is approximately one-third the width of the hysteresis loop in nanocrystals comprised of 2000 atoms (Fig. 2), consistent with the smaller number of atoms, and thus a small volume contribution to the kinetic barrier. However, changing the composition of the pressure medium substantially alters the width of the Cd32S50 hysteresis, demonstrating that the kinetic barrier also depends on the interface. In addition, there is a large shift in the center point of the hysteresis, indicating that the interface energy determines the thermodynamic transition point.

Figure 5

Hysteresis curves for the four- to six-coordinate transformation in Cd32S50 clusters dissolved in two different pressure media. The fraction of rock salt present was determined by integration of optical absorption features. Open squares are for a 3:1 mixture of tetrahydrofuran (THF) and methanol (CH3OH); filled circles are for a 3:1 mixture of THF and chloroform (CHCl3). Type II diamonds were used to measure absorption spectra down to 250 nm. The model at the bottom represents the structure of a CdS (dark circles are Cd atoms, light circles are S atoms) core cluster (18). The model at the top is the predicted rock-salt structure after transformation according to the proposed transition pathway (12).

These experiments allow us to understand the size evolution of the kinetic barriers to structural transformations in defect-free solids (Fig. 6). At very small sizes, the barriers are small, as in many molecular isomerizations, and the kinetics are dominated by the interface contribution (22). This is clearly the case for the Cd32S50 (15 Å in diameter) clusters. As the nanocrystals increase in size, the barriers become substantially larger, and a volume, or interior, contribution dominates. The crossover to interior-dominated kinetics will depend on the chemical composition of the interface but presumably takes place when there is an identifiable core (typically for diameter >20 Å). In this regime, a linear dependence on cluster volume is consistent with models in which a nanocrystal changes phase by coherent deformation of the entire cluster in one step (3). A cooperative mechanism of this type also is in accordance with the fact that nanocrystals of 102 to 103 atoms are smaller than the “critical nuclei” predicted for the bulk solid-solid phase transitions (6). At sufficiently large sizes (possibly in the clusters with diameters of 43 Å studied here) the barrier height no longer scales linearly with volume. At these sizes, the kinetics may involve separate nucleation and growth steps, although the barrier height is still large compared to the small cluster limit.

Figure 6

Illustration of the various size regimes of the kinetics of solid-solid phase transitions. Defects, which act as nucleation sites, are indicated by asterisks in the cartoon of the bulk solid.

This trend with size does not extrapolate to the bulk solid. Barrier heights in bulk solids are lower than in nanocrystals, mainly because of the influence of defects. For example, the hysteresis width of the transition from wurtzite to rock salt in bulk CdSe is approximately one-third of the width observed in CdSe nanocrystals (19,20). The size at which more complex bulklike kinetics set in will depend both on the material and on its crystallinity. Highly crystalline Si clusters as large as 500 Å in diameter still demonstrate single nucleation (13), and therefore, for the present class of materials, the likely crossover range is 100 to 1000 Å.

An analogy can be drawn between solid-solid phase transitions in nanocrystals and the well-known magnetic phase transitions in nanocrystals (23). Magnetic nanocrystals behave as single domains, which at high temperatures are superparamagnetic and respond to an applied field with no hysteresis. As the system is cooled below the “blocking temperature,” the magnetization versus applied field shows hysteresis, including remanence (residual magnetization after the applied field is turned off). The characteristic relaxation time for this hysteresis follows the simple equationEmbedded Image where K is the crystalline anisotropy and V is the volume of the crystal (24). In crystals above a certain size, multiple magnetic domains are observed, and this equation no longer applies. In solid-solid phase transitions, nanocrystals below a certain size behave as single structural domains, and the kinetic barrier “blocking” the transition can cause the system to be metastable (in analogy to remanence).

We conclude that there is an optimal size to achieve metastability in a solid. This optimal size will depend on the largest size at which single nanocrystals can be prepared defect-free and also on the judicious choice of interface. A much wider range of materials may therefore be metastable in nanocrystals than in extended solids. Examples include high-pressure rock-salt phases stable at ambient conditions in nanocrystalline GaN synthesized under pressure (25) and in CdS nanocrystals synthesized in an ionic polymer matrix (26). In contrast, high-pressure studies on both bulk GaN (27) and CdS (19) show that they completely revert from the rock-salt phase to the four-coordinate phase upon release of pressure.

Our experiments demonstrate that the kinetics of solid-solid transitions may be understood more clearly in single-domain nanocrystals than in extended crystals. In addition to allowing study of the kinetics, it seems likely that it will be possible to use this technique to investigate additional features of solid-solid phase transitions. For instance, if temperatures sufficiently high compared to the barrier height can be achieved, then the nanocrystals will rapidly fluctuate between the two stable structures on the time scale of the experiments, and the relative populations will be determined by thermodynamics only. It may also be possible to directly determine the detailed dynamics of solid-solid phase transitions in nanocrystals. Indeed, in the limit of finite size, solid-solid phase transitions behave like molecular isomerizations, and a full range of time-resolved techniques may be applied to them.

  • * Present address: Department of Chemistry, National Chung-Cheng University, Chia-Yi, Ming-Hsiung 621, Taiwan.


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