ANNUAL REVIEWS
Annu.
Rev. Phys. Chern. /995. 46: 595- E CD :::i:
I
.
,
800 600 15
20
25
30
Radius (Al
35
40
45
50
Figure 1 Melting temperature as a function of size for CdS nanocrystals. The solid line is a fit to a model that describes the decrease in melting temperature in terms of the difference in
surface energy between the solid and liquid phases. (Figure from Reference 2.)
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NANOCRYSTALS AT HIGH PRESSURE
597
point depressions of over 50% are observed for sufficiently small-sized nanocrystals. This depression can be understood by considering the factors that contribute to the total energy of a nanocrystal: In a system containing only a few hundred atoms, a large fraction of these atoms will be located on the surface. Because surface atoms tend to be coordinatively unsaturated, a large energy is associated with this surface. The key to understanding this melting point depression is the fact that the surface energy is always lower in the liquid phase than in the solid phase. In the dynamic fluid phase, surface atoms move to minimize surface area and unfavorable surface interactions. In the solid phase, rigid bonding geometries cause stepped surfaces with high-energy edge and corner atoms. On melting, the total surface energy is thus reduced, and this stabilizes the liquid phase. The smaller the nanocrystal, the larger the contribution made by the surface energy to the overall energy of the system and thus the more dramatic the melting point depression ( l a,b). As melting is believed to start on the surface of a nanocrystal, this surface stabilization is an intrinsic and immediate part of the melting process (3, 4). Using the known l /radius drop-off in cohesive energy per atom and the resulting depression in melting temperature as a starting point (5), it is worthwhile to consider the different regimes of behavior that are expected in ever smaller crystals. First, one encounters the regime in which only small numbers of nucleation events can occur in a given crystallite, leading to changes in the kinetics of melting. Next, the number of surface atoms becomes a large fraction of the total number of atoms, perturbing the relative stability of the liquid and the solid phases. This is the nanocrystal regime of ·100 to 1 0,000 atoms, in which there is a well-defined interior with a single defined structure at low temperature. Finally, as the number of surface atoms exceeds the number in the interior, the quantum mech anical nature of the chemical bonding changes, and entirely new structures may emerge. This is in many ways the most interesting regime because it is here that one might find entirely new structures with properties that are not simple extrapolations from the bulk material. We may not expect the size dependence of the melting temperature to extend into this regime because the intrinsic bonding of the atoms can change. For example, calculations show that there may be no single well-defined structure for Si clusters of 1 0-45 atoms (6). More ionic semiconductors, in contrast, seem to retain unique crystalline structures down to very small sizes (7, 8). Thus one of the most important questions in cluster physics and chemistry is are there a small number of well-defined low-temperature structures in nanocrystals? To extend our understanding of the effects of size on phase stability and to address the issues discussed above, we have studied solid-solid phase
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598
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& ALIVISATOS
changes in tetrahedral inorganic semiconductors as a function of the size of the crystal from 1 ,000,000 down to 1 00 atoms. In the extended solids, these materials exist in a small number of well-defined crystal structures; interconversion between these structures is usually accompanied by a discontinuous change in unit cell volume. As with melting, there tends to be very different bonding in different structures, e.g. covalent versus metallic. The mechanics of transformation, however, are quite different from melting. In an order-disorder transition such as melting, there is diffusive motion of atoms that starts at the surface and moves inward. In a solid-solid phase transition, in contrast, atoms must move along a well defined transition path in a cooperative manner. Although transformations of this sort can be induced by changes in temperature in some solids, many more materials will transform between solid structures upon the application of high pressures. In particular, tetrahedral semiconductors tend to exhibit solid-solid phase changes at relatively modest pressures. Because of the directionality of the bonding, tetrahedral semiconductors have very open structures with large void volumes. Under pressure, the energy of the system can be easily lowered by a more volume-efficient packing of atoms. As a result, diamond-, zinc blende-, and wurtzite-phase semiconductors undergo transformations to rock salt or f3-Sn phases at pressures between 2 and 1 5 GPa. The rock salt phase exhibits octahedral bonding with large Madelung stabilization and can occur in ionic systems such as I-VII, II-VI, and some III-V semi conductors. The p-Sn phase is a distorted octahedral structure. This metal lic phase is observed at high pressures in IV-IV and less-ionic III-V semi conductors. Our goal is to develop a general rule for the effect of size on first-order solid-solid phase transitions, comparable to the well-known l /r dependence of the melting temperature (5). Because of their large volume change and moderate transition pressures, we will use tetrahedral semi conductor nanocrystals as model systems. A wide variety of semiconductor nanocrystals can be prepared. In par ticular, advances in wet chemical synthetic techniques now allow for the preparation of virtually any II-VI and some I-VII semiconductor nano crystals in a variety of sizes with high crystallinity and narrow size dis tributions (9, 1 0). Gas pyrolysis reactions have been successfully used to produce highly crystalline IV-IV nanocrystals in a variety of sizes, although size distributions are not nearly as narrow as for the II-VI systems ( l l a,b). Finally, organometallic syntheses, similar to those used for II-VI semi conductors, are now being utilized to make III-V and IV-IV nanocrystals ( l2a-d, 1 3). All of these methods produce crystallites with some type of surface passivation layer that prevents agglomeration of the nanocrystals and allows the crystallites to be dissolved in various organic solvents. Here
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NANOCRYSTALS AT HIGH PRESSURE
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we concentrate on three different types of nanocrystals: the closely related II-VI systems, CdS and CdSe, and the IV-IV system Si. These systems are exemplary because they span the range of possible ionicities in tetrahedral systems: from dominantly ionic to completely covalent. Preparation of CdS (14, 1 5), CdSe (9, 1 6), and Si ( 1 1 a,b) nanocrystals is now well docu mented and thus will not be discussed here. For the experiments reviewed here, CdS nanocrystals were synthesized with a radius of 20 A, a zinc blende structure, moderate crystallinity, and moderate size distribution. Si nanocrystals in the diamond structure were produced with high crys tallinity and moderate size distributions; median sizes ranged from 50 to 250 A in radius. Finally, CdSe nanocrystals with a wurtzite structure were produced with high crystallinity and narrow size distributions; sizes ranged from 10 to 30 A in radius. A TEM micrograph of a field 25 A radius CdSe nanocrystals arrayed on an amorphous carbon grid is presented in Figure 2. The individual lattice planes visible in most crystallites indicate the highly ordered nature of each discrete cluster. Bulk CdSe transforms from a wurtzite structure to a rock salt structure at 3.0 GPa with applied hydrostatic pressure ( l 7a-c). CdS undergoes an analogous transition between 2.7 and 3. 1 GPa ( l 8a-j). Bulk silicon transforms from the diamond structure to the f3-Sn phase at approximately 1 1 GPa and then further transforms to a primitive hexagonal structure at about 1 6 GPa ( l9a-h). ! As we show below, in all cases and for all sizes examined, these phase transition pressures are significantly elevated in nanocrystals compared to the bulk material (20). 2 Further, the elevation is a function of crystallite size with smaller diameter crystallites undergoing transitions at higher pressures. In this paper, we review the available data on pressure-induced phase transformations in nanocrystals and explore the various methods that can be used to study phase transitions in finite size. We then combine these data to produce an understanding of the elevation in phase transition pressure and to propose a general rule for the effects of size on first-order solid-solid phase transitions.
TECHNIQUES AND SYSTEMS Conventional diamond anvil cell techniques can be used to study phase transitions at high pressure ( 2 1 ) . The surface passivation of these crysI This is actually somewhat of a simplification. Bulk Si appears to transform from the p. Sn phase to the closely related Imma structure at 13 GPa. The Imma phase then transforms to a primitive hexagonal structure near 16 GPa. A variety of references combine to explain this complicated phase behavior (see 19a-h). 2No elevation in transition pressure was observed in one experiment performed on 100300 A CdS nanocrystals. This result is to be expected, given the size of these crystallites and the results presented here on CdS and CdSe (see 20).
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600
Figure 2
TOLBERT & ALIVISATOS
TEM micrograph of approximately 5-nm diameter CdSe nanocrystals. The nano
crystals are sitting on an amorphous carbon film. Clear faceting of the crystallites can be observed.
tallites allows them to be dissolved at high concentration in suitable quasi hydrostatic pressure media. At high pressure this generally results in an optically clear sample composed of nanocrystals in an organic glass. This glass is moderately soft and can distort to transform the applied uniaxial pressure to an isotropic pressure around each individual crystallite. Because the anvils of the pressure cell are diamonds, they allow for both optical and X-ray access to the sample. Experiments that probe electronic, vibrational, and physical structure of the crystallites at high pressure are thus possible.
Raman Scattering on CdS and CdSe The earliest high-pressure experiments on CdS (20, 22, 23) and CdSe (24) nanocrystals all utilized Raman scattering. Resonance enhancement of the Raman signal is quite strong for these polar semiconductors (25a,b; 26), so Raman signals can be easily obtained on the nanoliter sample volumes used with diamond anvil cells. In addition, Raman does not require an optically clear sample, so expe riments can also be performed on nano crystallites grown in a glass matrix (22). More importantly, however, the observed Raman modes can be directly correlated to the interior structure of the nanocrystals. All of the experiments referenced above show clearly that the tetrahedral structure present at atmospheric pressure persists to pressures well above the bulk stability limit. An example of this type of
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NANOCRYSTALS AT HIGH PRESSURE
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data is shown in Figure 3 for 40 A diameter CdS nanocrystals (23). Figure 3a shows the zinc blende LO phonon fundamental and overtone at low pressure (0.5 GPa) and well above the bulk phase transition pressure of 2.7 GPa (scan taken at 5.0 GPa). There is clearly no change in the zinc blende structure of the nanocrystals. High-pressure luminescence measure ments have also been used to make this conclusion (22): No change in the fluorescence spectra is seen at pressures well above the bulk phase tran sition pressure. Figure 3b further shows that the shift in Raman frequency with pressure (ow/oP) is essentially identical to the bulk LO phonon shift. The linear fit to the data agrees well with the value of ow/oP reported for bulk CdS (23, 27). Because the Raman shift can be directly related to the volume compressibility through the mode Griineisen parameter, this result indi cates that the potential energy as a function of unit cell volume is identical in both the harmonic and the first anharmonic term in tetrahedrally bonded bulk and nanocrystal systems. This is a very important result a)
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4 High-pressure Se EXAFS obtained on SA-nm diameter CdSe nanocrysta\s. (a) Background-corrected X-ray absorption data presented as k*X(k). A clear shift in the EXAFS oscillations can be seen at the phase transition. (b) (Top) Shift in Cd-Se bond length with pressure. A dramatic increase in bond length is observed at the phase transition. (Bottom) Volume change with pressure calculated from the changes in Cd-Se bond length assuming a wurtzite to rock salt phase transiti on. The solid line is a fit to the Murnaghan equations of state with Bo 37 ± 5 GPa and Bo' 11 ± 3. (D ata from Reference 32.) Figure
=
=
yields a measure of the bulk modulus (inverse volume compressibility) and its derivative with respect to pressure: Bo 37 ± 5 GPa, Bo' 1 1 ± 3 . The values obtained are somewhat different than those for bulk CdSe: Bo 55 GPa. This is because deviations from linear compressibility are small in the region between 0 and 3 GPa, where bulk CdSe is stable. Nanocrystals, however, continue to persist in the wurtzite structure to much higher pressures. As the Cd-Se bond length decreases, repulsive forces dominate, and the compressibility decreases. This information is interesting for two reasons: First, compressibility data has proven very important in thermo dynamic analyses of high-pressure nanocrystal phase transition data. Second, nanocrystals provide a unique opportunity to study the tetra hedrally bonded Cd-Se potential in the region beyond the bulk stability limit. =
=
=
X-ray Diffraction on CdSe and Si For the same reasons that EXAFS is ideally suited for studying very small nanocrystals, it is not the best technique for studying phase transitions in general. EXAFS probes only short-range order, so although it can provide
605
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NANOCRYSTALS AT HIGH PRESSURE
information about the high-pressure phase structures, it cannot provide information about the degree of crystallinity in the newly transformed nanocrystals. For this, high-pressure X-ray diffraction (HP-XRD) is required. Because of the small sample volume and limited crystallinity of semi conductor nanocrystals at high pressure, HP-XRD experiments were car ried out at a synchrotron. The experiments were performed in an angle dispersive geometry with a modified diamond anvil cell equipped with either a diffraction slit or a Be backing plate to allow the diffracted X rays out of the cell. Figure 5 shows the quality of X-ray diffraction data that can be obtained on CdSe nanocrystals (36, 37). Diffraction peaks are broadened by the finite size of the crystallites (Debye-Scherrer broadening). Despite this fact, low-pressure phase peaks can still be easily indexed to a wurtzite structure (38). As with the other techniques presented here, the elevation in phase transition pressure can be clearly seen: The wurtzite phase is (111) (200)
a)
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(220)
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J
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t 8
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16
20
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36
40
Figure 5 High-pressure X-ray diffraction data obtained on 4.2-nm diameter CdSe nano crystals. (a) Diffraction patterns collected with increasing pressure. The system transforms from a wurtzite structure to a rock salt structure at approximately 6.3 GPa. (b) Diffraction patterns collected with decreasing pressure. The system transforms from the rock structure to a mixed zinc blende-wurtzite phase at approximately 1 GPa. Double-headed arrows mark diffraction from the metal gasket of the high-pressure cell. Pressures and diffraction peak indexing are indicated on the figure. (Figure from Reference 36.)
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606
TOLBERT & ALIVISATOS
stable until approximately 6 GPa, well above the bulk transition of 3 GPa observed with increasing pressure (upstroke). In contrast to other techniques, however, much new information can be gained about the high pressure phase. In the first place, the diffraction peaks in this new phase unambiguously index to a rock salt structure (39a,b). Although this con clusion has been suggested by other techniques, only diffraction proves it. Upon release of pressure, the transformation is marked by significant hysteresis. The tetrahedral phase does not recover until approximately 1 GPa, and then it appears to be a mixture of the closely related zinc blende (40) and wurtzite structures.3 The recovery of this mixed phase can be observed only by diffraction. More importantly, a careful analysis of diffraction peak widths in both the wurtzite and rock salt structures shows that no broadening of the diffraction peaks, and thus no decrease in the crystalline domain size, has occurred upon phase transition. This result has two important consequences: In the first place, it has implications for the mechanism of transformation. This is discussed in some detail below. In addition, it indicates that high pressure can be used to synthesize highly crystalline particles in new bonding geometries. Electronic properties of these new samples can then be studied to learn about the effects of finite size of a variety of nanocrystal systems (42). The real power of X-ray diffraction, however, can be seen in samples such as Si nanocrystals. Bulk Si undergoes three solid-solid phase tran sitions between 1 1 and 16 GPa [diamond to f3-Sn to Imma to primitive hexagonal ( 1 9a-h)], with a fourth transition to a metastable phase upon release of pressure [f3-Sn to Be8 (19a-h)]. All of the structures other than the semiconducting diamond phase are metals, so resonance Raman (43t or optical techniques are not useful for studying these phases. Si EXAFS is not in an energy range accessible in the diamond cell. Only diffraction can be used to fully sort out the high-pressure phase behavior in Si nano crystals. Diffraction data, presented in Figure 6a, shows the complicated phase behavior displayed by 490 A diameter Si nanocrystals coated with Si02 (44). Like the metal chalcogenides, the phase transition pressure is much higher than that observed in bulk Si, a particularly striking result given the large diameter of the nanocrystals in question. The system can, however, be converted from the diamond structure (45) (Figure 6a, section 1) to the primitive hexagonal structure (Figure 6a, section 2) ( 19a-h). Whether this transition proceeds through the f3-Sn and Imma phases is 3 This
effect has been observed in recovered bulk CdSe samples as well (see 1 7b,c). Raman data has been obtained on metallic phases of bulk Si at high pressure. It is unclear whether it would be possible to collect data of this sort on nanophase materials (see 43). 4 Off-resonance
607
NANOCRYSTALS AT HIGH PRESSURE
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a)
b)
(111)
10
12
14
16 18 2 theta
(wavelength
=
20 0.62
22
A)
24
o
5
10
15
20
Pressure (GPa)
25
30
Figure 6 High-pressure data obtained on Si nanocrystals coated with Si02• (a) High pressure X-ray diffraction data obtained on 49-nm diameter nanocrystals. The crystallites are stable in the diamond phase to well above the bulk stability limit of 11 OPa (section 1 ). With sufficient pressure, the crystallites do transform to a primitive hexagonal phase (section 2). Upon partial release of pressure, the Imma phase is observed (section 3). Upon full release of pressure, the bulk BC8 phase is not observed. Instead, amorphous Si is recovered (section 4). Indexing and pressures are indicated on the figure. (b) Phase behavior for IO-nm diameter Si crystallites measured by optical absorption. The increase in 0.0. at 22 OPa marks the diamond to fJ-Sn (primitive hexagonal) transformation. The decrease in 0.0. at 5 OPa marks the fJ-Sn to amorphous Si transformation. The arrow marks the bulk Si upstroke diamond to fJ-Sn phase transition pressure. (Figure from Reference 44.)
unclear from the data. Upon partial release of pressure, an Imma phase is observed (Figure 6a, section 3) and a f3-Sn phase (not shown) (1 9a-h). Although the transition pressures are altered in the nanocrystal sample with respect to bulk Si, all of these phases can be observed in bulk as well as nanocrystalline Si. Upon full release of pressure, however, instead of recovering a Be8 phase (46), the sample appears to form amorphous Si (Figure 6a, section 4). This complicated phase behavior would be almost impossible to unravel with any technique other than diffraction. Although X-ray diffraction is not ideally suited for examining very small nanocrystals, interpretable data on CdSe nanocrystals as small as 10 A in radius can, in fact, be collected (37). This data shows the same type of structural changes observed in larger samples in Figure 5. The system transforms from a wurtzite structure to a rock salt structure and back to a tetrahedral structure (wurtzite and zinc blende structures cannot be distinguished in this size range) with no apparent loss of crystalline domain
608
TOLBERT & ALIVISATOS
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size. Further, data (not shown) indicate that the phase transition pressure is somewhat higher in these very small crystallites than in the larger sample presented in Figure 5. Obviously, the next goal in understanding the elevation in phase transition pressure in semiconductor nanocrystals is to map out the nanocrystal size dependence of the phase transition pressure. Unfortunately, diffraction is not the ideal experiment for this due to the expense and time constraints associated with working at a synchrotron. A simpler experiment that can be performed in the laboratory is needed.
Optical Absorption on CdSe, CdS, and Si For nanocrystal samples that can be homogeneously dissolved to form a clear solution, optical absorption is probably the simplest high-pressure analysis technique available. Unlike diffraction and EXAFS, electronic absorption experiments can be performed with relatively simple equipment in the lab. Unlike Raman scattering, there is usually some signature of both high- and low-pressure phases apparent in the absorption spectrum, at least for the first tetrahedral to octahedral (or quasi octahedral) trans formation. Most importantly, however, the experiments can be efficiently
repeated on a wide variety of sample sizes with small pressure steps to accurately determine phase transition pressures as a function of nano crystal size. A sample of the type of available optical absorption data is presented in Figure 7 for CdSe nanocrystals (37). In the wurtzite phase, the absorption spectra show a series of discrete features due to quantum confinement of the direct gap optical excitations (47a,b; 48). These features shift to the blue with applied pressure. At the phase transition pressure, the discrete features disappear and are replaced by a slowly rising absorption that can be assigned to the indirect gap of the rock salt phase (49, 49a). 5,6 Upon release of pressure, the discrete features are recovered. The data in Figure 7 have been corrected for changes in optical density (O.D.) caused by deformation of the diamond anvil cell metal gasket. Integration of the direct gap absorption features provides a simple method for quantitative mapping of the transition hysteresis. These data, presented in Figure 8 for three sizes of CdSe nanocrystals, show wide hysteresis curves with rela tively sharp phase transitions (36, 37). The transition pressures can be assigned to the 50% transformed point for both upstroke and downstroke transformations. 5 High-pressure optical absorption data on rock salt phases of both CdS and CdSe indicate that they are indirect gap semiconductors (see 17a--c and 18a-g). 6 Simulations of the electronic structure of rock salt phase bulk CdSe and CdS also indicate that both are indirect gap semiconductors (see 49, 49a).
NANOCRYSTALS AT HIGH PRESSURE
a)
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Figure 7 High-pressure optical absorption obtained on 2.S-nm diameter CdSe nanocrystals. (a) Shift in the wurtzite phase absorption with pressure. (b) Disappearance of the discrete wurtzite absorption features (solid line) at the phase transition pressure and appearance of the indirect gap spectra of the rock salt phase (large dash line). (c) Recovery of the discrete zinc blende-wurtzite features upon release of pressure (solid line). The original atmospheric pressure spectra is included for comparison (small dash line). Pressures are indicated on the figure. All spectra have been corrected for changes in O.D. due to deformation of the high pressure cell gasket. (Figure from Reference 37.)
For indirect gap semiconductors such as Si, the optical absorption experiment is somewhat more difficult. The lack of discrete features in either the low- or high-pressure phases makes the phase transition pressure somewhat less obvious. The large increase in 0.0. at the semiconductor to metal transition pressure is, however, a clear indication of structural change. Figure 6b shows data obtained on 100 A diameter Si nanocrystals (44). A clear upstroke transition, well above the bulk Si upstroke phase transition, is observed. Note that the recovery of a semiconducting state upon release of pressure is consistent with high-pressure diffraction data obtained on these samples: Amorphous Si is a semiconductor, while the metastable phase BC8 recovered from the bulk is a semi-metal. These Si absorption data were not corrected for changes in 0.0., resulting from deformation of the diamond cell gasket. Because of its simplicity, optical absorption has been applied to a wide variety of samples [e.g. CdSe (24, 36, 37, 50), CdS (22, 23), and Si (44)). Owing to past synthetic constraints, however, a systematic size dependent
610
TOLBERT & ALIVISATOS
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Figure 8 Hysteresis curves for three sizes of CdSe nanocrystals. The data are obtained by integration of the direct gap absorption features like those presented in Figure 7. Changes in both the upstroke and downstroke transition pressures are observed with variations in the nanocrystal size. Phase transition pressures are assigned to the midpoints of these hysteresis curves (average of upstroke and downstroke 50% transformed pressure). (Figure from References 36 and 37.)
set of data had been collected only for CdSe. The hysteresis shown in Figure 8 adds some difficulty to determining the size dependence of the phase transition pressure. Should the transition pressure be assigned to the upstroke transition pressure? The midpoint of the hysteresis curves? The downstroke transition pressure? Although there is some argument in the literature over which is the most meaningful point, the existence of broad hysteresis curves indicates the presence of a significant barrier to phase transformation and thus suggests that the midpoint of the curves is the most appropriate. Under this assumption, optical absorption can be used to map out the size dependence of the phase transition pressure. The data presented in Figure 9 for CdSe nanocrystals in pyridine show a monotonic increase in phase transition pressure with decreasing nano crystal size (36, 37). In the next section, we examine this trend and its implications for phase transitions in nanocrystals in general.
SURFACE ENERGIES AND PATH EFFECTS In the section above, we examined a variety of methods that can be used to observe phase transformations in nanocrystals. By combining the results
NANOCRYSTALS AT HIGH PRESSURE
611
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4.8
3.6 +-...--,....,"-"-.--T-r--,----r--T>-...,.---l 3.4 10 12 14 16 18 20 22 radius (A) Figure 9
Crystallite size dependence of the wurtzite to rock salt phase transfonnation
pressure for CdSe nanocrystals. The data are obtained from the midpoints of hysteresis
curves like those shown in Figure 8. The line is a fit to a thennodynamic model that describes the elevation in transformation pressure in tenns ofthe differences in surface energies between the wurtzite and rock salt phases. (Figure from References 36 and 37.)
of all these experiments, it should be possible to develop a general under standing of phase transitions in finite systems.
Surface Energies By analogy with the theories for melting in finite systems, the elevation in
solid-solid phase transition pressure observed in semiconductor nano crystals can be interpreted in terms of changes in surface energies. In the case of melting, the reduced surface energy in the liquid phase caused a decrease in melting temperature. The increase in solid-solid transition pressure can, by analogy, be assigned to an increase in surface energy in the high-pressure phase nanocrystals. The reality of a surface energy associated with solid phases of semi conductor nanocrystals can be seen experimentally from wurtzite-phase X-ray power diffraction at atmospheric pressure (51). Figure 10 presents the fractional change in lattice constant (relative to bulk CdSe) versus size for CdSe nanocrystals (37). The data show a small but systematic decrease in lattice constant with decreasing nanocrystal size. This change in lattice constant can be related to a surface energy through the Laplace Law. Simply put, the Laplace Law results from a macroscopic theory that relates a surface tension to a surface pressure. This pressure is then converted to a lattice change through a knowledge of the volume compressibility. Although the Laplace Law assumes only radial forces, a large sim plification for a structure with directional bonding like a wurtzite-phase
612
TOLBERT & ALIVISATOS
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Annu.
Rev. Phys. Chern. /995. 46: 595- E CD :::i:
I
.
,
800 600 15
20
25
30
Radius (Al
35
40
45
50
Figure 1 Melting temperature as a function of size for CdS nanocrystals. The solid line is a fit to a model that describes the decrease in melting temperature in terms of the difference in
surface energy between the solid and liquid phases. (Figure from Reference 2.)
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NANOCRYSTALS AT HIGH PRESSURE
597
point depressions of over 50% are observed for sufficiently small-sized nanocrystals. This depression can be understood by considering the factors that contribute to the total energy of a nanocrystal: In a system containing only a few hundred atoms, a large fraction of these atoms will be located on the surface. Because surface atoms tend to be coordinatively unsaturated, a large energy is associated with this surface. The key to understanding this melting point depression is the fact that the surface energy is always lower in the liquid phase than in the solid phase. In the dynamic fluid phase, surface atoms move to minimize surface area and unfavorable surface interactions. In the solid phase, rigid bonding geometries cause stepped surfaces with high-energy edge and corner atoms. On melting, the total surface energy is thus reduced, and this stabilizes the liquid phase. The smaller the nanocrystal, the larger the contribution made by the surface energy to the overall energy of the system and thus the more dramatic the melting point depression ( l a,b). As melting is believed to start on the surface of a nanocrystal, this surface stabilization is an intrinsic and immediate part of the melting process (3, 4). Using the known l /radius drop-off in cohesive energy per atom and the resulting depression in melting temperature as a starting point (5), it is worthwhile to consider the different regimes of behavior that are expected in ever smaller crystals. First, one encounters the regime in which only small numbers of nucleation events can occur in a given crystallite, leading to changes in the kinetics of melting. Next, the number of surface atoms becomes a large fraction of the total number of atoms, perturbing the relative stability of the liquid and the solid phases. This is the nanocrystal regime of ·100 to 1 0,000 atoms, in which there is a well-defined interior with a single defined structure at low temperature. Finally, as the number of surface atoms exceeds the number in the interior, the quantum mech anical nature of the chemical bonding changes, and entirely new structures may emerge. This is in many ways the most interesting regime because it is here that one might find entirely new structures with properties that are not simple extrapolations from the bulk material. We may not expect the size dependence of the melting temperature to extend into this regime because the intrinsic bonding of the atoms can change. For example, calculations show that there may be no single well-defined structure for Si clusters of 1 0-45 atoms (6). More ionic semiconductors, in contrast, seem to retain unique crystalline structures down to very small sizes (7, 8). Thus one of the most important questions in cluster physics and chemistry is are there a small number of well-defined low-temperature structures in nanocrystals? To extend our understanding of the effects of size on phase stability and to address the issues discussed above, we have studied solid-solid phase
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598
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& ALIVISATOS
changes in tetrahedral inorganic semiconductors as a function of the size of the crystal from 1 ,000,000 down to 1 00 atoms. In the extended solids, these materials exist in a small number of well-defined crystal structures; interconversion between these structures is usually accompanied by a discontinuous change in unit cell volume. As with melting, there tends to be very different bonding in different structures, e.g. covalent versus metallic. The mechanics of transformation, however, are quite different from melting. In an order-disorder transition such as melting, there is diffusive motion of atoms that starts at the surface and moves inward. In a solid-solid phase transition, in contrast, atoms must move along a well defined transition path in a cooperative manner. Although transformations of this sort can be induced by changes in temperature in some solids, many more materials will transform between solid structures upon the application of high pressures. In particular, tetrahedral semiconductors tend to exhibit solid-solid phase changes at relatively modest pressures. Because of the directionality of the bonding, tetrahedral semiconductors have very open structures with large void volumes. Under pressure, the energy of the system can be easily lowered by a more volume-efficient packing of atoms. As a result, diamond-, zinc blende-, and wurtzite-phase semiconductors undergo transformations to rock salt or f3-Sn phases at pressures between 2 and 1 5 GPa. The rock salt phase exhibits octahedral bonding with large Madelung stabilization and can occur in ionic systems such as I-VII, II-VI, and some III-V semi conductors. The p-Sn phase is a distorted octahedral structure. This metal lic phase is observed at high pressures in IV-IV and less-ionic III-V semi conductors. Our goal is to develop a general rule for the effect of size on first-order solid-solid phase transitions, comparable to the well-known l /r dependence of the melting temperature (5). Because of their large volume change and moderate transition pressures, we will use tetrahedral semi conductor nanocrystals as model systems. A wide variety of semiconductor nanocrystals can be prepared. In par ticular, advances in wet chemical synthetic techniques now allow for the preparation of virtually any II-VI and some I-VII semiconductor nano crystals in a variety of sizes with high crystallinity and narrow size dis tributions (9, 1 0). Gas pyrolysis reactions have been successfully used to produce highly crystalline IV-IV nanocrystals in a variety of sizes, although size distributions are not nearly as narrow as for the II-VI systems ( l l a,b). Finally, organometallic syntheses, similar to those used for II-VI semi conductors, are now being utilized to make III-V and IV-IV nanocrystals ( l2a-d, 1 3). All of these methods produce crystallites with some type of surface passivation layer that prevents agglomeration of the nanocrystals and allows the crystallites to be dissolved in various organic solvents. Here
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NANOCRYSTALS AT HIGH PRESSURE
599
we concentrate on three different types of nanocrystals: the closely related II-VI systems, CdS and CdSe, and the IV-IV system Si. These systems are exemplary because they span the range of possible ionicities in tetrahedral systems: from dominantly ionic to completely covalent. Preparation of CdS (14, 1 5), CdSe (9, 1 6), and Si ( 1 1 a,b) nanocrystals is now well docu mented and thus will not be discussed here. For the experiments reviewed here, CdS nanocrystals were synthesized with a radius of 20 A, a zinc blende structure, moderate crystallinity, and moderate size distribution. Si nanocrystals in the diamond structure were produced with high crys tallinity and moderate size distributions; median sizes ranged from 50 to 250 A in radius. Finally, CdSe nanocrystals with a wurtzite structure were produced with high crystallinity and narrow size distributions; sizes ranged from 10 to 30 A in radius. A TEM micrograph of a field 25 A radius CdSe nanocrystals arrayed on an amorphous carbon grid is presented in Figure 2. The individual lattice planes visible in most crystallites indicate the highly ordered nature of each discrete cluster. Bulk CdSe transforms from a wurtzite structure to a rock salt structure at 3.0 GPa with applied hydrostatic pressure ( l 7a-c). CdS undergoes an analogous transition between 2.7 and 3. 1 GPa ( l 8a-j). Bulk silicon transforms from the diamond structure to the f3-Sn phase at approximately 1 1 GPa and then further transforms to a primitive hexagonal structure at about 1 6 GPa ( l9a-h). ! As we show below, in all cases and for all sizes examined, these phase transition pressures are significantly elevated in nanocrystals compared to the bulk material (20). 2 Further, the elevation is a function of crystallite size with smaller diameter crystallites undergoing transitions at higher pressures. In this paper, we review the available data on pressure-induced phase transformations in nanocrystals and explore the various methods that can be used to study phase transitions in finite size. We then combine these data to produce an understanding of the elevation in phase transition pressure and to propose a general rule for the effects of size on first-order solid-solid phase transitions.
TECHNIQUES AND SYSTEMS Conventional diamond anvil cell techniques can be used to study phase transitions at high pressure ( 2 1 ) . The surface passivation of these crysI This is actually somewhat of a simplification. Bulk Si appears to transform from the p. Sn phase to the closely related Imma structure at 13 GPa. The Imma phase then transforms to a primitive hexagonal structure near 16 GPa. A variety of references combine to explain this complicated phase behavior (see 19a-h). 2No elevation in transition pressure was observed in one experiment performed on 100300 A CdS nanocrystals. This result is to be expected, given the size of these crystallites and the results presented here on CdS and CdSe (see 20).
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600
Figure 2
TOLBERT & ALIVISATOS
TEM micrograph of approximately 5-nm diameter CdSe nanocrystals. The nano
crystals are sitting on an amorphous carbon film. Clear faceting of the crystallites can be observed.
tallites allows them to be dissolved at high concentration in suitable quasi hydrostatic pressure media. At high pressure this generally results in an optically clear sample composed of nanocrystals in an organic glass. This glass is moderately soft and can distort to transform the applied uniaxial pressure to an isotropic pressure around each individual crystallite. Because the anvils of the pressure cell are diamonds, they allow for both optical and X-ray access to the sample. Experiments that probe electronic, vibrational, and physical structure of the crystallites at high pressure are thus possible.
Raman Scattering on CdS and CdSe The earliest high-pressure experiments on CdS (20, 22, 23) and CdSe (24) nanocrystals all utilized Raman scattering. Resonance enhancement of the Raman signal is quite strong for these polar semiconductors (25a,b; 26), so Raman signals can be easily obtained on the nanoliter sample volumes used with diamond anvil cells. In addition, Raman does not require an optically clear sample, so expe riments can also be performed on nano crystallites grown in a glass matrix (22). More importantly, however, the observed Raman modes can be directly correlated to the interior structure of the nanocrystals. All of the experiments referenced above show clearly that the tetrahedral structure present at atmospheric pressure persists to pressures well above the bulk stability limit. An example of this type of
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NANOCRYSTALS AT HIGH PRESSURE
601
data is shown in Figure 3 for 40 A diameter CdS nanocrystals (23). Figure 3a shows the zinc blende LO phonon fundamental and overtone at low pressure (0.5 GPa) and well above the bulk phase transition pressure of 2.7 GPa (scan taken at 5.0 GPa). There is clearly no change in the zinc blende structure of the nanocrystals. High-pressure luminescence measure ments have also been used to make this conclusion (22): No change in the fluorescence spectra is seen at pressures well above the bulk phase tran sition pressure. Figure 3b further shows that the shift in Raman frequency with pressure (ow/oP) is essentially identical to the bulk LO phonon shift. The linear fit to the data agrees well with the value of ow/oP reported for bulk CdS (23, 27). Because the Raman shift can be directly related to the volume compressibility through the mode Griineisen parameter, this result indi cates that the potential energy as a function of unit cell volume is identical in both the harmonic and the first anharmonic term in tetrahedrally bonded bulk and nanocrystal systems. This is a very important result a)
·····
300
5 GPa
900
700
500
Wavenumbers (em·')
60
b) �
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50
a . .
2LO
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3
5
11
40
. .
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pressure (GPa)
.
8
4 High-pressure Se EXAFS obtained on SA-nm diameter CdSe nanocrysta\s. (a) Background-corrected X-ray absorption data presented as k*X(k). A clear shift in the EXAFS oscillations can be seen at the phase transition. (b) (Top) Shift in Cd-Se bond length with pressure. A dramatic increase in bond length is observed at the phase transition. (Bottom) Volume change with pressure calculated from the changes in Cd-Se bond length assuming a wurtzite to rock salt phase transiti on. The solid line is a fit to the Murnaghan equations of state with Bo 37 ± 5 GPa and Bo' 11 ± 3. (D ata from Reference 32.) Figure
=
=
yields a measure of the bulk modulus (inverse volume compressibility) and its derivative with respect to pressure: Bo 37 ± 5 GPa, Bo' 1 1 ± 3 . The values obtained are somewhat different than those for bulk CdSe: Bo 55 GPa. This is because deviations from linear compressibility are small in the region between 0 and 3 GPa, where bulk CdSe is stable. Nanocrystals, however, continue to persist in the wurtzite structure to much higher pressures. As the Cd-Se bond length decreases, repulsive forces dominate, and the compressibility decreases. This information is interesting for two reasons: First, compressibility data has proven very important in thermo dynamic analyses of high-pressure nanocrystal phase transition data. Second, nanocrystals provide a unique opportunity to study the tetra hedrally bonded Cd-Se potential in the region beyond the bulk stability limit. =
=
=
X-ray Diffraction on CdSe and Si For the same reasons that EXAFS is ideally suited for studying very small nanocrystals, it is not the best technique for studying phase transitions in general. EXAFS probes only short-range order, so although it can provide
605
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NANOCRYSTALS AT HIGH PRESSURE
information about the high-pressure phase structures, it cannot provide information about the degree of crystallinity in the newly transformed nanocrystals. For this, high-pressure X-ray diffraction (HP-XRD) is required. Because of the small sample volume and limited crystallinity of semi conductor nanocrystals at high pressure, HP-XRD experiments were car ried out at a synchrotron. The experiments were performed in an angle dispersive geometry with a modified diamond anvil cell equipped with either a diffraction slit or a Be backing plate to allow the diffracted X rays out of the cell. Figure 5 shows the quality of X-ray diffraction data that can be obtained on CdSe nanocrystals (36, 37). Diffraction peaks are broadened by the finite size of the crystallites (Debye-Scherrer broadening). Despite this fact, low-pressure phase peaks can still be easily indexed to a wurtzite structure (38). As with the other techniques presented here, the elevation in phase transition pressure can be clearly seen: The wurtzite phase is (111) (200)
a)
J J
(220)
b)
J
(111) (200)
(220)
� �
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t 8
12
16
20
24 28 2 theta
32
40 8 12 16 (wavelength 0.9939 A)
36
t
t
(220) (311) 20
24 28 2 theta
32
36
40
Figure 5 High-pressure X-ray diffraction data obtained on 4.2-nm diameter CdSe nano crystals. (a) Diffraction patterns collected with increasing pressure. The system transforms from a wurtzite structure to a rock salt structure at approximately 6.3 GPa. (b) Diffraction patterns collected with decreasing pressure. The system transforms from the rock structure to a mixed zinc blende-wurtzite phase at approximately 1 GPa. Double-headed arrows mark diffraction from the metal gasket of the high-pressure cell. Pressures and diffraction peak indexing are indicated on the figure. (Figure from Reference 36.)
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606
TOLBERT & ALIVISATOS
stable until approximately 6 GPa, well above the bulk transition of 3 GPa observed with increasing pressure (upstroke). In contrast to other techniques, however, much new information can be gained about the high pressure phase. In the first place, the diffraction peaks in this new phase unambiguously index to a rock salt structure (39a,b). Although this con clusion has been suggested by other techniques, only diffraction proves it. Upon release of pressure, the transformation is marked by significant hysteresis. The tetrahedral phase does not recover until approximately 1 GPa, and then it appears to be a mixture of the closely related zinc blende (40) and wurtzite structures.3 The recovery of this mixed phase can be observed only by diffraction. More importantly, a careful analysis of diffraction peak widths in both the wurtzite and rock salt structures shows that no broadening of the diffraction peaks, and thus no decrease in the crystalline domain size, has occurred upon phase transition. This result has two important consequences: In the first place, it has implications for the mechanism of transformation. This is discussed in some detail below. In addition, it indicates that high pressure can be used to synthesize highly crystalline particles in new bonding geometries. Electronic properties of these new samples can then be studied to learn about the effects of finite size of a variety of nanocrystal systems (42). The real power of X-ray diffraction, however, can be seen in samples such as Si nanocrystals. Bulk Si undergoes three solid-solid phase tran sitions between 1 1 and 16 GPa [diamond to f3-Sn to Imma to primitive hexagonal ( 1 9a-h)], with a fourth transition to a metastable phase upon release of pressure [f3-Sn to Be8 (19a-h)]. All of the structures other than the semiconducting diamond phase are metals, so resonance Raman (43t or optical techniques are not useful for studying these phases. Si EXAFS is not in an energy range accessible in the diamond cell. Only diffraction can be used to fully sort out the high-pressure phase behavior in Si nano crystals. Diffraction data, presented in Figure 6a, shows the complicated phase behavior displayed by 490 A diameter Si nanocrystals coated with Si02 (44). Like the metal chalcogenides, the phase transition pressure is much higher than that observed in bulk Si, a particularly striking result given the large diameter of the nanocrystals in question. The system can, however, be converted from the diamond structure (45) (Figure 6a, section 1) to the primitive hexagonal structure (Figure 6a, section 2) ( 19a-h). Whether this transition proceeds through the f3-Sn and Imma phases is 3 This
effect has been observed in recovered bulk CdSe samples as well (see 1 7b,c). Raman data has been obtained on metallic phases of bulk Si at high pressure. It is unclear whether it would be possible to collect data of this sort on nanophase materials (see 43). 4 Off-resonance
607
NANOCRYSTALS AT HIGH PRESSURE
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a)
b)
(111)
10
12
14
16 18 2 theta
(wavelength
=
20 0.62
22
A)
24
o
5
10
15
20
Pressure (GPa)
25
30
Figure 6 High-pressure data obtained on Si nanocrystals coated with Si02• (a) High pressure X-ray diffraction data obtained on 49-nm diameter nanocrystals. The crystallites are stable in the diamond phase to well above the bulk stability limit of 11 OPa (section 1 ). With sufficient pressure, the crystallites do transform to a primitive hexagonal phase (section 2). Upon partial release of pressure, the Imma phase is observed (section 3). Upon full release of pressure, the bulk BC8 phase is not observed. Instead, amorphous Si is recovered (section 4). Indexing and pressures are indicated on the figure. (b) Phase behavior for IO-nm diameter Si crystallites measured by optical absorption. The increase in 0.0. at 22 OPa marks the diamond to fJ-Sn (primitive hexagonal) transformation. The decrease in 0.0. at 5 OPa marks the fJ-Sn to amorphous Si transformation. The arrow marks the bulk Si upstroke diamond to fJ-Sn phase transition pressure. (Figure from Reference 44.)
unclear from the data. Upon partial release of pressure, an Imma phase is observed (Figure 6a, section 3) and a f3-Sn phase (not shown) (1 9a-h). Although the transition pressures are altered in the nanocrystal sample with respect to bulk Si, all of these phases can be observed in bulk as well as nanocrystalline Si. Upon full release of pressure, however, instead of recovering a Be8 phase (46), the sample appears to form amorphous Si (Figure 6a, section 4). This complicated phase behavior would be almost impossible to unravel with any technique other than diffraction. Although X-ray diffraction is not ideally suited for examining very small nanocrystals, interpretable data on CdSe nanocrystals as small as 10 A in radius can, in fact, be collected (37). This data shows the same type of structural changes observed in larger samples in Figure 5. The system transforms from a wurtzite structure to a rock salt structure and back to a tetrahedral structure (wurtzite and zinc blende structures cannot be distinguished in this size range) with no apparent loss of crystalline domain
608
TOLBERT & ALIVISATOS
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size. Further, data (not shown) indicate that the phase transition pressure is somewhat higher in these very small crystallites than in the larger sample presented in Figure 5. Obviously, the next goal in understanding the elevation in phase transition pressure in semiconductor nanocrystals is to map out the nanocrystal size dependence of the phase transition pressure. Unfortunately, diffraction is not the ideal experiment for this due to the expense and time constraints associated with working at a synchrotron. A simpler experiment that can be performed in the laboratory is needed.
Optical Absorption on CdSe, CdS, and Si For nanocrystal samples that can be homogeneously dissolved to form a clear solution, optical absorption is probably the simplest high-pressure analysis technique available. Unlike diffraction and EXAFS, electronic absorption experiments can be performed with relatively simple equipment in the lab. Unlike Raman scattering, there is usually some signature of both high- and low-pressure phases apparent in the absorption spectrum, at least for the first tetrahedral to octahedral (or quasi octahedral) trans formation. Most importantly, however, the experiments can be efficiently
repeated on a wide variety of sample sizes with small pressure steps to accurately determine phase transition pressures as a function of nano crystal size. A sample of the type of available optical absorption data is presented in Figure 7 for CdSe nanocrystals (37). In the wurtzite phase, the absorption spectra show a series of discrete features due to quantum confinement of the direct gap optical excitations (47a,b; 48). These features shift to the blue with applied pressure. At the phase transition pressure, the discrete features disappear and are replaced by a slowly rising absorption that can be assigned to the indirect gap of the rock salt phase (49, 49a). 5,6 Upon release of pressure, the discrete features are recovered. The data in Figure 7 have been corrected for changes in optical density (O.D.) caused by deformation of the diamond anvil cell metal gasket. Integration of the direct gap absorption features provides a simple method for quantitative mapping of the transition hysteresis. These data, presented in Figure 8 for three sizes of CdSe nanocrystals, show wide hysteresis curves with rela tively sharp phase transitions (36, 37). The transition pressures can be assigned to the 50% transformed point for both upstroke and downstroke transformations. 5 High-pressure optical absorption data on rock salt phases of both CdS and CdSe indicate that they are indirect gap semiconductors (see 17a--c and 18a-g). 6 Simulations of the electronic structure of rock salt phase bulk CdSe and CdS also indicate that both are indirect gap semiconductors (see 49, 49a).
NANOCRYSTALS AT HIGH PRESSURE
a)
609
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ci 0.8 0.4 c)
a 2
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/
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----
2
2.4
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1.6 recovered
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d 0.8 0.4 0
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-'
2.4 2.8 Energy (eV)
3.2 " .' ' ..
3.2
Figure 7 High-pressure optical absorption obtained on 2.S-nm diameter CdSe nanocrystals. (a) Shift in the wurtzite phase absorption with pressure. (b) Disappearance of the discrete wurtzite absorption features (solid line) at the phase transition pressure and appearance of the indirect gap spectra of the rock salt phase (large dash line). (c) Recovery of the discrete zinc blende-wurtzite features upon release of pressure (solid line). The original atmospheric pressure spectra is included for comparison (small dash line). Pressures are indicated on the figure. All spectra have been corrected for changes in O.D. due to deformation of the high pressure cell gasket. (Figure from Reference 37.)
For indirect gap semiconductors such as Si, the optical absorption experiment is somewhat more difficult. The lack of discrete features in either the low- or high-pressure phases makes the phase transition pressure somewhat less obvious. The large increase in 0.0. at the semiconductor to metal transition pressure is, however, a clear indication of structural change. Figure 6b shows data obtained on 100 A diameter Si nanocrystals (44). A clear upstroke transition, well above the bulk Si upstroke phase transition, is observed. Note that the recovery of a semiconducting state upon release of pressure is consistent with high-pressure diffraction data obtained on these samples: Amorphous Si is a semiconductor, while the metastable phase BC8 recovered from the bulk is a semi-metal. These Si absorption data were not corrected for changes in 0.0., resulting from deformation of the diamond cell gasket. Because of its simplicity, optical absorption has been applied to a wide variety of samples [e.g. CdSe (24, 36, 37, 50), CdS (22, 23), and Si (44)). Owing to past synthetic constraints, however, a systematic size dependent
610
TOLBERT & ALIVISATOS
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� 0.4 0 .2 0.1
2
4
6 P(GPa)
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10
Figure 8 Hysteresis curves for three sizes of CdSe nanocrystals. The data are obtained by integration of the direct gap absorption features like those presented in Figure 7. Changes in both the upstroke and downstroke transition pressures are observed with variations in the nanocrystal size. Phase transition pressures are assigned to the midpoints of these hysteresis curves (average of upstroke and downstroke 50% transformed pressure). (Figure from References 36 and 37.)
set of data had been collected only for CdSe. The hysteresis shown in Figure 8 adds some difficulty to determining the size dependence of the phase transition pressure. Should the transition pressure be assigned to the upstroke transition pressure? The midpoint of the hysteresis curves? The downstroke transition pressure? Although there is some argument in the literature over which is the most meaningful point, the existence of broad hysteresis curves indicates the presence of a significant barrier to phase transformation and thus suggests that the midpoint of the curves is the most appropriate. Under this assumption, optical absorption can be used to map out the size dependence of the phase transition pressure. The data presented in Figure 9 for CdSe nanocrystals in pyridine show a monotonic increase in phase transition pressure with decreasing nano crystal size (36, 37). In the next section, we examine this trend and its implications for phase transitions in nanocrystals in general.
SURFACE ENERGIES AND PATH EFFECTS In the section above, we examined a variety of methods that can be used to observe phase transformations in nanocrystals. By combining the results
NANOCRYSTALS AT HIGH PRESSURE
611
5 T-------�
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4.8
3.6 +-...--,....,"-"-.--T-r--,----r--T>-...,.---l 3.4 10 12 14 16 18 20 22 radius (A) Figure 9
Crystallite size dependence of the wurtzite to rock salt phase transfonnation
pressure for CdSe nanocrystals. The data are obtained from the midpoints of hysteresis
curves like those shown in Figure 8. The line is a fit to a thennodynamic model that describes the elevation in transformation pressure in tenns ofthe differences in surface energies between the wurtzite and rock salt phases. (Figure from References 36 and 37.)
of all these experiments, it should be possible to develop a general under standing of phase transitions in finite systems.
Surface Energies By analogy with the theories for melting in finite systems, the elevation in
solid-solid phase transition pressure observed in semiconductor nano crystals can be interpreted in terms of changes in surface energies. In the case of melting, the reduced surface energy in the liquid phase caused a decrease in melting temperature. The increase in solid-solid transition pressure can, by analogy, be assigned to an increase in surface energy in the high-pressure phase nanocrystals. The reality of a surface energy associated with solid phases of semi conductor nanocrystals can be seen experimentally from wurtzite-phase X-ray power diffraction at atmospheric pressure (51). Figure 10 presents the fractional change in lattice constant (relative to bulk CdSe) versus size for CdSe nanocrystals (37). The data show a small but systematic decrease in lattice constant with decreasing nanocrystal size. This change in lattice constant can be related to a surface energy through the Laplace Law. Simply put, the Laplace Law results from a macroscopic theory that relates a surface tension to a surface pressure. This pressure is then converted to a lattice change through a knowledge of the volume compressibility. Although the Laplace Law assumes only radial forces, a large sim plification for a structure with directional bonding like a wurtzite-phase
612
TOLBERT & ALIVISATOS
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