J . Phys. Chem. 1990, 94, 4356-4363
4356
are consistent with HPC aggregation below the LCST.31 Application of similar techniques to PNIPAAM reveals no such aggregation? a result consistent with the smaller cooperative unit calculated for the PNIPAAM transition on the basis of calorimetric measurements.
T
*CP
l 38.4 "C
l
l
l
43.4 "C
l 48.4
"c
l
l
l
53.4 'C
TEMPERATURE ("C) Figure 7. Microcalorimetric endotherms for HPC (4.0 mg/mL). Samples D. E, and F are identified in Table 11.
as the molecular weight rises from IO5 to lo6. The enthalpy and width of the transition appear to be chain length independent. In each case, the cloud point agrees more closely with the temperature of the onset of the endotherm than with the peak value. The size of the cooperative units calculated for these samples increases with increasing molecular weight from ca. 3 to 8 to 35 polymer chains, suggesting an important role for chain aggregation. Winnik has reported results of nonradiative energy-transfer experiments that
Conclusions Calorimetry has been successfully applied to study LCST phenomena in polymer blend^;^^,^^ its extension to polymer solutions is facilitated through application of very sensitive instrumentation previously used to study phase transitions in proteins, lipids, and nucleic a ~ i d s . ' ~ - Good ~ ~ , ~agreement ~ - ~ ~ is typically found between transition temperatures and widths determined by cloud point and calorimetric methods; moreover, the availability of thermodynamic parameters permits a more thorough description of the aqueous solution behavior of these polymers and the perturbation of that behavior by ionic cosolutes. The transition enthalpies found for PNIPAAM, PVME, PPG, and HPC are all consistent with the loss of ca. 1 hydrogen bondlrepeating unit on phase separation. The sizes of the cooperative units of these transitions appear to be of the order of the size of the polymer chain, although the uncertainty of the estimated molecular weights and the polydispersities of the polymers preclude precise comparison. The exception is HPC, where the size of the cooperative unit exceeds that of the chain, consistent with preaggregation of HPC below the LCST. Acknowledgment. This work was supported by a National Science Foundation Predoctoral Fellowship to H.G.S. and by a grant from the U S . Army Research Office (DAAL03-88-K0038). Registry No. PVME, 9003-09-2; PPG, 25322-69-4; HPC, 9004-64-2; PNIPAAM, 25 189-55-3; Na2S04,7757-82-6;NaBr, 7647-1 5-6; NaSCN , 540-72-7. (32) Ebert, M.; Garbella, R. W.; Wendorff, J. H. Makromol. Chem., Rapid Commun. 1986, 7 , 65. ( 3 3 ) Percec, V.; Schild, H. G.; Rodriguez-Parada, J . M.; Pugh, C. J . Polym. Sci., Part A: Polym. Chem. 1988, 26, 935.
Dynamics of Electron-Hole Pair Recombination in Semiconductor Clusters M. O'Neil, J. Marohn, and G. McLendon* Department of Chemistry, University of Rochester, Rochester, New York 14627 (Received: September 8, 1989; In Final Form: December I, 1989)
The kinetics of radiative electron-hole pair recombination in CdS and Cd3As, clusters (where the radius of the cluster is smaller than the de Broglie wavelength of photogenerated excitons) were studied with picosecond photon counting luminescence decay measurements over wide temperature and energy ranges. The decay profiles were quantitatively examined with several models. The decays are composed of two distinct time regimes, each with very different temperature and emission energy dependence. The first (fast) regime is attributed to an unusually efficient thermal repopulation mechanism. The second (slow) component is well described by a distributed kinetic model. The kinetic behavior of wide (CdS) and narrow (Cd3As2) band gap materials was remarkably similar when composed of clusters in the quantum confined regime.
Introduction The study of the transition between molecular and bulk properties in semiconductors has been facilitated by the recent introduction of chemical synthesis of stable clusters with controllable mean diameter of less than 50 Numerous reports ( I ) Rossetti, R.: Elison, J.; Gibson, J.; Brus, L. J . Chem. Phys. 1984, 80, 4464. ( 2 ) Fojtik, A.; Weller, H.; Koch, U.;Henglein. A. Ber. Bunsen-Ges. Phys. Chem. 1984, 88,969. (3) Weller, H.; Koch, U.:Gutitrrez, M.; Henglein, A. Ber. Bunsen-Ges. Phys. Chem. 1984, 88, 649.
0022-3654/90/2094-4356$02.50/0
of the photophysical and optical properties of these materials have appeared recently, with particular emphasis paid to the quantum restriction effect on the static optical absorption and luminescence of cadmium sulfide and related material^.^-^ (4) Dannhauser, T.; O'Neil, M.; Johansson, K.; Whitten, D.; McLendon, G . J . Phys. Chem. 1986, 90,6074. (5) Brus, L. J . Chem. Pkys. 1983, 79,5566. (6) Brus. L. J . Chem. Phvs. 1984. 80. 4403. ( 7 ) Nedeljkovic, J.; Nenahovic, M'.; Micic, 0.; Nozik, A. J . Phys. Chem. 1986. 90. 12.
(8) AIiv&tos, A. P.; Harris, A.; Levinos, N.; Steigerwald, M.; Brus, L. E. J . Ckem. Phys. 1988, 89,4001. (9) Mills, G.; Meisel, D. J . Colloid Interface Sci. 1987, I20, 540.
0 1990 American Chemical Society
Electron-Hole Pair Recombination When the overall dimensions of a lattice become smaller than the de Broglie wavelength of photogenerated excitons, at least two effects become increasingly important. First, the exciton energy is perturbed due to quantum confinement. Second, the proportion of cluster surface area to volume greatly increases. For example, in a 30-A diameter CdS cluster, at least 15% of the total atoms are on the surface. This latter effect is compounded by the irregular nature of the cluster surface. Prepared with the classical colloid chemistry technique of arrested precipitation, the CdS cluster lattice must terminate with a large number of sulfur vacancies. In fact, the surface might be viewed as an ensemble of dangling and terminated bonds, as well as screw and other point defects, all in various forms of coordination with the solvent and associated ionic species. Therefore, the cluster may not be viewed as a perfect, electronically neutral and spherically confined lattice subject only to the excitonic effects seen in bulk single crystals. Several qualitative reports of the dynamics of e-/h+ pair recombination in clusters have been p r e ~ e n t e d . ~ ~Among ’ ~ ~ ’ ~the important observations are luminescence decay curves which are strongly multiexponential. This behavior closely resembles a distributed kinetic model, such as Hopfield’s model of donoracceptor emission in the bulk crystal.I2 Several different physical models have been proposed to account for these kinetics, ranging from thermal repopulation to Hopfields “distant pair” model. However, detailed tests of any model are not available, and no attempt to test the decay line shape in terms of quantitative predictions of these models has yet been presented. Therefore in the present work, we have measured photogenerated e-/h+ pair recombination kinetics in several CdS clusters with picosecond resolution, over a wide range of temperatures. The data cannot be fit to any purely distributed kinetic model. At least two fundamentally different mechanisms must exist for the radiative annihilation of the photogenerated e-/h+ pair. Two clearly differentiable time regimes, with very different temperature and surface environment dependence, can be distinguished in the decay spectra. The first is very fast (5 ns).
Luminescence Decay of Cadmium Arsenide Room Temperature. Qualitatively, the dynamical behavior of Cd3As2 is remarkably similar to CdS. Both types of clusters exhibit two regimes of decay; both show decay curves that become independent of emission energy when the cluster is surface modified. Both have nearly identical temperature dependence. (17) OConnor, D.; Sumitani, M.; Takagi, Y.; Nakashima, N.; Kamogawa, K.; Udagawa. Y . : Yoshihara, K. J . Phys. Chem. 1983,87, 4848.
TABLE VI: Decay Rate of a Function of Energy for Surface-Modified Cd3As2 L l fast regime (x109) slow regime 520 1.48 3.8 x 107 560 1.40 3.6 x 107 600 1.31 2.9 x 107 640 1.22 2.5 X lo6 680 0.18 2.6 X IO6 720 1.10 2.3 X IO6 TABLE VII: Decay Rate of Both Regimes as a Function of Temperature for 150-A C d d s z in 2-Propanol fast regime (all x109) slow regime T, K 550nm 650 nm 550nm 650 nm 4.8 1.73 1 .oo 17 I .73 1.03 45 1.85 1.17 1IO 2.02 1.22 3.5 x 106 5.9 x 105 1.38 1.1 x 107 130 2.13 156 2.21 1.49 3.5 x 107 5.5 x 106 195 2.27 1.70 8.3 X IO7 1.8 X IO7 2.2 x 107 1.75 1.6 x IO* 225 2.33
Cadmium arsenide has a much lower bulk band gap than CdS; thus the onset of the confinement effect occurs at a much larger radius. To observe the same HOMO LUMO transition seen in 45-A CdS, Cd3As2need be 180 A in diameter. Table V presents decay rate as a function of emission wavelength for 180-A Cd3As2 clusters prepared in H,O/HMP. Effect of Surface Treatment. Arsenides are unstable in acidic media. The pH of a typical aqueous preparation is 11.5, so a reagent other than KOH must be chosen to modify the surface trap energy. Triethylamine (TEA) works very well, increasing &,,, by a factor of 4.5 at 2 pM TEA. Such a preparation generated the values in Table VI. The cluster size was 180 A. Temperature Dependence. In this case only two emission wavelengths were monitored; 550 and 650 nm. The emission maximum of the sample was at 600 nm. The observed decay rates computed with the Williams-Watts function are presented in Table VII.
-
Quantitative Decay Models The Williams- Watts Model. Previous attempts to describe the kinetics of recombination emission in semiconductor clusters Not relied on simple double- (or triple-) exponential surprisingly, theses attempts are largely unsuccessful, as noted by the authors. Standard statistical tests,14 e.g., xz and the Durbin-Watson parameters, indicate inadequate fits. This failure surely stems in part from an inadequate physical model. For example, while Brus3 et al. invoke the Hopfield “distant pairs” modelI2 to explain the observed kinetics, this model requires distributed kinetics, rather than the two (or three) exponentials reported. Important in this context are Ware’s detailed experiments, which showis that triple-exponential fits may provide a seemingly good fit to data which actually are composed of a distribution of rates. Thus, Ware et al. suggest that in the absence of independent proof for several independent (chemical) components, such data should be treated with distributed kinetic models. This approach reduces the number of variable parameters required to describe that data. Furthermore, such models provide better tests of the underlying physics of the kinetic process(es). Therefore, as a first approach to apply distributed kinetic models to such luminescence data, the Williams-Watts functionig (a stretched exponential) was used: I ( t ) = ae-(r/r)’
(1)
Here a is a preexponential constant and (3 is a real number such (18) James, D.; Ware, W. R. Chem. Phys. Lett. 1985, 120, 455 (19) Lindsey, C.; Patterson, G. J Chem. Phys. 1980, 73, 3349.
The Journal of Physical Chemistry, Vol. 94, No. 10, 1990 4361
Electron-Hole Pair Recombination
And we may now define the average relaxation time (here the average radiative lifetime), ( T ~as~ ) (7ww) = Pww =r( Y ,
417
133.
'
22'4.
31'4.
'
'
4d3.
'
Channel Number (0.078nr / channel) 0!2
Figure 9. Reconvolved fit (with residuals) to the Williams-Watts function for the A,, = 600 nm data from Figure 4. Values: /3 = 0.369; A = 3.14; T* = 1.69; x 2 = 6.5.
k)
Throughout, ( f w w ) has been used when a numerical estimate of decay times is required. This empirical model provides excellent compactness and precision over a wide time range. Distributed Kinetics-Thomas-Hopfield Model. The problem of e-/h+ pair recombination from trap sites in bulk semiconductors has been extensively reported, and at least one satisfactory theory due to Hopfield has been developed to treat the low-temperature radiative kinetics in large single-crystal G a P doped with AI and Si donors and acceptors.2' This model, outlined below, has been most commonly assumed to explain the complex kinetics observed for cluster emission. The authors suggest that the model is fully applicable to large single-crystal CdS. Briefly, Thomas et al. proposed that for an isolated initially neutral donor surrounded by a distribution of neutral acceptors (or vice versa)z1
( Q ( t ) )=
it)"Jv
J...!exp(-EW(n,)t)J
d3r,...d3rj...d3rn (7)
which is equivalent to
-
,me--
me.
i
where ( Q ( t ) )is the ensemble average of the probability that an electron is on the donor as a function of time, N is the number of acceptors, Vis the volume, and W(r) =
Wmax
exp[-r/Rdl
(9)
with W,,, a constant and Rd half the donor Bohr radius. Observing that N / Vis finite (corresponding to the concentration of acceptors), one may write
( Q ( t ) )= exp[4xnJm(exp[-W(r)r] - 112dr]
(10)
The intensity of light emitted as a function of time is then d I ( t ) = -z(Q(t)) (1 1)
= ( 4 n n A - ~ ( r )exp[-W(r)t]r2 dr) X /exp[4anLm{exp[-W(r)t] - l J r Zdr]) (12) For a quantum restricted system, where the volume is not infinite, we propose that one may write
I ( t ) = 14nnJ
b
W(r) exp[-W(r)t]rz dr) X (exp[4nnJb(exp[-W(r)t] -l)rz dr]l (1 3)
where r ( n ) is the gamma function and distribution function. Further 4ww(t)
2
p(7)
is a characteristic
exP[-(t/7WW)BWWl
which allows the definition of pww as (20) Kopelman, R. Science 1988, 241, 1620.
(4)
where b is the cluster diameter. This parameter was varied in an attempt to find an ideal fit, as discussed below. One other important case of the Thomas-Hopfield model occurs when donor and acceptor concentrations are equal. Conveniently, under this condition the functional form of the solution for I ( t ) is identical with eq 13 for times less than 10" s. Notice also that eq 13 is independent of temperature. In other words, it applies only to a system where thermal detrapping does not occur. Experimentally, this has been verified for bulk CdS crystals cooled to below 20 K. Therefore all attempted fits to eq 13 were for spectra obtained at 4.8 K. At this temperature, the initial decay component is broadened by a factor of 1.7 relative to room tem(21) Thomas, D.G.; Hopfield, J. J.; Augustyniak, W. Phys. Rev. A 1965, 202, 140.
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The Journal of Physical Chemistry, Vol. 94, No. 10. 1990
O'Neil et al.
LUMO
-t 0
073
1 2
Figure 11. Fit (after the initial 590 ps) using the Thomas-Hopfield model (eq 3.20) with the A,, = 550 nm at 12 K data presented in Table 111. The value of xz was minimized (=1.9) when b = 38 A. The fit parameters correspond to an oscillator strength of 8.3.
perature. This is in marked contrast to the slower decay regime which slowed from ( rww)= lo8 s-I at room temperature to ( T ~ > lo6 s-] at 4.8 K. Even so, eq 13 varies too slowly at short times and does not decay to a virtual plateau as does the data. The parameter b was varied widely, and as expected, the sharpest initial decay with the most significant plateau was found for 30 8, < b < 65 A. One may fit the tail with eq 13, but not the initial decay, which is extremely resistant to any type of distributed fit presented here. Using the A,, = 560 nm data from Figure 4, the Thomas-Hopfield model works well if one fits only after the initial 590 ps, which corresponds to the fast decay regime. The fit is shown in Figure 1 1 .
Analysis and Conclusions Qualitatively and at low temporal resolution, radiative decay curves of the type presented here are fully consistent with a Hopfield-type distant pair model. Such a model has electrons and holes trappped at lattice sites at an ensemble of distances r. Only certain values of r are allowed. This suggests discrete lines in the emission spectrum where none are seen, but phonon broadening and sample inhomogeneity could easily account for their absence. Quantitatively, and at high temporal resolution, the pure distant pair model breaks down. The fast decay component, seen at all emission wavelengths, is inconsistent with pure donor-acceptortype recombination. The Williams-Watts model is sufficient to quantify the multiexponential decays presented here. The function is compact and computationally facile. It has been used throughout as an expedient, but it is not assigned physical significance in the present context. Purely excitonic decay behavior has been modeled with a fractal reaction scheme that is functionally identical with the stretched exponential.20 Thus, we have seen that the cluster dynamics are composed of two distinct regimes with different physical behaviors. The Thomas-Hopfield model2' has physical significance when applied to a bulk single crystal. The model considers a distribution of separated pairs. However, clusters are irregular. Their surface is composed of a distribution of trap energies, as well may be the lattice interior. Applying the Thomas-Hopfield model to the total luminescence decay of clusters is risky for all times, but the behavior observed here and by other workers does strongly suggest distributed kinetics. We see that applying the Thomas-Hopfield model fails at short times but succeeds for the slow decay regime. This does provide some very useful information. We now have several physical and analytical indicators separating the two regimes as two different physical processes. The first question to address is the nature of the fast component of the decay. The first possibility is that it is an excitonic emission. In a bulk lattice, emission from an excitonic state is fast (> c149 c16. (For a given guest, c14 and c16 are the same and the C,,in the two structures are generally the same.) Mixed hydrates'are formed when cages OF the same kind are occupied by two or more species. If one of the components in a gas mixture is large enough to form structure 11, then the mixed hydrate most likely will also be structure 11 as the structure 11 forming comDonent cannot fit in the relativelv smaller cages of structGe I. i t has been reported that, in t h i c a s e of theyCH4 + C3H8)mixture, the presence of
