Size Dependence of the Spontaneous Emission Rate and Absorption

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J. Phys. Chem. C 2009, 113, 6511–6520

6511

Size Dependence of the Spontaneous Emission Rate and Absorption Cross Section of CdSe and CdTe Quantum Dots Celso de Mello Donega´* and Rolf Koole Condensed Matter and Interfaces, Debye Institute for Nanomaterials Science, Utrecht UniVersity, Princetonplein 5, 3508 TA Utrecht, The Netherlands ReceiVed: December 22, 2008; ReVised Manuscript ReceiVed: March 4, 2009

In this paper, the size dependence of the band gap, of the spontaneous emission rate, and of the absorption cross section of quantum dots is systematically investigated over a wide size range, using colloidal CdSe and CdTe QDs as model systems (diameters ranging from 1.2 to 8 nm and from 2 to 9.5 nm, respectively). The size dependence of the band gap is well-described by theoretical models, and is dominated by the quantum confinement contribution (1/d2 scaling). The spontaneous emission rate increases linearly with the emission frequency for both CdSe and CdTe QDs, in good agreement with theoretical predictions. By extrapolating the frequency dependence of the emission rates to the bulk band gap values, the exciton radiative lifetime in bulk CdSe and CdTe could be estimated for the first time (viz., 18 and 20 ns, respectively). Comparison between the empirical trends and theoretical predictions provides new fundamental insights into the size dependence of the 1S(e)1S3/2(h) oscillator strengths of QDs, both for emission and absorption. The results highlight the importance of the balance between quantum confinement and coulomb interaction contributions to the size dependence of the exciton properties in QDs and offer an explanation to the long-standing discrepancies observed between the empirical size-dependent trends and the theoretical predictions. The difference between the size dependence of the radiative decay rates and of the absorption cross sections is shown to be due to the fundamental differences between the emission and absorption transitions (viz., spontaneous versus stimulated). The results are also relevant from a practical viewpoint, since they show that the molar extinction coefficients at energies far above the band gap are better suited for analytical purposes. Moreover, more extended and accurate sizing curves are provided for CdSe and CdTe QDs. 1. Introduction The remarkable size-dependent properties and chemical versatility of colloidal semiconductor nanocrystals, also known as quantum dots (QDs),1-3 have turned them into promising materials for many applications, such as light-emitting diodes,4,5 lasers,6,7 nonlinear optics,8-11 photovoltaics,12,13 and biomedical imaging.14,15 The knowledge of the concentration of QDs is of crucial importance for these applications and also for the chemical preparation of heteronanocrystals,16,17 silica-encapsulated QDs,18 nanocrystal superlattices,19,20 and QD-polymer composites.21 Optical absorption spectroscopy is probably the most convenient way to determine the concentration of chromophores in general, provided the molar extinction coefficient of the absorbing species is accurately known. This has motivated several groups to investigate the size dependence of the extinction coefficient of colloidal QDs for a number of materials, such as CdS,22,23 CdSe,23-27 CdTe,23,28 InAs,29 PbS,30 and PbSe.31 Understanding the size dependence of the absorption cross section of colloidal QDs is also important from a fundamental viewpoint. Theoretical models32-34 predict that the oscillator strength per unit (i.e., per single ion pair) of the lowest energy exciton transition for QDs in the strong confinement regime scales as 1/r3 (r is the QD radius), which implies a sizeindependent oscillator strength per QD. However, most of the experimental results23-26,30,31 are in contradiction with these theoretical predictions. Further, large discrepancies are also observed between the empirical size-dependent trends reported * To whom correspondence should be addressed. Phone: +31-302532226. Fax: +31-30-2532403. E-mail: [email protected].

by different groups, even for the same material. The size dependence of the molar extinction coefficient (ε) per QD of the lowest energy exciton transition is usually expressed in the form of a power law (ε ) crn, where c is a constant), with exponents varying from zero (i.e., size-independent, as theoretically predicted) to ∼3 for different materials or groups (e.g., n ) 0 for CdTe,28 CdS,22 and InAs;29 n ) 1 for CdSe26 and PbSe;31 and 2 < n < 3 for CdS,23 CdTe,23 CdSe,23-25 and PbS30). The discrepancies between the results reported by different groups are widely recognized in the literature23,27,29-31 and have been attributed to the inherent experimental difficulties associated with the accurate determination of sizes and concentrations of QDs. The variety of techniques and methodologies employed further complicates the issue, since it is often difficult to precisely evaluate the combined uncertainties of all the measurements involved in the determination of molar extinction coefficients and absorption cross sections of QDs (viz., sizing by TEM or XRD techniques, sample digestion and measurement of elemental concentrations, etc.).23,27,29,30 Further, the reported values are not always fully comparable, since some authors have used peak intensities,23,24 but others employed the integrated intensities either in wavelength26 or energy.25,29 The size dependence of the exciton radiative lifetime of QDs is also of great scientific interest from both fundamental and applied viewpoints but has been much less investigated35-38 than the absorption properties, probably due to the difficulty of synthesizing QDs yielding purely radiative exciton decay. The recent availability of high-quality QDs showing singleexponential photoluminescence (PL) decay has made the

10.1021/jp811329r CCC: $40.75  2009 American Chemical Society Published on Web 03/31/2009

6512 J. Phys. Chem. C, Vol. 113, No. 16, 2009 detailed investigation of the radiative exciton decay rates possible, particularly for the model systems CdSe and CdTe. By carefully analyzing the deviations from the single exponential behavior, it has been possible to deduce quantitative information about, for example, inter-QD energy transfer rates,39-42 thiolinduced quenching rates,43 or local-field effects on the spontaneous emission rate of CdTe and CdSe QDs.44 The size and temperature dependence of the exciton lifetimes has been investigated over a broad temperature range (1.3-300 K) for CdSe QDs ranging from 1.7 to 6.3 nm in diameter.38 Three temperature regimes were identified: the “radiative regime” (below ∼50 K), the “quenching regime” (between ∼50 and ∼200 K), and the “antiquenching regime” (above ∼200 K). The latter two regimes are size-independent.38 The temperature antiquenching regime is characterized by an increase in both the PL intensity and the exciton lifetime with increasing temperature.38 This unusual behavior has been shown to be due to a phase transition in the surfactant layer attached to the QD surface.45,46 The lifetimes in the radiative regime increase with decreasing temperature and size.38 This behavior can be modeled by a thermal distribution between two states, the lower one being optically passive (the so-called “dark exciton”47), with a sizedependent energy separation and “dark” exciton lifetime.38 The frequency dependence of the room temperature exciton lifetimes of CdTe and CdSe colloidal QDs has been investigated in detail by Van Driel and co-workers.37 The authors concluded that the spontaneous exciton emission rates of both CdTe and CdSe QDs increase with the emission frequency in a supralinear way, which was explained by the thermal occupation of dark exciton states.37 The emission transition of a QD is in first approximation the reverse of the lowest energy exciton absorption transition. Therefore, it can be expected that the size (and frequency) dependence of the oscillator strengths of the emission transition and of the lowest absorption transition of a QD will be similar. To date, however, no experimental work has been reported comparing the size (or frequency) dependence of the emission rates and absorption cross sections of QDs. In this work, we investigate the frequency and size dependence of the spontaneous emission rate and of the energy integrated absorption cross section of the 1S3/2(h) f 1S(e) transition of CdSe and CdTe QDs over a wide size range (viz., 1.2-8 nm for CdSe QDs and 2-9.5 nm for CdTe QDs). The size dependence of the band gap is also addressed. The absorption cross sections (per ion pair unit and per QD) are derived from normalized absorption and excitation spectra. This allows the absolute size-dependent trends to be unraveled and quantified while avoiding the lengthy and potentially inaccurate procedures needed to obtain the absolute absorption cross sections. The values obtained from the normalized spectra can be converted into absolute values by using the bulk optical constants. The paper is organized as follows. First, the experimental methods are described, with particular emphasis on presenting and justifying the normalization procedure. Subsequently, the results are presented and discussed, starting with the size dependence of the band gap and then following with the analyses of the spontaneous emission rates and the absorption cross sections per QD and per CdSe or CdTe ion pair unit. Finally, some general conclusions of both fundamental and applied relevance are presented. 2. Experimental Methods 2.1. Sample Preparation. The QD samples investigated in this work consist of colloidal suspensions of organically capped CdSe and CdTe nanocrystals in toluene, with diameters (d) ranging from 1.2 to 8 nm for CdSe and from 2 to 10 nm for

Donega´ and Koole CdTe (size dispersion ) 4-10%). The synthesis of the CdSe QDs has been reported previously38,48 (d ) 1.2-2.7 nm and 2.5-8 nm, respectively). The preparation of the CdTe QD samples in the size range d ) 2-4 nm has been described before.46,49,50 CdTe QDs larger than 4.5 nm were grown by a modified SILAR51 procedure, using ∼3.5 nm CdTe QDs as cores (see Supporting Information for details). This method allows the growth of CdTe QDs ranging from 4.5 to 13 nm in diameter, depending on the number of “CdTe monolayers” added. Samples for optical measurements were prepared by directly dissolving the crude reaction mixture in anhydrous toluene under argon or nitrogen. Samples purified by dissolution and reprecipitation (two to three times) were also investigated for comparison and observed to differ from the as-prepared samples only in terms of the PL quantum yields (lower) and PL decay curves (faster and multiexponential). The differences between as-prepared and purified samples can be ascribed to the decreased surface passivation caused by the removal of surfactant molecules from the QD surface.52,53 Absorption spectra were affected only after excessive purification (viz., more than five cycles), in which case a small blue shift is observed, occasionally accompanied by decreased solubility. To avoid reabsorption, energy transfer between the QDs, and deviations from linearity in the absorption and PL excitation spectra, all optical measurements were carried out on samples with a low optical density (e0.05 at the lowest energy absorption maximum and e0.2 at 400 nm). 2.2. Optical Spectroscopy and Spontaneous Emission Rates. Absorption spectra in the UV-visible range, photoluminescence (PL) and photoluminescence excitation (PLEx) spectra were acquired as described before.44-46 The spectra were corrected for the instrumental response and the spectral dependence of the excitation lamp intensity. PL decay curves were obtained by time-correlated, single-photon counting via timeto-amplitude conversion, as previously described.38,44 Very low excitation fluences were used ( aB. In the strong confinement regime, the increase in Θ will induce a redistribution of the total oscillator strength per unit volume (i.e., fTOT/V) so that the fraction taken by the lowest exciton state (fSS/V) becomes strongly size-dependent,32,34

( )

fSS(r) fSS(∞) aB 3 ωSS(∞) = V V r ωSS(r)

(15)

where fSS(r) and fSS(∞) are the 1S(e)1S3/2(h) oscillator strength for a QD with radius r and for bulk, respectively, and ωSS is the 1S(e)1S3/2(h) emission frequency. The net result is that relative intensity shifts into the lowest exciton state as the QD size decreases below aB. Because fTOT per QD remains directly proportional to the QD volume, the values of fSS per QD in the strong confinement regime should depend only on the emission frequency, as long as the effective mass approximation remains valid and Coulomb interactions are negligible. Although this theoretical prediction is in good agreement with the frequency dependence of ΓRAD observed above (Figure 3 and eqs 12 and 13), the shortcomings of this simple model become evident when the frequency and size dependence of the absorption oscillator strengths are taken into account, as will be made clear below. It is interesting to note that expressions 12 and 13 allow the bulk radiative exciton lifetimes to be estimated by making ESS equal to Eg(∞) (viz., 1.75 eV for CdSe70 and 1.56 eV for CdTe71), which yields 60 ns for CdSe and 79 ns for CdTe. The radiative lifetime of an exciton in a QD is longer than in bulk because the internal electric field is reduced by dielectric screening. Therefore, the values estimated by extrapolating expressions 12 and 13 to the bulk band gap values have to be corrected by a factor [3εm/(ε1 + 2εm)]-2, where εm is the dielectric constant of the surrounding medium (viz., 2.25 for toluene) and ε1 is the dielectric constant of the semiconductor (7.8 for CdSe72 and 9 for CdTe73). This yields a radiative lifetime of 18 ns for bulk CdSe and 20 ns for bulk CdTe. These values are about 1 order of magnitude longer than those experimentally observed in bulk samples of CdSe and CdTe, consistent with the fact that the exciton decay in bulk semiconductors is dominated by nonradiative recombination processes. 3.3. Absorption Cross Sections. Figure 1 clearly shows that the relative absorption strength of the exciton transitions is redistributed as the QD size decreases, resulting in an increase in the 1S(e)1S3/2(h) oscillator strength per CdSe (or CdTe) unit, in qualitative agreement with the theoretical model discussed above.32,34 To assess whether the theoretical predictions are also

Figure 4. Size dependence of the relative energy integrated absorption cross section per unit of the 1S3/2(h) f 1S(e) transition (µ*SS) of CdSe and CdTe QDs. The solid lines are fits to the data using a power law function.

quantitatively correct, the size dependence of the integrated 1S(e)1S3/2(h) absorption cross section per unit (µ*SS) is analyzed (Figure 4). The frequency dependence of µ*SS is less suitable for a quantitative comparison to the theoretical prediction, since it follows a rather complex trend (Figure SI-3, Supporting Information). It should be noted that, although µ*SS is the relatiVe integrated 1S(e)1S3/2(h) absorption cross section per CdSe (CdTe) unit, its size and frequency dependence will reflect the absolute trends, since µ3.1eV is constant and size-independent. Moreover, the absolute values can be easily obtained by using bulk optical constants (see Section 3.4, below). The size dependence of µ*SS (Figure 4) can be well-described by a power law, although two different sets of parameters are needed to fit the whole size range of CdTe QDs. The fitting results are as follows:

µ*SS(CdSe) ) 0.001 + 0.625d-(2.07(0.10) in the d ) 1.5-8 nm range

(16)

µ*SS(CdTe) ) 0.001 + 1.22d-(2.29(0.02) in the d > 3.5 nm range (red curve in Figure 4)

(17)

µ*SS(CdTe) ) 2.75d-(3.08(0.02) in the d < 3.5 nm range (blue curve in Figure 4)

(18)

The 1/d2 dependence observed for CdSe QDs and for CdTe QDs larger than ∼3.5 nm is not consistent with the theoretical prediction, but is in good agreement with the empirical size dependence reported for the 1S(e)1S(h) absorption cross section per unit of CdSe QDs in the 2-7 nm diameter range,26 PbSe QDs in the 3-8 nm diameter range,31 and CdS QDs in the 2.4-5.6 nm diameter range.22 Interestingly, the 1/d3 dependence observed for CdTe QDs smaller than ∼3.5 nm is in good agreement with both the theoretical prediction and the experi-

Emission and Absorption of CdSe and CdTe QDs

J. Phys. Chem. C, Vol. 113, No. 16, 2009 6517 to compare the frequency dependence of σ*SS (Figure 5) with that observed for ΓRAD (Figure 3). Considering that the lowest energy absorption transition is to first approximation the reverse of the emission transition, it can be expected that the size dependence of σ*SS (and σSS) would be similar to that observed for the spontaneous emission rate, ΓRAD. The inadequacy of this simple assumption is clearly demonstrated by the dramatic difference between the trends observed in Figures 3 and 5. There are, however, several important differences between the spontaneous emission transition and the lowest energy absorption transition, the most fundamental of which being that absorption is a stimulated optical transition. According to Fermi’s Golden rule, the spontaneous emission rate depends on the density of photon modes, whereas the rates of stimulated emission and absorption do not. Therefore, it can be inferred that the absorption transition rate, Γij, is related to the spontaneous emission rate Γ** ji by (see Supporting Information for details)78

Γij = Figure 5. The relative energy integrated absorption cross section per QD of the 1S3/2(h) f 1S(e) transition (σ*SS) of CdSe and CdTe QDs as a function of the transition energy. The solid lines are fits to the data using a biexponential function.

mental dependence reported for CdTe QDs in the 2-4 nm diameter range28 and for CdS QDs in the 1.3-2.4 nm diameter range.22 The fact that µ*SS scales as 1/d2 implies that the 1S(e)1S3/2(h) absorption cross section per QD (σSS) will increase linearly with the QD diameter, as is indeed observed for CdSe QDs26,55 (see also Figure SI-4, Supporting Information) and for PbSe QDs.31 Two different size regimes can be identified for CdTe QDs (Figure SI-4, Supporting Information), consistent with the trend observed for µ*SS: a (nearly) size-independent regime for d e 6 nm (σ*SS = 24 ( 4) and an exponential regime for larger sizes. Different scaling regimes become quite evident for both CdSe and CdTe QDs if the σ*SS values are plotted as a function of the 1S(e)1S3/2(h) transition energy (Figure 5). The data in Figure 5 can be fit to a biexponential function, yielding

σ*SS(CdSe) ) 10 + 2.4 × 104e-(ESS/0.28) + 2 × 109e-(ESS/0.10)

(19)

σ*SS(CdTe) ) 23 + 3.0 × 103e-(ESS/0.29) + 1 × 1014e-(ESS/0.06)

(20)

It should be noted that fitting the data to a biexponential law is not justified a priori and is intended only as a way to capture the empirical trend defined by the data. The essential feature of the frequency (or energy) dependence of σ*SS is that it clearly consists of two regimes: a (nearly) independent regime at higher ESS and a strongly dependent regime at lower ESS. The transition between the two regimes is more abrupt for CdTe than for CdSe QDs, suggesting that the mechanisms responsible for this surprising behavior are more pronounced in CdTe. This is further supported by the fact that two different regimes are also observed in the size dependence of µ*SS (Figure 4) and σ*SS (Figure SI-4 of the Supporting Information) for CdTe QDs. Before discussing possible explanations for this observation, it is useful

gj Γji** Aji = gi ωij

(21)

Considering that ΓRAD is well-described by eq 14 (proportional to ωSS, see Section 3.2, above), eq 21 implies that σSS should be frequency (and size)-independent for QDs in the strong confinement regime, in good agreement with the behavior observed in the high-energy regime (Figure 5), particularly for CdTe QDs. Another point that should be considered is the fine structure of the lowest-energy exciton. The lowest-energy electronic absorption transition, 1S3/2(h) f 1S(e), creates an electron-hole pair occupying the lowest exciton state, 1S(e)1S3/2(h). The 8-fold degeneracy of this state is partially lifted by the combined effects of crystal field asymmetry, shape anisotropy, and electron-hole exchange interaction, yielding five levels, two of which are optically passive in the electric dipole approximation.47 The absorption transition involves all five levels, but the emission transition involves only the lowest two levels (the “dark” and the “bright” excitons47). Since most of the 1S(e)1S3/2(h) oscillator strength is carried by the two upper states,47 it follows that the oscillator strength of the 1S(e)1S3/2(h) absorption transition will be larger than that of the emission transition for all QD sizes. Further, the size (and frequency)-dependent trends will not necessarily be similar, since the oscillator strength per QD of the upper two levels may increase more rapidly with increasing size than that of the lower two levels. As discussed above, the theoretically predicted size independence of fSS per QD (neglecting the frequency dependence) is the result of a cancellation between two different effects (viz., the increase of fSS/V with increasing overlap integral Θ and the increase in the total transition dipole moment with increasing volume, see Section 3.2). This cancellation is perfect only if the increase in fSS/V with decreasing size follows exactly the theoretical 1/r3 dependence, which is predicted under the assumption that Coulomb interactions are negligible. This approximation should become less appropriate as the QD size increases, due to the enhancement of the relative contribution of the Coulomb interaction term to the exciton energy (see eqs 8 and 9, above). As Figure 2 clearly shows, the quantum confinement regime (i.e., d e 2aB) can be divided into two distinct regimes: a moderately (weakly) size-dependent regime for d smaller than 2aB but still larger than 2ae and 2ah (ae and ah are the electron and hole Bohr radii, respectively) and a strongly size-dependent regime in which both the electron and the hole are confined.

6518 J. Phys. Chem. C, Vol. 113, No. 16, 2009

Donega´ and Koole

The reasoning above explains both the trend observed in Figure 5 and the difference between CdSe and CdTe QDs. In both materials, the high-energy regime corresponds to the stronger confinement regime (i.e., r < ae and ah), in which σSS should be frequency-independent (according to eq 21 above), and the low energy range corresponds to the moderate confinement regime (i.e., ah, ae < reaB), in which the Coulomb contributions become progressively larger with increasing size, inducing an increasingly larger deviation from the expected trend. Consequently, σSS will increase with the QD size (and thus, with decreasing frequency). The trend is more abrupt for CdTe than for CdSe QDs due to stronger quantum confinement effects and weaker coulomb interactions in the former material with respect to the latter (aB, ae, and ah are 7.3, 5.7, and 1.6 nm, respectively, for CdTe; and 4.9, 3.8, and 1.1 nm, respectively, for CdSe; whereas the dielectric constants are 12 and 8 for CdTe and CdSe, respectively). The differences between CdTe and CdSe QDs are also clearly evidenced by the size dependence of the band gap (see Section 3.1, above). The results above provide new insight into the size dependence of the absorption cross section of QDs and offer a possible explanation for the long-standing discrepancies between theory and experiment that were mentioned in the Introduction. As suggested by several authors,23,27,29-31 the discrepancies can be (partially) explained by the inherent experimental uncertainties associated with the accurate determination of sizes and concentrations of QDs and also by the fact that the reported values are not always fully comparable. The present work shows that there is also a fundamental reason for the discrepancy: the size dependence of the band edge absorption strength will be quantitatively different for different semiconductor materials and for different size ranges (even for the same material) because it is determined by a subtle balance between the relative contributions of quantum confinement effects and electron-hole coulomb interactions to the total exciton energy. The theoretically predicted size independence of the absorption cross section per QD will be observed only under conditions in which quantum confinement effects dominate; namely, for sizes small enough to confine both the electron and the hole and for materials that are well-described within the effective mass approximation (i.e., for which the energy dispersion near k ) 0 is parabolic). Conversely, the increase in the relative contribution of coulomb effects (i.e., larger sizes, materials with smaller dielectric constants, or with band structures that deviate from the parabolic approximation) will enhance the size dependence. Furthermore, it is demonstrated that the differences between the frequency and size dependence of the radiative emission rates and the absorption cross sections are due to the fact that the absorption is a stimulated optical transition, whereas the emission is a spontaneous transition. 3.4. Implications for the Use of Molar Absorption Coefficients As Analytical Tools for the Determination of QD Concentrations. The quantification of the concentration of QDs in colloidal suspensions by means of optical absorption measurements requires the knowledge of the absolute molar extinction coefficient per QD (εQD). Although the integrated 1S(e)1S3/2(h) absorption cross sections discussed above are relative values, they can easily be converted into absolute µSS values, provided µ3.1eV is known. The absolute molar extinction coefficient per QD (εSS) can then be obtained by combining eqs 4 and 5, as follows:

εSS ) (µSSNunits/QDNA)/2303

(22)

The absorption cross section of QDs at high energy can be directly determined from the bulk optical constants, as demonstrated for both CdSe55 and InAs29 QDs. The absolute µ3.1eV values can thus be obtained by using

µ3.1eV )

n1 R3.1eV |F | 2 nsolv loc Nunit

(23)

where R3.1eV is the bulk absorption coefficient at 3.1 eV (2.0 × 105 cm-1 and 3.0 × 105 cm-1 for CdSe72 and CdTe,73 respectively), Nunit is the number of ion pair units per cubic centimeter, and n1 and nsolv are the real parts of the semiconductor and the solvent refractive index, respectively (viz., 2.6 for CdSe,72 2.7 for CdTe,73 and 1.5 for toluene). Equation 23 is obtained from the expression derived by Ricard et al.26,29,79 for the absorption cross section of a QD embedded in a solvent by dividing it by VQD and Nunit. The local-field correction factor, Floc, is given by79

Floc )

3msolv2

(24)

m21 + 2msolv2

where m1 and msolv are the complex refractive indices of the semiconductor and the solvent, respectively (viz., 2.8 for CdSe,72 3.5 for CdTe73). Since the absorption of toluene at 3.1 eV is negligible, msolv can be replaced by nsolv (viz., 1.5). The absolute µ3.1eV values for CdSe and CdTe QDs are then calculated to be equal (viz., µ3.1eV ) (5.8 ( 0.1) × 10-18 cm2), which implies that the relative µ*SS values obtained from the normalized absorption and excitation spectra of CdTe and CdSe QDs can be directly compared. The calculated µ3.1eV is in good agreement with the value experimentally determined by Leatherdale et al.26 for CdSe QDs (7 × 10-18 cm2). This confirms that the absorption cross section of QDs at high energies can be obtained from bulk parameters and, therefore, that the relative 1S(e)1S3/2(h) absorption cross sections reported here can easily be converted into absolute values (viz., µSS ) µ*SSµ3.1eV, σSS and εSS can be calculated using eqs 5 and 22, respectively). Nevertheless, this is not necessary for the analytical determination of QD concentrations, since the use of the molar extinction coefficient at 3.1 eV is more advantageous. The molar extinction coefficient at 3.1 eV (ε3.1eV, in M-1 cm-1), for both CdTe and CdSe QDs, is given by

[

ε3.1eV ) (5.8 ( 0.1) × 10-18 × NA × z

( )

]

VQD /2303 ) VUC

zVQD × (1.52 ( 0.03) × 103 VUC

(25)

where NA is Avogadro’s number, VQD is the QD volume in cubic centimeters, z is the number of unit ion pairs per unit cell (viz., four for zincblende and two for wurtzite crystal structures), and VUC is the volume of the unit cell (viz., CdSe: 2.217 × 10-22 cm3 for zincblende and 1.296 × 10-22 cm3 for wurtzite;80 CdTe: 2.723 × 10-22 cm3 for zincblende80 and 1.568 × 10-22 cm3 for wurtzite81). The advantages of using the extinction coefficient at high energy (i.e., far above the band edge) have been recognized by several authors26,29,31 but are not yet widely accepted. The results presented in the present work strongly support the use of ε3.1eV rather than εSS for analytical purposes, and highlight that caution

Emission and Absorption of CdSe and CdTe QDs should be exercised when using the band-edge molar extinction coefficient to determine the concentration of QDs. The discussion above clearly shows that σSS is inherently sensitive to the size dispersion of the QD ensemble, and this introduces potentially large errors in the concentrations determined, especially if peak intensities and wavelength scales are used (as proposed in, e.g., ref 23). The absorption cross section per QD at energies far above the band edge scales linearly with the QD volume, making it a more reliable parameter, irrespective of size dispersion. Therefore, the use of ε3.1eV rather than εSS is strongly recommended. This has the additional advantage of increasing the detection sensitivity, since ε3.1eV is much larger than εSS. It should, however, be emphasized that the optical densities must be very low (e0.15 at 3.1 eV) to avoid errors due to deviations from linearity. 4. Conclusions The results presented in this paper provide new insights into the size dependence of several fundamental properties of QDs, using CdSe and CdTe QDs as model systems. The size dependence of the band gap is dominated by the contribution of quantum confinement effects for both materials, with a small contribution of Coulomb interactions in the case of CdSe QDs. The spontaneous emission rate is observed to increase linearly with the emission frequency for both CdSe and CdTe QDs, in good agreement with theoretical predictions based on the effective mass approximation. By extrapolating the frequency dependence of the emission rates to the bulk band gap values, the exciton radiative lifetime in bulk CdSe and CdTe was estimated to be 18 and 20 ns, respectively. The size and frequency dependence of the integrated 1S(e)1S3/2(h) absorption cross section per unit (µSS) and per QD (σSS) follows a more complex trend, which agrees with the theoretical predictions only for sizes small enough to confine both the electron and the hole. The deviation from the theoretically expected trend is ascribed to the contribution of Coulomb interactions, which is larger for CdSe than for CdTe QDs, due to the smaller values of aB, ae, ah, and ε for CdSe. The deviation can be ascribed to the progressive increase in the relative contribution of Coulomb interaction terms (viz., effective coulomb interaction, polarization energies, and electron-hole correlation), with increasing QD size. The difference between the size and frequency dependence of the radiative decay rates and of the absorption cross sections is shown to be due to the fact that the absorption is a stimulated optical transition, whereas the emission is a spontaneous transition. The results reported here have relevant implications also from a practical viewpoint, since they imply that the molar extinction coefficients at high energies (i.e., far above the band edge) are better suited for analytical purposes. Acknowledgment. The authors thank Andries Meijerink (Utrecht University) for stimulating discussions. Financial support from the division of Chemical Sciences (CW) of The Netherlands Organization for Scientific Research (NWO) (TOPGrant 700.53.308) is gratefully acknowledged. Supporting Information Available: Details concerning the preparation of larger CdTe QDs, derivation of the relation between absorption and spontaneous emission rates, examples of multigaussian fits to absorption spectra, and figures showing the frequency dependence of µ*SS and the size dependence of σ*SS are provided. This material is available free of charge via the Internet at http://pubs.acs.org.

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