J. Phys. Chem. C 2008, 112, 9261–9266
9261
Linear Absorption and Molar Extinction Coefficients in Direct Semiconductor Quantum Dots Jinjun Sun and Ewa M. Goldys* Department of Physics and Engineering, Macquarie UniVersity, TalaVera Road, North Ryde, NSW, 2109, Australia ReceiVed: January 24, 2008; ReVised Manuscript ReceiVed: April 9, 2008
We present a simple theoretical analysis of the linear absorption coefficient due to interband transition in quantum dots of direct bandgap semiconductor materials aimed at quantification of the quantum dot concentration in colloidal suspensions. We establish the relationship of the linear absorption coefficient thus calculated with the universally used molar extinction coefficient to be able to compare our theoretical predictions with experimental studies. On the basis of this foundation, we develop a simple approach to calculate the value of the coefficient using empirical values taken from the extinction measurements and basic physical parameter of materials. We explain the long-standing discrepancies between the theories of quantum dot absorption and experiments as well as the trends of the extinction coefficient with varying quantum dot size. 1. Introduction Colloidal semiconductor quantum dots have recently gained prominence because of an increasing range of applications, such as in hybrid chips, LEDs, lasers, solar cells, and biomedical fluorescence labeling.4 In most of these applications, it is essential but difficult to accurately determine the actual concentration (number per unit volume) of the quantum dots in solution or in a medium. Optical determination of the extinction coefficient of a quantum dot solution could be an important method for assessing this concentration due to its convenience and accuracy, as the concentration may be easily obtained from the UV-vis absorption spectra of the sample. The challenge is, however, to accurately establish the relationship between quantum dot concentration and its extinction coefficient. The most common method so far involved empirical calibration of UV-vis transmission by using atomic absorption.19 In this method, quantum dots are dissolved in a suitable acid and the concentration of ions is measured. The quantum dot concentration is then calculated by using the average size of the quantum dots. Another method to determine the concentration is through the mass of reagents and the reaction yield (see for example ref 8). The accuracy of these methods is limited, with major discrepancies observed between various works. These can be clearly seen in several recent publications describing the results of UV-vis measurements of the molar extinction coefficient in colloidal quantum dots.8,11,13,19 Thus, the problem of calibration of the molar extinction coefficient needs to be addressed in a different way. We approached this problem through quantum mechanical modeling of the molar extinction coefficient. Our work focuses at establishing a quantitative relationship between the absorption coefficient of a single quantum dot in a simple model and the experimentally observed extinction spectra. Here, we draw on the strength of the theory of absorption, which not only gives the spectral dependence of the linear absorption and molar extinction coefficient, but also predicts their absolute magnitudes without any adjustable parameters. Moreover, the magnitude of the absorption coefficient thus obtained depends on values of certain semiconductor material parameters which are now well-established, such as the momentum matrix element for dipole transitions as well as on empirical parameters, which can
be easily measured from the spectra such as peak energies and peak widths. This is an important step forward compared to earlier results. We finally arrive at a quantum mechanical expression for the extinction coefficient which makes it possible to predict its characteristic values and theoretically calibrate the quantum dot concentration. It also helps explain the observed trends in the molar extinction coefficient with varying quantum dot diameter and in different quantum dot materials. The comparison of our calculated values and trends with those empirically observed in earlier studies made it possible to give better predictions of the former and explain the observed discrepancies. 2. Absorption of a Quantum Dot The linear absorption coefficient R of a bulk material commonly used in physics and material science is an intrinsic material parameter measured directly by passing light through a plane-parallel slab of known thickness and measuring how quickly it decays. In analytical chemistry, a similar experiment of passing light through a solution of particles or molecules is described in terms of the molar extinction coefficient ε which implies a certain value of absorbance of each individual particle or molecule. One also needs to be conscious of scattering effects, which for colloidal quantum dots can be ignored, as well as possible absorption by the solvent, which can be fully accounted for when necessary.8,14 The linear absorption coefficient of a material is calculated using the Fermi Golden rule which gives the probability of a transition from an initial to a final state
Tiff )
2π |〈f|H ′ |i〉|2F p
(1)
where Tiff is the transition probability, p is the Plank constant, i and f are the initial and final states, respectively, H′ expresses the interaction between light and electron states causing the transition, F is the density of final electronic states, and the value of |〈f|H′|i〉| is related to the interband momentum matrix element which varies for different materials. Such transitions remove incident photons from the incoming light beam which is experimentally observed as absorbance.
10.1021/jp800700m CCC: $40.75 2008 American Chemical Society Published on Web 06/03/2008
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Since we are often interested in absorbance at a specific wavelength, one needs to sum over all possible states which contribute to absorbance at a given energy; such a consideration leads to a known expression for absorption which includes the density of states in the material. The density of states can be easily described analytically for ideal three-, two-, one-, and zero-dimensional materials. For realistic QDs which are neither fully 3D nor fully 0D, it is too complicated to be expressed in analytical form, although a simplified relationship can be produced in the free electron gas approximation.17 In this work, we have chosen a spherical infinite well model of a QD which implies a simple 0D expression:
N0D(E) )
1 VQD
∑ (2l + 1)δ(E - En)
(2)
πe2Epap 1 RQD(pω) ) 2m0cnrε0ω VQD
πe2Epap 1 RQD(pω) ) 2m0cnrε0ω VQD
∑ (2l + 1)δ(pω - Enl)
(3)
n,l
where ap is the average over directions of light polarization (2/3 for lamp light), pω is the energy of the incident light, e is the electron charge, Ep is the momentum matrix element of the bulk material, m0 is electron mass, c is the speed of light in vacuum, nr is the refractive index of the quantum dot material which varies with wavelength, ε0 is the permittivity of vacuum, p is Planck’s constant divided by 2π, En is the nth transition energy, and l is the corresponding degeneracy. The delta function can be replaced by a Gaussian distribution which reflects the size nonuniformity of QDs. This leads to a distribution of quantized state energies,1,16 characterized by the widths ∆Enl
n,l
×
nl
(
exp -
(pω - Enl)2 2 2∆Enl
)
(4)
The relationship between the molar extinction ε(pω) and linear absorption coefficients for a homogeneous solution of QDs is (see Appendix):
ε(pω) )
NAVQDRQD(pω) ln(10)
(5)
We can thus obtain ε(pω), which can be expressed as
n,l
N0D(E) is the density of states for a zero-dimensional material, and VQD is the volume of a single quantum dot which, in practice, can be evaluated by averaging the volumes in the batch of quantum dots to account for size dispersion. Such a spherical infinite well model is reasonably well justified for colloidal quantum dots, which, in the first approximation, can be regarded as small spheres of bulk semiconductor material. The charge carriers in such QDs are confined by potential barriers given by the material work function which is large for common QD materials such as CdSe (5.35 eV 9). Such a model is more suitable for larger QDs compared to the lattice constant, which can be viewed rather as spheres of a bulk material confined by infinite potential boundaries than molecular clusters. When adopting this model, we must be conscious however that its predictions of confined-state energies are only approximate; thus, instead, we will be using the empirically observed transition energies. The alternative approach based on tight binding calculations has been extensively discussed in the literature.10 This method is more accurate compared to ours and it can yield correct confinement energies; however, it is not easily applied to the absorption coefficient whose calculations require not just accurate energies but also accurate wave functions, which cannot be directly compared with experiment. Extensive numerical calculations based on a number of poorly known parameters are also necessary in tight binding calculations; thus, its superiority for calibrating the experimental data is unclear in principle. Within our infinite barrier model, we use the envelope function approximation.2,5,12 Because the lattice constant of the quantum dot is almost the same as in a bulk semiconductor, we assume that the momentum matrix element, Ep, in a quantum dot and the bulk semiconductor are the same. Thus, the linear absorption coefficient of single quantum dot RQD(pω) can be written as1,12
+ 1) ∑ E(2l √2π∆E
ε(pω) ) A
∑ n,l
(
(pω - Enl)2 (2l + 1) exp 2 2∆Enl E√2π∆Enl where
)
πNAe2Epapp A) (6) 2 ln(10)m0cnrε0
where NA is the Avogadro constant. 3. Relationship between Molar Extinction Coefficient and Particle Size The expression for the extinction coefficient describes superimposed Gaussian peaks centered at confined energies Enl whose width ∆Enl is a reflection of QD size distribution. In high-quality quantum dots which have a narrow size distribution the peak overlap is limited and contributions to the peak absorption at the first confined state from higher states may be neglected. In such a case, the maximum intensity at the first peak is given by the following equation:
(
πNAe2Epapp 1 1 ε1st ) 2 ln(10)m0cnrε0 E1st √2π∆E 1st
)
(7)
where E1st and ∆E1st are the energy at the first absorption peak and the width of this peak, both of which are easily observed in the UV-vis absorption spectra. Thus, the theoretical value of the molar extinction coefficient at the first absorption peak depends on these empirically accessible parameters and wellknown material constants such as Ep. The approximation which consists of ignoring the higher-order peaks is less accurate in poorer-quality quantum dots with broader peaks and also in larger QDs where the separation between energy levels compared with the peak widths is smaller. However, the experimental values of energies and peak widths for the higher-order peaks are practically impossible to extract from the spectra. Thus, in the following we use the first peak only for characterization of the QDs. When comparing this theory with experiments, it is important to be aware of their confinement regime. Three such confinement regimes for QDs have been identified in the literature depending on the relationship between the QD radius and its Bohr radius aB ) ah + ae where ah and ae are, respectively, the hole and electron Bohr radii:17 these are strong (R , ah and R , ae), intermediate (R . ah and R , ae), and weak confinement (R . aB). The values of the Bohr radii as well as the relevant constants for common quantum dot semiconductor material are shown in Table 1.20 The predictions of eq 7 as a function of the observed first transition energy are now compared with the data published in
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J. Phys. Chem. C, Vol. 112, No. 25, 2008 9263
TABLE 1: Bohr Radii and Underpinning Physical Constants for Common Quantum Dot Material material
m/h
me/
ε2
ah
ae
CdSe CdS CdTe PbS
0.45 0.8 0.4 0.25
0.13 0.21 0.11 0.25
9.4 8.6 10.3 17
2.2 1.14 2.73 7.2
7.66 4.34 9.93 7.2
TABLE 2: Constant for the Theoretical Calculation of the Molar Extinction Coefficient QD
Ep (eV)
nr ) ε2
A/e2 (eV2 L/M cm)
CdSe CdS CdTe PbS
20.0 21.0 18.5 2.5
3.06 2.93 3.2 4.12
1.25 × 105 1.37 × 105 1.10 × 105 1.12 × 104
literature.3,19
earlier The summary of the constants used to draw the theoretical curves is given in Table 2.21 First, we concentrate on CdSe, CdS, and CdTe quantum dots, which typically cross over from strong to intermediate confinement regime as their size is varied. The results for CdSe quantum dots are shown in Figure 1 where the peak width ∆λ (∆λ is related to ∆E1st through ∆λ ) λ2 ∆E1st/hc) used in Figure 1 was taken to be 25 nm as deduced from QD size dispersion given in ref.19 Figure 1 shows that the calculated values of the peak extinction coefficient reproduce quite closely the experimental results. They exhibit the same trend and the agreement from around 2.08 eV onward is good, which is at the onset of the strong confinement regime as shown in the figure. We note that more pronounced differences between our theory and the data are observed for transition energies lower than 2.08 eV, for intermediate confinement. Thus, within a broad range of QD sizes our model describes the behavior of CdSe QDs quite well, with the accuracy within 50% for peak energies corresponding to strong confinement. Our calculations have also been compared with experimentally observed results for CdS QDs with ∆λ ) 18 nm, as shown in Figure 2. The theoretical results for CdS are in agreement with experiments at higher energies, but the two curves come apart at the lower energy end, from around 3.45 eV. We have also plotted a similar comparison for CdTe quantum dots with ∆λ ) 29 nm (shown in Figure 3). It shows close agreement between theory and experiment after
Figure 1. The value of molar extinction coefficient in CdSe quantum dots as a function of the lowest observed transition energy based on this theory and experimental results.19 The extinction coefficient is shown in units of L/molQD · cm. R is the dot radius in nm, which is equal to the Bohr radius for the hole, and E1st is the first absorption peak energy in eV for the quantum dot with radius of R . The arrows indicate different confinement regimes.
Figure 2. The value of the molar extinction coefficient of CdS QDs as a function of the energy of the first confined transition energy based on this theory and experimental results.19 The extinction coefficient is presented in units of 105 L/MolQD · cm. R is the dot radius in nm, which is equal to the Bohr radius for the hole, and E1st is the first absorption peak energy in eV for the quantum dot with radius of R . The arrows indicate different confinement regimes.
Figure 3. The value of the molar extinction coefficient of CdTe QDs as a function of the energy of the first confined transition energy based on this theory and experimental results.19 The extinction coefficient is presented in units of 105 L/MolQD · cm. R is the dot radius in nm, which is equal to the Bohr radius for the hole, and E1st is the first absorption peak energy in eV for the quantum dot with radius of R. The arrows indicate different confinement regimes.
1.8 eV and departures before this energy. The agreement even improves somewhat around 2.1 eV. We have thus established that the peak molar extinction coefficient in CdSe, CdS, and CdTe quantum dots is reasonably closely reproduced by our theory but some discrepancies exist especially at lower energies. These can be attributed to increasing influence of the higher excited states for larger and less confined QDs and by the increasing difference between the real QD DOS and the 0D expression used here. This indicates the applicability regime for the present theory. It is well-known17 that only QDs with QD radius smaller than their excitonic Bohr radius are in good agreement with conventional quantum confinement models. It is thus instructive to examine a material which, unlike the previous three, fully meets this condition across the full energy range, such as PbS. In Figure 4, we show the comparison of our calculation with the experimental data from ref3 with the size dispersion ∆λ of 5%. In this case, the experimental data and our calculations are in much closer agreement. This is because, in PbS QDs whose sizes are always much smaller than the hole Bohr radius of 7.2 nm, the first absorption peak is very pronounced and the infinite well
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Jinjun and Goldys
E1st ) Eg +
πp2 2µR2
(8)
In this case, the width of the absorption peak can be approximated by the first derivative of the equation (eq 8) as
∆E1st )
πp2 ∆R µR3
(9)
where µ is the reduced mass of electron and hole. Considering only the first, leading term in the Taylor expansion of the molar extinction coefficient in eq 7, ε1 can be expressed in terms of the average dot radius, R,
Figure 4. The molar extinction coefficient of PbS QDs from the theory and experimental results.3 The extinction coefficient is presented in units of 105 L/MolQD · cm. R is the dot radius in nm, which is equal to the Bohr radius for the hole, and E1st is the first absorption peak energy in eV for the quantum dot with radius of R. The line with arrow indicates the strong confinement regime.
TABLE 3: Relationship between the Extinction Coefficient and QD Size from Various Literature Sourcesa type
D vs ε
Yu et al. Yu et al. 19 Yu et al. 19 Striolo et al. 13 Schmelz et al. 11
CdSe CdS CdTe CdSe CdSe
Leatherdale et al. 8 Cademartiri et al. 3
CdSe PbS
ε1 ) 5857(D)2.65 ε1 ) 21536(D)2.3 ε1 ) 10043(D)2.12 ε1 ∝ D3 ε1 ∝ D3 D < 4.5, const D > 4.5 nm ε1 ∝ D3 ε1 ) 19600R2.32
reference 19
a ε1 is the molar extinction coefficient at the first confined peak in units of L M-1 cm-1 . D is the average diameter of the QD. R is the average radius of the QDs.
approximation describes them well. Conversely, dots with larger ratio of size to Bohr radius experience poorer quantum confinement and are better described by more complicated models of density of states. Consequently, this simple quantum-mechanical theory cannot predict their absorbance as well as for the strongly confined QDs. Therefore, we conclude that a simple quantum mechanical theory of quantum dot linear absorption in an infinite spherical well model can accurately and quantitatively describe the extinction coefficient of quantum dots. In cases of strongly confined QDs whose size is much smaller than the hole Bohr radius, such as PbS, the accuracy is especially good, but also QDs which are bordering on the intermediate confinement can be reasonably well described. Our results for both these types of QDs may be used for concentration calibration, with due account of correction factors for those energies where the departures are observed. Moreover, our theory explains the discrepancies between the observed trends of QD extinction coefficient at the first peak with varying size listed in Table 3 for QDs whose size places them in the strong confinement regime. The relationships are of the form ε ∝ Dβ with β generally varying between 2 and 3 and different in separate studies. Our theory predicts that the molar extinction coefficient ε1 at the first peak is directly related to the first absorption peak energy (the energy of 1se - 1sh transition) as well as the peak width ∆E1st which is related to QD size distribution. In order to explain the observed trends, we need to use the approximate relationship between E1st and R given by:10
ε1 )
(
A 2
πp √2π Eg + 2µR2
)
µR2 πp2δR
(10)
where A is given in Table 2, δR is the standard deviation of dot radius divided by R . Equation 10 explains why the empirical curve follows Rβ with 2 < β < 3 especially for larger radii R. Such larger radii R cause the denominator to be dominated by Eg for constant δR . In these conditions, the apparent relationship is close to the square of the radius, as shown in Table 3. For smaller R, the relationship will tend toward R4, and intermediate values of R will produce curves in between these two. 4. Conclusion We have shown that a simple quantum mechanical calculation of linear absorption which uses an infinite spherical quantum dot model describes well the experimental results of extinction coefficient obtained for frequently studied high-quality colloidal quantum dots of direct semiconductor materials. We compared our calculations with the experimental results for CdSe, CdS, and CdTe which, depending on the QD size, cross over from a strong to intermediate confinement regime with radius R greater than the hole Bohr radius ah, and for PbS QDs which fall into the strong confinement regime. The departures of the theoretical curves from the experimental data are more pronounced at lower energies than for larger QDs which are more weakly confined. This is due to model limitations such as the use of 0D density of states, which is only approximate for larger QDs, ignoring the influence of the second and higher confined states and the treatment of QD size dispersion. Therefore, although the present theory is only strictly valid for ideal quantum dots which are in the strong confinement regime, it can be applied to a range of nanoparticles in the moderate confinement regime R j aB,17 and it gives acceptable accuracy for particles with sizes somewhat larger than the hole Bohr radius. The presented results can be of assistance, for example, for testing the concentration of commercial QDs during biological experiments such as conjugation to specific ligands or biomolecules, as in such protocols, the bound and unbound QDs are separated, so the concentration changes. They can also be helpful for those who use QDs to visualize various processes in cells and tissues where local chemical variations and other factors can change the local QD concentration. Our method can be used in microscopy where it makes it possible to estimate the local concentration of QDs, thus opening the way for quantitative studies. Appendix: Linear Absorption and Molar Extinction The purpose of this section is to clarify the relationship of the experimentally measured molar extinction to the linear absorption coefficient for a single quantum dot RQD (in units of
Linear Absorption and Molar Extinction Coefficients in QDs
J. Phys. Chem. C, Vol. 112, No. 25, 2008 9265
Ii ) rIi-1 exp{-RQDD} + (1 - r)Ii-1 ) (r exp{-RQDD} - r + 1)Ii-1 i ) 1, 2, · · · , j
(13)
Consequently
Ii ) I0(r exp{-RQDD} - r + 1)i
(14)
Light irradiance emerging from the jth layer is given by
Ij ) I0(r exp{-RQD D} - r + 1) j Figure 5. The diagram of the absorption of a quantum dots solution. I0 is the incident light irradiance, Ii is the light irradiance inside the cuvette and Ij is light irradiance leaving the cuvette.
m-1).
The literature typically gives the absorption of a quantum dot solution in terms of its molar extinction coefficient, , commonly used in chemistry and sometimes called molar absorptivity.
The extinction coefficient has units of L mol-1 cm-1 or cm2 mol-1 which are the same as the absorption cross section, m-2 or cm-2 per mole. We will assume, following ref,8 that scattering has a negligible influence on the value of extinction coefficient in quantum dots. In such a case, the absorbance, A, of a sample containing a single species is dependent on the light path length, l, and the molar concentration c of the species via the Beer-Lambert Law
A ) εcl
(11)
We now establish the relationship of the two coefficients RQD and . To simplify the problem, we assume the quantum dots are approximated by identical cubes of size D and homogeneously distributed inside the solvent. We assume that quantum dots within the solvent are located in individual thin layers of thickness D perpendicular to the incident light beam with a cross section area s . In each such layer, a number k of quantum dots illuminated by the beam is the same. Because the molar concentration, cQD, of a quantum dot solution is normally low (or they can be diluted), the distance among the quantum dots is much larger than D and the likelihood of two different quantum dots located precisely behind one another is negligible. The QD solvent is divided into j such layers (see Figure 5) which add up to a total path length of l . In each layer, inside the beam, k quantum dots absorb the light, which reduces its intensity. We denote the initial incident light irradiance as I0 . The first layer will attenuate this light to I1, the second to I2, and so on until the light leaves the cuvette with irradiance of IjThe relationship between the irradiances, Ii, i ) 1, 2, · · · , j is given by
I1 ) k
(
)
D2 -RQD D D2 + 1-k I I0e s s 0
(12)
The first term on the right side of eq 12 describes light absorbed by the dots inside the layer. The second is the light which misses the QDs. To simplify, we denote kD2/s ) r . The irradiance Ii is then given by
(15)
In eq 15, RQD is very small, normally on the order m-1, -8 and D is below 10 m for typical QDs. The experimentally measured absorbance, A, is given by 106
Ij ) I010-A
(16)
A ) εcQDl
(17)
where
Considering that r is very small, combining eqs 15 and 16 and retaining only first-order terms of RQDD, we get
-A ln(10) ) jrRQD D
(18)
Thus
εcQDl ln(10) ) jk
D2 R D s QD
(19)
where s is the cross-sectional area of the light beam, and s multiplied by l (light path length) is the volume of the light beam inside the solution. The product jk is equal to the number of quantum dots inside the beam volume, jk/NAsl is the molar concentration of the quantum dot solution, which is equal to cQD, and D3 is the volume of quantum dots, VQD. Therefore, eq 19 leads to
ε)
NAVQDRQD ln(10)
(20)
Equation 20 gives the relationship between the linear absorption coefficient, RQD, and the molar extinction coefficient for the colloidal QDs solution with relatively low molar concentration. Acknowledgment. The research is supported by the iMURS scholarship of Macquarie University. References and Notes (1) Ferreira, D. L. A. U.; Alves, J. L. A. The effects of shape and size nonuniformity on the absorption spectrum of semiconductor quantum dots. Nanotechnology 2004, 15, 975–981. (2) Burt, M. G. The evaluation of the matrix element for interband optical transitions in quantum wells using envelope functions. J. Phys.: Condens. Matter 1993, 5 (24), 4091–4098. (3) Cademartiri, L.; Montanari, E.; Calestani, G.; Migliori, A.; Guagliardi, A.; Ozin, G. A. Size-dependent extinction coefficients of pbs quantum dots. J. Am. Chem. Soc. 2006, 128 (31), 10337–10346. (4) Chan, W. C. W.; Nie, S. Quantum Dot Bioconjugates for Ultrasensitive Nonisotopic Detection. Science 1998, 281 (5385), 2016. (5) Haug, H.; Koch, S. W.; NetLibrary, I. Quantum Theory of the Optical and Electronic Properties of Semiconductors; World Scientific: Singapore, 2004. (6) Hermann, C.; Weisbuch, C. k [over] · p [over] perturbation theory in III-V compounds and alloys: a reexamination. Phys. ReV. B 1977, 15 (2), 823–833. (7) Kang, I.; Wise, F. W. Electronic structure and optical properties of PbS and PbSe quantum dots. J. Opt. Soc. Am. B 1997, 14 (7), 1632– 1646.
9266 J. Phys. Chem. C, Vol. 112, No. 25, 2008 (8) Leatherdale, C. A.; Woo, W.-K.; Mikulec, F. V.; Bawendi, M. G. On the absorption cross section of cdse nanocrystal quantum dots. J. Phys. Chem. B 2002, 106 (31), 7619–7622. (9) Magnusson, K. O.; Karlsson, U. O.; Straub, D.; Flodstro¨m, S. A.; Himpsel., F. J. Angle-resolved inverse photoelectron spectroscopy studies of cdte(110), cds(1120), and cdse(1120). Phys. ReV. B 1987, 36 (12), 6566– 6573. (10) Ramaniah, L. M.; Nair, S. V. Optical absorption in semiconductor quantum dots: A tight-binding approach. Phys. ReV. B 1993, 47 (12), 7132– 7139. (11) Schmelz, O.; Mews, A.; Basche, T.; Herrmann, A.; Mullen, K. Supramolecular complexes from cdse nanocrystals and organic fluorophors. Langmuir 2001, 17 (9), 2861–2865. (12) Singh. J. Electronic and Optoelectronic Properties of Semiconductor Structures; Cambridge University Press: Cambridge, 2003. (13) Striolo, A.; Ward, J.; Prausnitz, J. M.; Parak, W. J.; Zanchet, D.; Gerion, D.; Milliron, D.; Alivisatos, A. P. Molecular weight, osmotic second virial coefficient, and extinction coefficient of colloidal cdse nanocrystals. J. Phys. Chem. B 2002, 106 (21), 5500–5505. (14) van de Hulst, H. C. Light Scattering by Small Particles; 1957. (15) Woggon. U. Optical properties of semiconductor quantum dots; Springer: New York, 1997. (16) Wu, W.-Y.;.; Schulman, J. N.; Hsu, T. Y.; Efron, U. Effect of size nonuniformity on the absorption spectrum of a semiconductor quantum dot system. Appl. Phys. Lett. 1987, 51 (10), 710–712.
Jinjun and Goldys (17) Yoffe, A. D. Low-dimensional systems: quantum size effects and electronic properties of semiconductor microcrystallites (zero-dimensional systems) and some quasi-two-dimensional systems. AdV. Phys. 2002, 51 (2), 799–890. (18) Yoffe, A. D. Semiconductor quantum dots and related systems: electronic, optical, luminescence and related properties of low dimensional systems. AdV. Phys. 2001, 50 (1), 1–208. (19) Yu, W. W.; Qu, L.; Guo, W.; Peng, X. Experimental determination of the extinction coefficient of cdte, cdse, and cds nanocrystals. Chem. Mater. 2003, 15 (14), 2854–2860. (20) Here me is the electron effective mass and mh is the average effective mass of holes. Such average takes into account that heavy hole and light hole transitions are unresolved. Further discussion of the effective mass for the heavy hole or light hole under strong confinement can be referred to Woggon15 and Yoffe.17,18 All values of dielectric constant ε2 are cited from ref.17 The values of effective masses in CdSe, CdS, and PbS are cited from http://www.tf.uniiel.de/matwis/amat/semi_en/kap_2/backbone/ r2_3_1.html. The effective masses in CdTe are cited from http://en.wikipedia.org/wiki/Cadmium_telluride. (21) Dielectric constant ε2 are cited from ref.17 Ep of CdSe, CdS, and CdTe are cited from work of Hermann.6 PbS Ep value is cited from work of Kang.7 A ) πNAe2Epapp/2 ln 10m0cnrε0.
JP800700M
