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Dielectric Functions of Semiconductor Nanoparticles from the Optical Absorption Spectrum: The Case of CdSe and CdS Marcelo Alves-Santos,*,† Rosa Di Felice,†,§ and Guido Goldoni†,‡,§ Centro S3, CNR-Istituto di Nanoscienze, Via Campi 213/A, 41100 Modena, Italy, and Dipartimento di Fisica, UniVersita` di Modena e Reggio Emilia, Via Campi 213/A, 41100 Modena, Italy ReceiVed: October 2, 2009; ReVised Manuscript ReceiVed: January 21, 2010
We propose a new method to extract the complete size-dependent complex dielectric function ε(E) of semiconductor nanoparticles (NPs) from spectroscopic data. Typical experimental absorption spectra are available in a limited energy range near the absorption edge. We show that this limitation can be overcome by matching the NP dielectric function with the bulk dielectric function at high energies beyond the experimental range. Thus, using the Kramers-Kro¨nig relations that link the real and imaginary parts, it is possible to extract the complete ε(E) of the NP. This is achieved through an iterative procedure that systematically improves the trial ε(E) until the absorption spectrum of the NP is accurately reproduced in the experimental range. Here we describe the methodology and we obtain the ε(E) of CdSe and CdS NPs of selected sizes from published data sets. Quantum confinement effects at the nanoscale induce strong size and shape dependence of the dielectric function. In addition, the large surface-to-volume ratio leads to a significant influence of the environment. Therefore, nanoparticles (NPs) with specific optical properties might be synthesized by proper combinations of size, shape, and environment, offering new perspectives in nanodevice applications. The knowledge of the size-dependent ε(E) of NPs is an essential ingredient, for example, to predict the optical excitations of composite systems from those of the isolated components. This is an increasingly attracting task, due to the growing effort to disclose the optical response of hybrid metal-semiconductor nanoparticles.1-6 Hybrid nano-objects are particularly appealing for the development of nanotechnology applications that may exploit the size dependence and the combined properties of semiconductors (e.g., sharp optical emission in the visible) and metals (e.g., ability to anchor organic and biological supramolecular motifs). Ab initio methods are often numerically too demanding in assisting the design of the optical properties of such complex systems, and macroscopic theoretical modeling takes on a prominent role in this context. For example, methods based on classical electromagnetism can use the information about the dielectric function of each component, commonly expressed as a function ε(E) dependent only on the photon energy of the incident field, which implicitly includes the quantum size effects responsible for the specific features arising in the optical response of the material. These methods include the Mie Theory7,8 for spherical homogeneous NPs, generalized by Aden and Kerken9 to spherical core-shell NPs, the boundary element method (BEM)10-13 and the discrete dipole approximation (DDA),14,15 the latter two suitable for composite NPs of arbitrary shape and composition. Therefore, * To whom correspondence should be addressed. E-mail:
[email protected]. † CNR-S3. ‡ Universita` di Modena e Reggio Emilia. § E-mail addresses: (R.D.F.)
[email protected]; (G.G.)
[email protected] dielectric functions that describe the materials at the nanoscale are needed, opening the way to the interpretation of experimental data and design of the composites to obtain desired optical behavior. Despite the great interest in the research on NPs, however, there is presently no available information about dielectric functions of nanosized semiconductors that could be readily used for macroscopic modeling applications. Obtaining the dielectric function from the optical absorption spectrum is not a trivial task. In general, it implies the use of one of the classical methods mentioned above for calculating optical spectra within a trial-and-error procedure in which the absorption cross section is calculated from a parametrized dielectric function that is iteratively improved until the theoretical and the measured optical spectra match within a specified accuracy. It is in this framework that some methods were used for obtaining dielectric functions of small spherical particles, especially composed of materials with environmental relevance, such as aerosols.16,17 However, the micrometer-sized particles that were analyzed by these methods were large enough that the dielectric functions were not dependent on their dimensions within the size distribution considered. Our attention focuses on small semiconductor NPs for which low energy optical excitations arise from intrinsic quantum size effects, related to the change of the dielectric function with the particle size, which are far more important than effects due solely to the particle geometry. In this paper, we propose a method for extracting the dielectric function of spherical NPs from the experimental absorption spectra of particles with a narrow size distribution. We use the theoretical extinction cross section (which includes both the scattering and absorption effects), the standard Kramers-Kro¨nig (KK) relations, and a proper extension of the dielectric function for frequency ranges where absorption data are not available. We apply the proposed strategy to CdS and CdSe NPs of selected sizes and determine the dielectric functions from published data sets. One of the main steps of our method consists in obtaining the imaginary part of the dielectric function ε2(E), given its real exp . In principle, part ε1(E) and the experimental absorption σabs
10.1021/jp909466b 2010 American Chemical Society Published on Web 02/17/2010
Dielectric Functions of CdSe and CdS NPs
J. Phys. Chem. C, Vol. 114, No. 9, 2010 3777 exp is model function. Labeling Ea the highest energy at which σabs available for a given NP and Eb the highest energy at which ε2bulk is available in detail, because Eb > Ea we can use part of ε2bulk to obtain a short extension of ε2, that is
ε2(E) ) εbulk 2 (E), Ea e E e Eb
(3)
Above Eb, we extrapolate ε2 by means of the Drude-like function Figure 1. Scheme of the procedure for the calculation of the dielectric functions ε of semiconductor NPs. (a) Pictorial representation of the continuation of ε2, the imaginary part of ε, beyond the energy range in which the experimental absorption spectrum is available. The energy region in which ε2 is calculated from the experimental absorption is indicated as from exp. The dotted line represents the low energy part of the bulk ε2, which is excluded from the NP function. (b) Flow diagram of the iterative procedure.
this is easily achieved through the calculated extinction cross section σext(ε1, ε2), taking into account the size and geometry of the NP and the dielectric constant of the medium. Indeed, at a given photon energy E ) pω, the value of ε2 can be obtained by solving the equation exp σext[ε1(E), ε2] - σabs (E) ) 0
(1)
Since ε1(E) can be determined from ε2(E) by the KK integral
ε1(E) ) ε1(∞) +
2 π
E'ε (E')
∫0∞ dE' E′2 2- E2
(2)
one could iterate eqs 1 and 2 until convergence. However, the finite energy range of the available experimental absorption σexp abs , and therefore of the imaginary part ε2 computed by means of eq 1, constitutes the greatest obstacle in the solution of this integral which is extended to any frequency. This limitation requires that ε2 be properly extended beyond the experimentally available energy range for the absorption spectrum at both low and high energies, by matching to a suitable function. The basic idea is to use the bulk dielectric function ε2bulk in the energy ranges where this is known from experiments, which is anyway the reference for increasingly large NPs. We next distinguish two energy regions for the continuation of ε2 (below E0 and above Ea in Figure 1a). In the low energy region, below the absorption onset energy E0 characteristic of any NP, we set ε2(E) ) 0, so that the optical absorption intensity in the low-energy part of the spectrum vanishes for E < E0, in line with the dipole-mode approximation.18,19 We note that experimental data for bulk CdSe and CdS can assume large values close to 0 eV.20 However, show that εbulk 2 such nonvanishing parts of ε2bulk are restricted to a very narrow energy region quite well separated from the interband absorption region. Therefore, such contributions to ε2bulk can be safely neglected in the evaluation of the KK integral without compromising the shape of the converged computed absorption spectrum. In the high energy region, the experimental bulk dielectric function for most materials is reported in an energy range that is generally larger than, and includes that of the experimental absorption spectra of the NPs. This allows a good matching in a reasonably large range, rather than at a single point, with a
ε2(E) )
a , E g Eb E(E + b2) 2
(4)
with constants a and b calculated so that ε2(Eb) ) ε2bulk(Eb) and the sum rule ∫∞0 Eε2(E)dE ) π/2E2p applies.21 This sum rule must be obeyed for the function ε2(E) in the entire energy range (0 e E < ∞), with the bulk plasmon energy Ep. The continuation of ε2(E) to energies lower and higher than those of the experimentally available absorption spectrum is summarized in Figure 1a. The extended function is then used to calculate the KK integral. In practice, the numerical integration of ε2(E) is performed in the interval from 0 to Eb with addition of the analytical result of the integral of the function in eq 4 over the remaining energy range. Finally, in order to determine ε1(E) uniquely, the constant term ε1(∞) in eq 2 is adjusted so that the function presents the same value of the bulk at a chosen reference energy (as will be explained below). Equations 1 and 2 that connect ε1(E) to ε2(E), along with the continuation of ε2(E) outside the experimental absorption energy range, constitute the basis of the iterative method to determine the dielectric function. The flow diagram for the iterative procedure is shown in Figure 1b. In the first iteration, the bulk dielectric function is used throughout, and the continuation of ε2(E) implies only the extension for E e E0 and E g Eb, while at any following iteration it also includes the matching with ε2bulk within Ea e E e Eb. At the nth step, ε1(n)(E) is computed (E), while ε(n) through eq 2 from ε(n-1) 2 2 (E) is obtained by solving eq 1 via the secant method from ε1(n)(E) and the experimental absorption data. The calculation is interrupted when the maximum difference, in the interval [E0, Ea], between the components of the dielectric functions from two consecutive iterations is less than a given small tolerance value δ. The absorption spectrum used as input data for the solution of eq 2 must be given in absolute units of cross section. Since the available experimental data are normally given in arbitrary units, we normalized the spectra by imposing that the dielectric functions of the NP and of the bulk of the same material are equal at a chosen reference energy Eref (or the corresponding wavelength λref). According to Leatherdale et al.,22 the absorption spectra per unit volume of small CdSe NPs with different sizes tend to converge for wavelengths smaller than 400 nm, indicating that below this value the optical behavior of the particles would be indistinguishable from that of the bulk. This evidence suggests the choice of a suitable wavelength λref near the minimum wavelength λa of the experimental range, corresponding to Ea, generally 300 nm or somehow larger. Furthermore, we also imposed the condition that, in the interval [λa, λref], the integral of the normalized experimental absorption must be bulk , calculated equal to that of the NP extinction cross section σext from the bulk dielectric function, so that the two contributions to the oscillator strength would be equal in this wavelength region (see Figure 2). The matching condition then is
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Alves-Santos et al.
(5)
with λref given by
∫λ
λref a
exp σabs (λ)dλ
)
∫λ
λref a
bulk σext (λ)dλ
(6)
where we consider that the experimental absorption is already normalized for the spherical particles of interest here. In eq 6, the NP spectrum with the bulk dielectric function can be calculated either by means of the Mie Theory or by using the dipolar mode approximation (see below), which is good for the particle sizes considered in this work (smaller than 12 nm). If no wavelength is found to comply with eq 6, which is the case for the CdS particles examined here, or λa is larger than 350 nm, we simply take λref as the minimum experimental wavelength. The value of the energy Eref corresponding to this wavelength will then replace that of Ea in eq 3. The calculation of the extinction spectra for the spherical NPs, which is necessary to solve eq 1 in our iterative procedure, is carried out by the use of the Mie Theory, that we have implemented following an established formalism.18,19 The extinction cross section is basically obtained by summation of a series of Mie coefficients aL and bL, calculated in terms of Riccati-Bessel functions and their derivatives. These coefficients depend on the particle radius, the energy of the radiation E, the NP dielectric function ε(E) (or the corresponding complex refractive index), and the dielectric constant of the medium εm. The Mie extinction cross section can then be written as an explicit function of the energy ∞
Mie σext (E) )
∑
2π (2L + 1)Re[aL(E) + bL(E)] [k(E)]2 L)1
(7) where k is the wavenumber of the radiation. In the static regime, that is, for wavelengths much larger than the particle size, an alternative to the use of the Mie Theory is the dipole-mode approximation for the cross sections. In this approximation scattering is a negligible contribution to the extinction, and the
Figure 2. Normalized experimental absorption spectrum per unit volume for the 1.9 nm CdSe NP (solid line) and extinction cross section per unit volume calculated by using the bulk dielectric function (dashed line). The two curves assume equal values at the reference wavelength λref and the same integral over the interval from 300 nm to λref (represented by the gray area under the bulk curve). The deviation of the extinction spectrum computed with the bulk dielectric function from the experimental absorption spectrum manifests the need for a dielectric function that embodies the quantized nature of the NP in order to describe the optical properties of confined systems.
relation between the extinction cross section of the spherical particle and its dielectric function can be expressed as
ε2(E) E 3/2 d σext (E) ≈ 9 εm V0 c [ε1(E) + 2εm]2 + [ε2(E)]2
(8)
where V0 is the volume of the particle and c is the speed of light in vacuum. Therefore, in the dipole-mode approximation, eq 1, which gives ε2 in terms of ε1 and the experimental absorption, becomes a simple quadratic equation. We remark that our iterative procedure to obtain the dielectric function is not restricted to a specific method for the calculation of the absorption cross section. The Mie Theory adopted here is valid only for spherical particles. If one chooses, instead, a method that is valid for any shape, then also the whole procedure is shape-independent. We also note that the dielectric functions obtained with the proposed procedure depend on the medium where the NPs are dispersed. As can be seen explicitly in eq 8 for the dipole-mode approximation, this is especially true if the interaction between the semiconductor and the medium leads to a significant shift in the semiconductor exciton energy and therefore to a change of the absorption onset. Indeed, the ε2(E) functions obtained by fixing them to zero below a particular E0 will all result in the same onset absorption energy, whatever the medium dielectric constant considered. Following the above algorithm, we obtain the dielectric functions for CdSe and CdS NPs in hexane solution, characterized by a medium dielectric constant εm ) 1.89. We use the data by Murray et al.23 for the absorption spectra of CdSe NPs with diameters ranging from 1.6 to 11.5 nm (the 1.2 nm NP data is not used here, because it is not possible to obtain a proper normalization of the absorption spectrum) and the data by Steckel et al.24 for the absorption spectra of CdS NPs with diameters ranging from 3.7 to 5.2 nm. The bulk dielectric functions for wurtzite and zincblende CdSe and CdS are taken from ref 20. For the wurtzite phase, we use here a dielectric function determined by a weighted average of the functions corresponding to the extraordinary and the ordinary directions. The plasma frequencies are taken as 17.6 eV for CdSe and 18.2 eV for CdS.25 Generally, the convergence of the dielectric function is achieved in approximately 10 steps. Figure 3a demonstrates the convergence procedure: it shows the real and imaginary parts of the dielectric function of CdSe NPs with an average diameter of 1.9 nm23 after a few steps of the iterative procedure and until convergence is reached. We stop the iterative procedure when, at each energy, the value of the dielectric function is equal within 0.2% to the value found at the preceding step. Figure 3b shows the experimental absorption spectrum and the theoretical extinction cross section resulting from the calculated dielectric function. The two curves are identical in the energy region from the absorption onset to Eref, which is 3.84 eV for the 1.9 nm CdSe NPs. Above this energy, and up to the maximum experimental energy value, the curves differ by a small amount, usually less than 1% of the absorption value. The difference with Figure 2, where the extinction spectrum of the same NP computed using the bulk dielectric function deviates significantly from the experimental absorption spectrum, emphasizes the importance of choosing a good dielectric function, which respects the NP size and shape, to yield reliable results for the optical properties. The calculated dielectric functions of the CdSe and CdS NPs of different sizes are shown in Figure 4a-d (see also Supporting
Dielectric Functions of CdSe and CdS NPs
J. Phys. Chem. C, Vol. 114, No. 9, 2010 3779
Figure 3. Results from the application of the iterative procedure of Figure 1b to the CdSe NPs of 1.9 nm average size. (a) The plots show the real and imaginary parts of the dielectric functions calculated at selected iterations, along with the bulk dielectric function. The fast convergence is evident. (b) Experimental absorption spectrum and extinction cross section obtained with the calculated dielectric function. The gray area indicates the energy region where the spectra differ, but the agreement is quite good for E e Eref.
Figure 4. Real and imaginary parts of the dielectric functions calculated for the CdSe and CdS NPs by taking as reference the bulk dielectric functions of the wurtzite (a,c) and zincblend (b,d) phases.
Information). Both the real part ε1(E) and the imaginary part ε2(E) of the dielectric functions present sharp features that are directly related to peaks in the absorption spectra23,24 in the energy region between E0 and Ea, which contrasts with the much smoother components of the bulk dielectric function, indicative of the quantum confinement induced discrete density of states. The CdS ε2(E) curves exhibit much fewer oscillations than those of CdSe for particle sizes smaller than 5.2 nm. For each NP, the imaginary part ε2(E) presents the same onset energy E0 as the corresponding optical absorption spectrum, as established in the calculation procedure. The NP onset is systematically larger than that of the bulk, which is 1.7 and 2.4 eV for CdSe and CdS, respectively, as it incorporates quantum confinement effects. As the size of the particle is increased the exciton absorption energy near the onset of the optical absorption spectra decreases and the curves approach those of the bulk. For CdSe, the real and imaginary parts associated with the 8.3 and 11.5 nm sizes are already very close to those of the bulk, which indicates that the bulk dielectric function could be used to calculate the optical properties for CdSe particles larger than about 12 nm. We cannot determine a similar limit for CdS due
to the lack of absorption data for particles larger than 5.2 nm, for which the absorption onset energy still differs significantly from that of the bulk by ∼0.25 eV. For both CdSe and CdS the choice of the initial bulk dielectric function from the wurtzite or zincblende crystallographic phases does not introduce significant differences on the features of the final NP dielectric function. In fact, the difference of magnitude between the functions results from the normalization of the experimental absorption spectrum according to the value of the bulk dielectric function at the reference energy (or wavelength). We also find that the dielectric functions obtained within the Mie Theory and those obtained within the dipole-mode approximation are practically identical, meaning that the NP sizes and the wavelength range of the measured spectra guarantee the domain of validity of the approximation. Although the calculation with the dipole-mode approximation requires the execution of a smaller number of instructions (ε2 can be calculated directly by means of a quadratic equation, avoiding the use of the root finding method), the use of the procedure with the Mie Theory is still the best choice for precision, and the computational cost is quite small anyhow.
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In summary, we have reported a new method that allowed us to extract the full size-dependent dielectric functions of CdSe and CdS nanoparticles from experimental absorption spectra available in the literature. Our method uses an iterative procedure that combines extinction cross section calculations and KramersKro¨nig integrations, along with a suitable choice of the shape of ε2 outside the absorption measurement energy range. The dielectric functions obtained for CdSe and CdS describe reasonably well the optical behavior of the semiconductor NPs, mainly the change of the exciton absorption energy with the particle size. This accuracy supports the use of the obtained dielectric functions in more complex situations, such as the cases of semiconductor core-shell nanoparticles and hybrid metalsemiconductor nanoparticles of various shapes, although care should be taken in utilizing the dielectric function in an environment different from that in which they are computed. Note finally that our method is clearly extendable to different classes of semiconductors, given the existence of experimental absorption data of monodisperse NPs in a reasonable energy range. Although the presented implementation and application of the method is for spherical NPs, it is straightforward (although numerically more demanding) to generalize our approach to arbitrary geometries by employing one of the general methods mentioned in the introduction to calculate σext in eq 1. Acknowledgment. This work has been supported by the EU FP6 ERANET project “NanoSci-ERA: NanoScience in the European Research Area”, the Italian Ministery for Research through FIRB RBIN06JB4C and the regional Emilia Romagna net-lab PROMINER. The INFM-CNR supercomputing initiative is acknowledged for computing time at CINECA (Bologna). We are grateful to Fabrice Vallée, Natalia Del Fatti, Uri Banin, and Carsten Sönnichsen for fruitful discussions in the framework of the NanoSci-ERA subproject “Single Nano-Hybrid”. Supporting Information Available: Dielectric function tables of CdSe (1.6, 1.9, 2.3, 3.2, 3.7, 5.1, 6.5, 7.5, 8.3, and 11.5 nm) and CdS (3.7, 3.9, 4.2, 4.4, 4.7, 4.9, and 5.2 nm) nanoparticles obtained with wurtzite and zincblende crystal-
Alves-Santos et al. lographic phases of bulk as reference. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Kan, S.; Mokari, T.; Banin, U. Nat. Mater. 2003, 2, 155. (2) Mokari, T.; Rothenberg, E.; Popov, I.; Costi, R.; Banin, U. Science 2004, 304, 1787. (3) Mokari, T.; Sztrum, C.; Salant, A.; Rabani, E.; Banin, U. Nat. Mater. 2005, 4, 855. (4) Govorov, A. O.; Bryant, G. W.; Zhang, W.; Skeini, T.; Lee, J.; Kotov, N. A.; Slocik, J. M.; Naik, R. R. Nano Lett. 2006, 6, 984. (5) Vasa, P.; Pomraenke, R.; Schwieger, S.; Mazur, Y. I.; Kunets, V.; Srinivasan, P.; Johnson, E.; Kihm, J. E.; Kim, D. S.; Runge, E.; Salamo, G.; Lienau, C. Phys. ReV. Lett. 2008, 101, 116801. (6) Shevchenko, E. V.; Bodnarchitk, M. I.; Kovalenko, M. V.; Talapin, D. V.; Smith, R. K.; Aloni, S.; Heiss, W.; Alivisatos, A. P. AdV. Mater. 2008, 20, 4323. (7) Mie, G. Ann. Phys. 1908, 25, 377. (8) van de Hulst, H. C. Light Scattering by Small Particles; Dover: New York, 1957. (9) Aden, A. L.; Kerker, M. J. Appl. Phys. 1951, 22, 1242. (10) de Abajo, F. J. G.; Howie, A. Phys. ReV. B 2002, 65, 115418. (11) Hohenester, U.; Krenn, J. Phys. ReV. B 2005, 72, 195429. (12) Trügler, A.; Hohenester, U. Phys. ReV. B 2008, 77, 115403. (13) Hohenester, U.; Trügler, A. IEEE J. Sel. Top. Quantum Electron. 2008, 14, 1430. (14) Purcell, E. M.; Pennypacker, C. R. Astrophys. J. 1973, 186, 705. (15) Draine, B. T.; Flatau, P. J. 2003, arXiv:astro-ph/0309069v1 (accessed September 2009). (16) Clapp, M. L.; Miller, R. E.; Worsnop, D. R. J. Phys. Chem. 1995, 99, 6317. (17) Dohm, M. T.; Potscavage, A. M.; Niedziela, R. F. J. Phys. Chem. 2004, 108, 5376. (18) Kreibig, U.; Vollmer, M. Optical Properties of Metal Clusters; Springer-Verlag: Berlin, 1995; Vol. 25. (19) Bohren, C. F.; Huffmann, D. R. Absorption and Scattering of Light by Small Particles; Wiley-Interscience: New York, 1983. (20) Adachi, S. Optical Constants of Crystalline and Amorphous Semiconductors; Kluwer Academic Publishers: Cambridge, Boston, MA, 1999. (21) Altarelli, M.; Dexter, D. L.; Nussenzveig, H. M.; Smith, D. Y. Phys. ReV. B 1972, 6, 4502. (22) Leatherdale, C. A.; Woo, W.-K.; Mikulec, F. V.; Bawendi, M. G. J. Phys. Chem. B 2002, 106, 7619. (23) Murray, C. B.; Norris, D. J.; Bawendi, M. G. J. Am. Chem. Soc. 1993, 115, 8706. (24) Steckel, J. S.; Zimmer, J. P.; Coe-Sullivan, S.; Stott, N. E.; Bulovic, V.; Bawendi, M. G. Angew. Chem. Int. Ed. 2004, 43, 2154. (25) Hengehold, R. L.; Pedrotti, F. L. Phys. ReV. B 1972, 6, 2262.
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