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Experimental Determination of the Absorption Cross-Section and Molar Extinction Coefficient of CdSe and CdTe Nanowires† Vladimir Protasenko,‡ Daniel Bacinello,‡,§ and Masaru Kuno*,‡ Department of Chemistry and Biochemistry and Notre Dame Radiation Laboratory, UniVersity of Notre Dame, Notre Dame, Indiana 46556, and Department of Chemistry, UniVersity of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 ReceiVed: September 15, 2006; In Final Form: October 9, 2006
Absorption cross-sections and corresponding molar extinction coefficients of solution-based CdSe and CdTe nanowires (NWs) are determined. Chemically grown semiconductor NWs are made via a recently developed solution-liquid-solid (SLS) synthesis, employing low melting Au/Bi bimetallic nanoparticle “catalysts” to induce one-dimensional (1D) growth. Resulting wires are highly crystalline and have diameters between 5 and 12 nm as well as lengths exceeding 10 µm. Narrow diameters, below twice the corresponding bulk exciton Bohr radius of each material, place CdSe and CdTe NWs within their respective intermediate to weak confinement regimes. Supporting this are solution linear absorption spectra of NW ensembles showing blue shifts relative to the bulk band gap as well as structure at higher energies. In the case of CdSe, the wires exhibit band edge emission as well as strong absorption/emission polarization anisotropies at the ensemble and single-wire levels. Analogous photocurrent polarization anisotropies have been measured in recently developed CdSe NW photodetectors. To further support fundamental NW optical/electrical studies as well as to promote their use in device applications, experimental absorption cross-sections are determined using correlated transmission electron microscopy, UV/visible extinction spectroscopy, and inductively coupled plasma atomic emission spectroscopy. Measured CdSe NW cross-sections for 1 µm long wires (diameters, 6-42 nm) range from 6.93 × 10-13 to 3.91 × 10-11 cm2 at the band edge (692-715 nm, 1.73-1.79 eV) and between 3.38 × 10-12 and 5.50 × 10-11 cm2 at 488 nm (2.54 eV). Similar values are obtained for 1 µm long CdTe NWs (diameters, 7.5-11.5 nm) ranging from 4.32 × 10-13 to 5.10 × 10-12 cm2 at the band edge (689-752 nm, 1.65-1.80 eV) and between 1.80 × 10-12 and 1.99 × 10-11 cm2 at 2.54 eV. These numbers compare well with previous theoretical estimates of CdSe/CdTe NW cross-sections far to the blue of the band edge, having order of magnitude values of 1.0 × 10-11 cm2 at 488 nm. In all cases, experimental NW absorption cross-sections are 4-5 orders of magnitude larger than those for corresponding colloidal CdSe and CdTe quantum dots. Even when volume differences are accounted for, band edge NW cross-sections are larger by up to a factor of 8. When considered along with their intrinsic polarization sensitivity, obtained NW cross-sections illustrate fundamental and potentially exploitable differences between 0D and 1D materials.
Introduction How effectively a particular substance absorbs electromagnetic radiation can be characterized by two interrelated optical properties: the absorption cross-section and the molar extinction coefficient. Experimental estimates of these frequency-dependent optical constants represent an area of key interest in nanoscience and nanotechnology. Specifically, a direct measure of how well a nanostructure absorbs light holds wide use in fundamental studies as well as in applied interests, both involving low dimensional materials such as colloidal quantum dots (QDs) and, more recently, solution-based semiconductor nanowires (NWs). For example, such values are useful in quantifying properties such as sample concentrations, fluorescence quantum yields, QD/NW photodetector conversion efficiencies, external photovoltaic efficiencies, lasing thresholds, and multicarrier excitation regimes. Despite the recognized usefulness of these values, it has not been straightforward to extract cross-sections and extinction †
Part of the special issue “Arthur J. Nozik Festschrift”. * To whom correspondence should be addressed. E-mail:
[email protected]. ‡ University of Notre Dame. § University of Waterloo.
coefficients from chemically synthesized nanostructures. Unlike molecules or other systems with well-defined molecular weights, colloidal QDs and solution-based NWs do not possess such exact physical properties. This arises, in part, because current syntheses do not fully control sample size polydispersities. Even in the best preparations, colloidal nanocrystals show size distributions ranging from 4 to 5%.1-3 Likewise, more recent solution-based NW syntheses yield diameter distributions between 15 and 30%.4-8 Notably, no length control has been achieved in these latter preparations. The effects of polydispersity have been ameliorated, to some extent, by the development of size selection schemes (i.e., “size selective precipitation”)2 or by improved chemical syntheses that yield high-quality nanocrystals with narrow (as made) size distributions.1 Additional complications arise from the lack of suitable nanostructure “sizing” curves (peak band edge absorbance versus size) and has only recently been addressed by the development of such tables for CdS, CdSe, and CdTe QDs9 as well as for other model systems. Further hindering the definitive structural characterization of chemically grown nanostructures are organic surfactants used in the above-mentioned syntheses. These surfactants control the
10.1021/jp066034w CCC: $33.50 © 2006 American Chemical Society Published on Web 11/17/2006
CdSe and CdTe Nanowire Optical/Electrical Study size, shape, growth kinetics, and even crystallographic phase of solution-based nanomaterials.10 They also electronically passivate QD/NW surfaces, improve their fluorescence quantum yields, provide solubility in a variety of organic/aqueous solvents, and protect their surfaces from oxidation. However, as an unintended consequence, actual masses of solution-based nanostructures are greatly influenced by these surface bound species. Thus, the weight of dried QD/NW powders is not necessarily representative of the actual number of nanostructures present in a sample. In spite of these problems, absorption cross-sections and associated molar extinction coefficients have been evaluated using either thermal gravimetric analysis (TGA),11 acid digestion followed by inductively coupled plasma mass spectroscopy12 (alternatively, inductively coupled plasma atomic emission spectroscopy), atomic absorption spectroscopy,9,13,14 anodic stripping voltammetry,15 or other approaches,16,17 all in conjunction with absorption spectroscopy. Each approach brings with it certain advantages and disadvantages. For example, TGA measurements are susceptible to errors from insufficient “washing” of the material, allowing resulting sample masses to be corrupted by surfactant contributions. Furthermore, wellseparated, temperature-dependent weight drops are required to unambiguously differentiate surfactant from semiconductor core contributions.18 Potential errors in inductively coupled plasma (ICP)-based measurements originate from the inexact stoichiometry of many nanostructures, given that binary QDs and NWs often have slight excesses of one element over the other. This likely originates from the abrupt termination of nanomaterials at their surfaces. On the other hand, ICP-based measurements are less sensitive to surfactant issues, making the approach intrinsically more reproducible. With all methods, however, robust absorption cross-sections, in good agreement with prior theoretical estimates, can be obtained.11,12 In this study, absorption cross-sections (σabs) and corresponding molar extinction coefficients () of solution-synthesized CdSe and CdTe NWs are determined through concerted transmission electron microscopy (TEM), UV/visible extinction spectroscopy, and inductively coupled plasma atomic emission spectroscopy (ICP-AES). Specifically, TEM analyses of highquality NW ensembles provide a measure of average nanowire diameters and corresponding diameter size distributions. ICPAES measurements and subsequent comparisons to commercial Cd, Se, and Te atomic emission standards yield sample stoichiometries as well as overall NW concentrations. Accompanying solution linear absorption spectra of the same ensembles allow both NW absorption cross-sections and corresponding molar extinction coefficients to be determined. Experimentally determined NW concentrations and UV/ visible extinction data are used to relate the total absorption of a NW ensemble to the total cross-section of a hypothetical single wire, responsible for all of the absorption in solution. The length of this single NW is then calculated using ICP-AES-determined sample concentrations and TEM-determined average diameters. Given the total length of the effective NW and its complementary absorption cross-section, an absorption “cross-sectional density” (χabs) with units of cm2/µm can be determined. The product of χabs and the average NW length in an ensemble then provides intermediate values for the absorption cross-section, which reside between two limits, the case where the NW lies parallel to the incoming light polarization and the case where it is oriented perpendicularly. Complicating these measurements are contributions to the extinction from NW scattering. For nanoscale objects such as
J. Phys. Chem. B, Vol. 110, No. 50, 2006 25323 nanocrystals with diameters between 5 and 10 nm (much smaller than the wavelength, λ, of light), scattering can generally be neglected. This is due to the faster decay of the nanocrystal scattering efficiency with size (∝r6) relative to its absorption (∝r3)19 and justifies the direct use of UV/visible extinction data for calculating QD absorption cross-sections.11,12 However, when the nanostructure size exceeds 20-100 nm, scattering begins to dominate the extinction. A NW is therefore an interesting object from a scattering perspective since one dimension (the diameter) is much smaller than the wavelength of light but the other (its length) is generally >1 µm and hence exceeds λ by 1-2 orders of magnitude. Since the current study focuses on the NW absorption, measured UV/visible extinction data are corrected for scattering to properly determine the NW absorption cross-section. Toward this end, a model describing the interaction between a long, finite length, dielectric cylinder and a plane wave is used to calculate the scattering contribution to the measured UV/visible extinction spectra.20 Obtained absorption cross-sections compare well with previous theoretical estimates obtained by modeling the NW density of states as bulklike far from the band edge.21,22 Resulting values are nearly 5 orders of magnitude larger than those measured for comparable diameter CdSe and CdTe QDs.9,11,14 Practical (and immediate) implications of this study include a more quantitative measure of single NW fluorescence quantum yields (previously estimated to be ∼0.1% 23), better estimates of multicarrier excitation regimes in transient differential absorption experiments,24 and an estimate of external photoconversion efficiencies in recently developed NW photodetectors.25 Materials and Methods Materials. Toluene, methanol, chloroform, hexanes, and nitric acid were used as received from Fisher Scientific. In the synthesis of CdSe and CdTe NWs, trioctylphosphine oxide (TOPO, 99%, Aldrich or 90%, Strem), cadmium oxide (CdO, 99%, Aldrich), octanoic acid (98%, Lancaster), trioctylphosphine (TOP, 90%, Aldrich), tributylphosphine (TBP, 95%, Aldrich), decylphosphonic acid (DPA, 98%, Lancaster), dodecylamine (DDA, 98%, Acros), and tetraoctylammonium bromide (TOAB, 98%, Acros) were all purchased and used as received. Dimethylcadmium (CdMe2, 99.995%, Strem) was passed through a 0.2 µm PTFE filter in a glovebox and was stored at low temperatures prior to use. Tellurium shot (200 mesh, 99.8%, Aldrich) was used to prepare tributylphosphine telluride (TBPTe, ∼0.5 M). Specifically, Te powder (0.32 g, 2.5 mmol) was mixed with TBP (5 mL, 20 mmol) in a glovebox and was allowed to stir overnight. The resulting solution was then filtered through a 0.2 µm PTFE syringe filter, yielding a clear yellow solution. Selenium powder (100 mesh, Aldrich) was used to prepare trioctylphosphine selenide of various molarities (TOPSe, 0.5-2.0 M) by mixing known amounts (0.197-0.790 g, 2.5-10.0 mmol) with TOP (5.0 mL, 11.2 mmol). Commercial atomic emission standards for Cd (1000 ppm), Se (1000 ppm), and Te (1000 ppm) in 2-3% nitric acid were purchased from Fisher Scientific and were used as received. Calibration solutions with trace concentrations ranging from 0.5 to 25 ppm were obtained by successively diluting stock solutions. Synthesis of Straight CdSe Nanowires. Straight CdSe NWs were synthesized according to the procedure described in refs 5 and 6. Briefly, the process consists of preparing a solution of TOPO (90%, 2.5 g, 6.47 mmol), CdO (25 mg, 0.194 mmol), and octanoic acid (230 µL, 1.43 mmol) in a three-neck flask. A condenser and thermocouple are then added whereupon the apparatus is connected to a Schlenk line. The solution is dried
25324 J. Phys. Chem. B, Vol. 110, No. 50, 2006 and degassed under vacuum at ∼100 °C. When complete, the reaction vessel is back-filled with N2 and is heated to 330 °C. An “injection solution” consisting of standardized ∼1.5 nm diameter Au/Bi NPs (175-230 µL, 0.067-0.087 µmol)5 and 1.0 M TOP-Se (25 µL, 25 µmol) is then prepared in a glovebox. When the temperature of the reaction mixture stabilizes, the injection solution is rapidly introduced into the three-neck flask. This yields an immediate color change from clear to red/brown, indicating the formation of CdSe NWs. The resulting solution is then left at high temperature for approximately 1 min before being cooled to room temperature. Synthesis of Straight CdTe Nanowires. Straight CdTe nanowires were synthesized according to the procedure outlined in ref 8. Briefly, a mixture of TOPO (99%, 10 g, 28 mmol), TOP (90%, 11.5 mL, 26 mmol), TBP-Te (0.5 M, 0.05 mL, 25 µmol Te), and DPA (71 mg, 0.32 mmol) is heated under vacuum at ∼100 °C to dry and degas the mixture. The reaction vessel is then back-filled with N2 and is heated to 275 °C. At this point, a solution consisting of Au/Bi NPs (175 µL, ∼0.067 µmol) and CdMe2 (11 µL, ∼0.16 mmol) is prepared in a glovebox and is loaded into a 1 mL disposable syringe. When the temperature of the reaction mixture reaches 275 °C, the Au/ Bi solution is injected into the three-neck flask. After a short induction period, the solution undergoes a rapid color change from clear to dark brown, indicating the growth of CdTe NWs. The reaction vessel is kept at high temperature for approximately 1 min before being cooled to room temperature. Occasionally a precipitate is observed following the injection but has no bearing on the quality of the resulting NWs. Workup. Both CdSe and CdTe NWs are processed in the same manner. Toluene (∼10 mL) is added to the cooled reaction mixture to prevent TOPO from solidifying. Excess methanol is then introduced to precipitate the NWs. The resulting suspension is centrifuged at 4400 rpm to yield a clear supernatant, which is discarded. The recovered precipitate is subsequently resuspended in hexanes and centrifuged, and any supernatant is removed. This “washing” procedure is performed at least two more times to reduce the amount of excess surfactant attached to the NWs. Once complete, the wires can be resuspended in either toluene or chloroform. Small amounts of either DDA or TOAB may then be added to aid the solubility of the wires. Instrumentation TEM. NW diameters and size distributions were evaluated by analyzing transmission electron microscope images of NW ensembles. Survey TEMs were obtained during the synthesis with a JEOL 100SX TEM. All low- and high-resolution TEM micrographs were taken with a JEOL 2010 TEM operating at 200 kV. Images were subsequently scanned and analyzed using commercial software. Linear Absorption. All UV/visible extinction spectra for CdSe and CdTe NWs were obtained at room temperature using a Cary 50-Bio UV/visible spectrophotometer. NW solutions were prepared by diluting known amounts of a given stock solution with 3 mL of toluene. A 1 cm path length Suprasil cuvette was used in all experiments. Baseline corrections were also conducted prior to each measurement. Transient Differential Absorption. Ultrafast transient differential absorption experiments were conducted with a Clark MXR CPA 2010 laser system in conjunction with an Ultrafast Systems detection scheme, consisting primarily of a fiber-based spectrometer. The second harmonic of the fundamental at 387 nm (3.2 eV) was used to excite all samples. A fraction of the fundamental was simultaneously passed through a sapphire plate
Protasenko et al. to generate a white light continuum between 420 and 800 nm (1.5-2.9 eV). An optical delay line then provided pump-probe delays ranging from 150 fs to 1.5 ns. NW samples were prepared by drop casting wire solutions onto transparent fused silica substrates. Nominal excitation intensities were ∼80 mW/cm2, with peak intensities on the order of ∼500 MW/cm2 (∼150 fs pulse width). The chirp between 450 and 750 nm is approximately 250 fs. Acquired spectra were taken at delays ranging from 600 fs to 1.5 ns after the pulse and were not chirpcorrected. Representative spectra were selected to best illustrate the various subband positions, which were observed not to vary with delay. Inductively Coupled Plasma Atomic Emission Spectroscopy. Concentrations of Cd, Se, and Te in NW samples were determined by inductively coupled plasma atomic emission spectroscopy using a Perkin-Elmer Optima 3300XL instrument. Samples for ICP-AES were prepared by digesting NW samples (typically 300 µL) with concentrated nitric acid and diluting the resulting residue in 5% nitric acid (v/v, 10 mL). Specifically, adding concentrated nitric acid to NW solutions turns them orange. Subsequent heating at ∼70-100 °C eventually renders the mixtures colorless. The remaining solvent is evaporated, leaving behind a residue which is re-dissolved in 5% nitric acid. Elemental concentrations were quantified by comparing NW ICP-AES data to results obtained from known standards. Cd, Se, and Te calibration curves were prepared using solutions having trace concentrations ranging from 0.5 to 25 ppm. All experimental calibration curves could be fit to lines with R2 values greater than 0.999. Results and Discussion NW Characterization. For the current study, high-quality NWs were made using the above-outlined SLS approach. Figure 1 shows representative low- and high-resolution TEM images of the resulting CdSe and CdTe NWs. Low-resolution images (Figure 1a,b) reveal the overall uniformity of the wires as well as their long lengths, exceeding 10 µm in some cases. Additional low-resolution images are provided in the Supporting Information section. High-resolution micrographs (Figure 1c,d) illustrate that both CdSe and CdTe NWs are crystalline. Diameters range from 6 to 42 nm in the case of CdSe and from 7.5 to 11.5 nm for CdTe. Accompanying size distributions are 20-40% and 15-25%, respectively. In either case, intrawire diameter variations are on the order of 5%.6,8 Organic surfactants such at trioctylphosphine oxide passivate NW surfaces, rendering them soluble in common organic solvents such as toluene or chloroform. Although crystalline, both CdSe and CdTe NWs exhibit phase admixtures. Specifically, high-resolution TEM measurements reveal that both zincblende (ZB) and wurtzite (W) phases coexist in the NWs. These phase admixtures, as well as twinning within ZB sections, can be seen in properly oriented wires, viewed along a 〈110〉 zone.6,8 Along this line, all NWs grow along the 〈111〉 and 〈0001〉 directions of their corresponding ZB and W phases.5,6,8 The origin of these phase admixtures is not fully understood but can be rationalized because of the low overall energy difference between ZB and W in each material.26 Conceptually, the phase transition simply involves a 60° rotation of the bond between metal/chalcogen layers along the 〈111〉/ 〈0001〉 NW growth direction.26 As a consequence, temperature fluctuations during the synthesis could easily induce transitions from one phase to the other. More details about the structural characterization of the NWs can be found in refs 6 and 8. Representative solution extinction spectra of CdSe and CdTe NWs are provided in Figure 2 (solid lines). Band edge emission
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Figure 1. Low-resolution TEM images of (a) CdSe and (b) CdTe NWs. High-resolution TEM images of (c) CdSe and (d) CdTe NWs.
Figure 2. Linear absorption (solid line) and accompanying transient differential absorption spectra (dashed line) of (a) CdSe and (b) CdTe NW ensembles. Mean diameters and associated size distributions are indicated.
is also observed in CdSe NWs at the ensemble and single-wire levels with no sign of deep trap emission.23 In either CdSe or CdTe, however, Figure 2 illustrates clear blue shifts of the effective band edge relative to the bulk band gap of CdSe (1.74 eV, 300 K) and CdTe (1.50 eV, 300 K). Structure is also seen at higher energies, suggesting confinement-induced subbands. This is supported by accompanying transient differential absorption experiments (dashed lines), showing at least two peaks underlying the linear absorption of CdSe and CdTe NWs.8,24 Their presence is consistent with narrow NW diameters,
generally below twice the bulk exciton Bohr radius of each material (CdSe aB ) 5.6 nm; CdTe aB ) 7.5 nm).27 In all cases, the position of the first transient absorption-induced bleach agrees well with the estimated spectral position of the band edge from above extinction measurements. The second peak in CdSe is also in excellent agreement with the shoulder in accompanying UV/visible extinction spectra. Such agreement between the spectral position of structure in the UV/visible data and the TAinduced bleaches is an indication that NW scattering does not dominate the linear absorption. This will also be shown below
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Figure 3. Absorption polarization anisotropy of a single-branched CdSe NW having a v-shape. Typical anisotropy values are on the order of 0.75. The red circles (blue triangles) are data from the NW arm marked with the corresponding symbol in the inset. Solid/dashed lines are cos2 γ fits to the data, where γ is the angle between the NW axis and incident light polarization.
theoretically. For both CdSe and CdTe, band edge peak positions track the NW diameter, supporting the presence of confinement effects. The case of CdTe is slightly unusual though since the position of the first bleach appears systematically blue-shifted relative to the band edge as deduced from above extinction measurements. This phenomenon is not currently understood but is present in all CdTe samples we have studied. The optical properties of both CdSe and CdTe NWs show strong polarization anisotropies (p) in the absorption25,28 (and emission in the case of CdSe28) with typical single-wire anisotropies of p ∼ 0.75.28 An example involving a single CdSe NW is provided in Figure 3 and illustrates the one-dimensional (1D) nature of the wires. Specifically, two polarization response curves are observed, originating from each arm of the v-shaped wire. Both responses are identical, with absorption maxima (minima) occurring when the incident light polarization is aligned collinearly (perpendicularly) to each NW arm. Corresponding polarization anisotropies of the top and bottom arms are both ∼0.80. The ∼112° phase shift between the two curves originates from the 109.5° crystallographic angle separating both arms given their growth out of equivalent {111} faces of a common ZB core. More information about branched NWs can be found in refs 6-8. When considered along with the high degree of sample crystallinity, structured linear absorption, subband-resolved transient differential absorption spectra, and lack of defect-related emission, these properties all suggest that the absorption of the wires is intrinsic to the material as opposed to defect-related. Optical Constants. NW absorption cross-sections are of primary interest for quantifying fundamental ensemble and single-NW optical/electrical properties.23 For example, knowledge of σabs enables a more quantitative measure of NW fluorescence quantum yields. Absorption cross-sections also provide direct estimates for the number of electron-hole pairs generated with each absorption event. Excitation thresholds, demarking bimolecular exciton-exciton annihilation from complementary three-carrier Auger cooling regimes, are also better revealed.24 Corresponding molar extinction coefficients are also useful since they provide a ready way to measure
solution concentrations using Beer’s Law, Aabs ) clcuv, where Aabs is the absorbance or optical density of the sample, (M-1cm-1) is the molar extinction coefficient, c (M) is the desired concentration, and lcuv (cm) is the optical path length. Molar extinction coefficients and absorption cross-sections are interrelated through
σabs )
2.303(1000) NA
(1)
where NA is Avogadro’s number.29 In what follows, σabs and are determined using results from three experimental techniques: TEM, UV/visible extinction spectroscopy, and ICP-AES measurements. At the heart of the approach is an evaluation of χabs, an absorption cross-sectional density with units of cm2/µm. This is achieved by TEM sizing of NW diameters and ICP-determined concentrations, ultimately enabling one to calculate the total length (L) of a single hypothetical wire responsible for all of the absorption in the sample. The product of χabs and the typical NW length in the ensemble then provides estimates for σabs. This approach is applicable at all frequencies spanning the spectral region between the band edge and higher energies, provided that the sample’s size distribution is not unduly broad. Calculations. In our approach, average NW diameters and corresponding size distributions are measured by analyzing TEM images of CdSe and CdTe wires. For the evaluated CdSe ensembles, diameters range from 6 to 42 nm with associated diameter distributions between 20 and 40%. Analogous CdTe samples have diameters spanning from 7.5 to 11.5 nm with associated size distributions between 15 and 25%. Of note is the much tighter size distribution of the as-made CdTe NWs. This may result from the more favorable growth kinetics of the material, as noted previously.8 Resulting diameters and corresponding size distributions are summarized in a table provided in the Supporting Information. A small fraction of the initial NW stock (typically ∼300 µL) is then digested with nitric acid. The resulting residue is diluted with 10 mL of 5% nitric acid whereupon ICP-AES measure-
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ments are conducted on the sample. The results are subsequently compared to those from calibration standards, yielding sample stoichiometries. Specifically, ICP-AES results show an approximate 1:1 Cd:Se ratio [1:1.07 (std dev ) 0.15)] in CdSe ensembles and a similar 1:1 Cd:Te ratio [1:0.98 (std dev ) 0.06)] in CdTe NWs. A table of NW stoichiometries is provided in the Supporting Information. These ICP-AES-determined elemental compositions are consistent with previous results from energy dispersive X-ray spectroscopy (EDXS) measurements on both CdSe and CdTe NWs.6,8 Slight deviations from an exact 1:1 stoichiometry may arise from the abrupt termination of NWs at their surfaces. Concentrations of Cd, Se, and Te in each sample (NW stock) are simultaneously obtained when the volume (dilution) of the ICP-AES solution is known. This allows one to quantify the total moles of CdSe and CdTe in each ensemble, using formula weights for CdSe and CdTe. The total length (in micrometers) of a single NW with a diameter matching the TEM-measured mean ensemble diameter is then calculated using
L)
4Mnw Fπd2
× 104
(2)
where Mnw is the solution NW mass in 1 cm3, F is the density of CdSe (5.81 g/cm3) [CdTe, 5.85 g/cm3], and d is the average NW diameter (cm) determined from TEM measurements. Since experimental stoichiometric ratios typically differ slightly from 1:1, the smaller of the two elemental mole quantities is used for both elements in calculating Mnw. The extinction (Aext), which includes both absorption and scattering, from a given dilution of the original NW stock is then obtained using UV/visible spectroscopy. This value is corrected for scattering contributions (described in more detail below) to yield Aabs. With the solution concentration at hand, the angle- and crystallographic phase-averaged absorption crosssections of all wires in solution are now established. This assumes negligible absorption contributions from NW ends and is reasonable given the typical ∼1000:1 aspect ratios of the wires. Specifically, from Aabs ) σabsndenslcuv log e and eq 2 an absorption cross-section per unit length (or “cross-sectional density”), χabs, averaged over all angles and crystallographic phases is calculated using
χabs )
Aabs ndenslcuv(log e)LN
(3)
where N (a dilution factor) is the ratio of NW concentrations in the extinction and ICP-AES measurements ([NW]UV/vis/ [NW]ICP-AES), lcuv ) 1 cm and ndens ) 1/cm3. Light Scattering by NWs. Complicating estimates of the absorption cross-section are scattering contributions to the extinction (Aext). Hence, to obtain Aabs from measured Aext values, scattering contributions are accounted for theoretically using an expression that angle-averages the NW scattering crosssection, σscat, over all possible wire orientations relative to the incident (unpolarized) light. In the current study, angle-averaged σscat values from an ensemble of finite length cylinders, excited with depolarized light, is obtained using the results of refs 20 and 30. Specifically, in the limit of finite length cylinders with a diameter, d, less than the excitation wavelength λ [πn′d/λ , 1, where n′ ) Re(ns/nI) is the real part of the ratio between the cylinder’s refractive index, ns, and that of the surrounding medium, nI] and where |n′′/n′| , 1 (n′′ ) Im(ns/nI) as well as
when the cylinder’s length, l, satisfies l > λ/2πnI, the scattering cross-section can be angle-averaged for arbitrary orientations of the cylinder relative to the incoming light. An explicit mathematical expression for σscat is obtained
σscat )
π 128
(κd)4l2
|
n2 + 1
(
[ ( | ( )|)])
| ∫0π m)0∑ R11m (θ) Re 2F2 1,1;2,6-
n2 - 1
2
3
m;2iκl sin
θ 2
sin θ dθ (4)
where κ ) 2π/λ is the wavevector of the incident light, d (l) is the cylinder diameter (length), n ) ns/nI is again the ratio between the cylinder’s refractive index and that of the surrounding medium, and R11 m (θ) are coefficients defined in the Supporting Information as well as in ref 20. Re[2F2(1,1;2,6m;2iκl|sin(θ/2)|] is the real part of the hypergeometric function, 2F2(a,b;c,d;z), having generic arguments a, b, c, d, and z, where z ) 2iκl|sin(θ/2)|,31 m is a summation index, and θ (with the range of 0 f π) is the angle between the incident light and the scattering direction. Using eq 4, we calculate σscat for CdSe and CdTe NWs dispersed in toluene. Since the NW scattering efficiency depends on the excitation wavelength through ns and because such numbers are not currently available for CdSe and CdTe NWs, corresponding bulk values of the refractive index are used.32,33 The refractive index of toluene is ≈1.5.34 Two interesting features of eq 4 are noted. First, σscat appears to scale quadratically with l. It therefore grows much faster than σabs. In general, the absorption cross-section is proportional to the NW volume and hence grows linearly with length. This would, in turn, suggest that an accurate accounting of NW ensemble length distributions is needed in order to properly calculate scattering contributions to the overall extinction. Fortunately, a detailed analysis reveals that the integral in eq 4 is well-approximated by ∼1/l. This behavior is caused by an increasing localization of the hypergeometric function Re[2F2(1,1;2,6-m;2iκl|sin(θ/2)|)] about values near θ ) 0 with increasing l and values of sin θ f 0.20,35 Figure 4a illustrates the phenomenon by plotting the angular distribution of the normalized (relative to the θ ) 0 value) scattering intensity for three different NW lengths and for arbitrary NW/light propagation direction orientations. From the graph it is apparent that, in all three cases, light is preferentially forward-scattered along the incident light propagation direction and is increasingly concentrated about θ ) 0 with growing NW length. As a consequence, σscat from eq 4 scales linearly with l. This is further illustrated in Figure 4b, where eq 4 is plotted at 555 nm for an arbitrary 14 nm diameter CdSe NW as a function of length. Similar results are obtained at all other sizes and wavelengths between 300 and 1200 nm. The end result is that the ratio between the NW absorption and scattering cross-sections is independent of length. Thus an accurate accounting of ensemble NW length distributions is not needed in subsequent evaluations of either Aabs or σabs. Figure 5 illustrates the scattering contribution to the total extinction for representative 12.8 and 21.4 nm diameter, 1 µm long CdSe NWs. From the graphs, it is apparent that scattering contributes only a small amount to the overall extinction in either case. Absorption therefore dominates Aext in the spectral range of interest. However, scattering contributions steadily increase toward the blue. In this respect, eq 4 shows that σscat scales as 1/λ4. This is analogous to the behavior of small particles19 and, as a consequence, NWs are predicted to scatter blue light efficiently. While not significant for thin wires and for red
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Figure 5. Angle-averaged extinction (solid line) and scattering (circles) cross-sections of (a) 12.8 and (b) 21.4 nm diameter 1 µm long CdSe NWs (where, for example, 4.0e-11 represents 4.0 × 10-11). Inset: Resulting angle-averaged absorption coefficients at various wavelengths. Figure 4. (a) Polar plot of scattered light intensity for different NW lengths. All intensities are normalized to the θ ) 0 value, where θ is the angle between the scattered light and incident light propagation direction. (b) Scattering cross-section (where, for example, 2.5e-11 represents 2.5 × 10-11) of a 14 nm diameter CdSe NW excited at 555 nm, showing a linear dependence of σscat with NW length. The solid line is a linear fit to the data.
frequencies, the 1/λ4 scattering becomes important when dealing with large-diameter (>30 nm) NWs at near-UV frequencies. To illustrate, for the CdSe/CdTe samples considered, the scattering contribution to the total extinction at 300 nm ranges from 2.3 to 37% when d increases from 6.4 to 22.0 nm. A CdSe sample where d ) 42.4 nm and where scattering becomes important is considered separately below. In particular, for this sample direct application of eq 4 under the following conditions, λ ) 300 nm, d ) 42.4 nm, and n′ ≈ 2.8, yields πn′d/λ ≈ 0.78 and (unphysical) values of the scattering cross-section which exceed measured σext values (Supporting Information Figure 2a). This occurs because eq 4 originates from a Bessel function and Hankel function Taylor expansion which assumes πn′d/λ , 1. Thus it is no longer valid when NW diameters exceed ∼30 nm. Since an exact form of eq 4 is not available,20 we consider the effects of the Taylor expansion by comparing exact versus Taylor-expanded scattering cross-sections of a model system, consisting of an infinite oriented cylinder.35,36 Figure 2b of the Supporting Information illustrates this comparison and shows that eq 4 likely overestimates the NW scattering cross-section of the d ) 42.4 nm sample by a factor of ∼2 at 300 nm. At frequencies above 400 nm in either expression, differences are nearly negligible. In the ensuing graphs and calculations, however, the d ) 42.4 nm CdSe data point has been excluded since a more reliable estimate of σscat in the blue has not been obtained.
Our subsequent evaluation of σabs for all CdSe/CdTe NW ensembles, which accounts for scattering contributions to the extinction, proceeds in one of two ways. Either Aabs is obtained by subtracting Ascat from Aext [with relevant values of Ascat obtained by applying eq 3 with χscat (eq 4 for a 1 µm long NW) replacing χabs and Ascat in place of Aabs] or alternatively χabs is obtained directly by subtracting χscat from χext since in either case
Aext ) Aabs + Ascat
(5a)
χext ) χabs + χscat
(5b)
To obtain χabs at other frequencies, reported absorption crosssectional densities can be scaled to account for differences in the sample absorption since a linear proportionality exists between χabs and Aabs (eq 3). One therefore has
χabs,new ) χabs,edge (Aabs,new/Aabs,edge)
(6)
When χabs is multiplied by the average length of a NW in solution (l, in micrometers),
σabs ) χabsl
(7)
angle-averaged values of the NW cross-section are obtained. Results Measured χabs values for CdSe NWs with diameters between 6 and 42 nm range from 6.93 × 10-13 to 3.91 × 10-11 cm2/µm at the band edge (from 1.73 to 1.79 eV). In CdTe, band edge (from 1.65 to 1.80 eV) χabs values (diameters between 7.5 and 11.5 nm) range from 4.32 × 10-13 to 5.10 × 10-12 cm2/µm. Further to the blue at 488 nm, CdSe (CdTe) χabs increase to
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Figure 6. Absorption cross-section at the band edge, 488 nm (2.54 eV), and 387 nm (3.2 eV) for both (a, c, e) CdSe and (b, d, f) CdTe NWs (where, for example, 4.0e-11 represents 4.0 × 10-11). Solid lines are linear fits to the data. Dashed lines are theoretical predictions, assuming a bulklike density of states.
values between 3.38 × 10-12 and 5.50 × 10-11 cm2/µm (from 1.80 × 10-12 to 1.99 × 10-11 cm2/µm), depending on size. A summary of absorption cross-sectional densities for all CdSe/ CdTe NW ensembles is provided in the Supporting Information. Corresponding absorption cross-sections are obtained using eq 7, assuming l ) 1 µm. Resulting σabs values for randomly oriented 1 µm long, 6-42 nm (7.5-11.5 nm) diameter CdSe (CdTe) NWs range from 6.93 × 10-13 to 3.91 × 10-11 cm2 (from 4.32 × 10-13 to 5.10 × 10-12 cm2) at the band edge and from 3.38 × 10-12 to 5.50 × 10-11 cm2 (from 1.80 × 10-12 to 1.99 × 10-11 cm2) at 488 nm. Cross-sections at other frequencies are obtained using the outlined linear scaling in eq 6. By inverting eq 1, corresponding molar extinction coefficients of 1.81 × 108 to 1.02 × 1010 M-1 cm-1 (1.13 × 108 to 1.33 × 109 M-1 cm-1) and 8.84 × 108 to 1.44 × 1010 M-1 cm-1 (4.71 × 108 5.20 × 109 M-1 cm-1) are obtained at the band edge and at 488 nm, respectively. These numbers apply for l ) 1 µm NWs. Prior estimates of NW absorption cross-sections far from the band edge have employed a theoretical expression used to evaluate the cross-section of corresponding colloidal QDs.11,12,21,22 The model assumes a bulklike density of states at high energies, is appropriate only far from the band edge where size quantization effects do not impart strong discreteness to the spectrum, and predicts cross-sections which scale with volume. In the case of colloidal nanocrystals this has led to good agreement with experimentally measured values.11,12 When applied to NWs, the
expression is modified to take into account the different NW geometry and local field factors as described in refs 23 and 24. Figure 6 compares model predictions (dashed lines) to experimental values (closed circles) by plotting σabs versus d2. Solid lines are linear fits to the data, showing an apparent linear dependence between the two and a corresponding scaling of σabs with NW volume. In the case of CdTe, however, the lack of data near the origin only allows us to suggest this linear relationship for those points obtained over the size range d ) 7-12 nm. In either case, however, good agreement is found between experiment and theory at 488 nm and at 387 nm (a wavelength previously used in NW transient differential absorption experiments).24 In the former, predicted values are within a factor of 1.5. In the latter, the agreement is likwise within a factor of 2. The better agreement at 488 nm is likely due to the smaller error in calculating σscat using eq 4 since the πn′d/λ , 1 criteria is more closely satisfied in this region. Surprisingly, however, large discrepancies are not observed at the band edge (Figure 6a,b). Instead, less than a factor of 5 differences are seen. This may be due to the presence of subbands underlying the NW linear absorption in contrast to the discrete atomic-like transitions underlying corresponding quantum dot absorption spectra. Proportionality constants between σabs (cm2) and d2 (nm2), extracted from the above linear fits to the data in the region of interest are 2.33 × 10-14 (5.31 × 10-14) for CdSe (CdTe) at the band edge [σabs,edge(CdSe) ) 2.33 × 10-14d2; σabs,edge(CdTe)
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) 5.31 × 10-14d2 - 2.31 × 10-12]. Likewise, at 488 nm the proportionality constants for CdSe (CdTe) are 1.06 × 10-13 (2.04 × 10-13) [σabs,488nm(CdSe) ) 1.06 × 10-13d2 and σabs,488nm(CdTe) ) 2.04 × 10-13d2 - 8.42 × 10-12]. Corresponding proportionalities at 387 nm are σabs,387nm(CdSe) ) 1.40 × 10-13d2 and σabs,387nm(CdTe) ) 4.11 × 10-13d2 - 1.55 × 10-11. Absorption Cross-Section Limits. The above cross-sections include the effects of orientational averaging. However, to extract σabs values free of this, we explicitly consider the cos2 γ dependence of both χabs and σabs, where γ is the angle between the incident light electric field and the NW growth axis.23,28 Specifically, values of χ| (or χ⊥) can be extracted using above χabs estimates and the NW absorption polarization anisotropy p ) (I| - I⊥)/(I| + I⊥), where I| and I⊥ are the intensities of the absorbed light with polarization parallel and perpendicular to the NW axis. The resulting expression for χ|, derived in detail in the Supporting Information, is
χ| )
3(1 + p) χ 3 - p abs
(8)
An analogous expression for χ⊥, also derived in the Supporting Information, is
χ⊥ )
3(1 - p) χ 3 - p abs
(9)
Thus, when the NWs are aligned collinearly with the polarization of the incident light, all χ values increase by a factor of ∼7/3, given previously measured p values of ∼0.75.25,28 Relevant limits of the prefactor are 3 and 1 when p ) 1 and p ) 0, respectively. Given χ| (or χ⊥), values of the absorption crosssection, free of angle averaging, can then be determined using eq 7. These results help quantify observations of strong NW absorption/emission polarization anisotropies due to dielectric contrast effects37 as well as confinement-induced optical selection rules.38 They also show that the wires absorb most effectively when the incident light polarization is aligned along the NW growth axis. Nanowire to Colloidal Quantum Dot Comparison (1D versus 0D). With values of χabs and σabs for CdSe and CdTe NWs in hand, absorption cross-sections can be scaled to account for volume differences between nanowires and corresponding colloidal quantum dots. Comparison of these values may then highlight fundamental differences between the two, stemming from their intrinsic 0D or 1D nature. To illustrate, absorption cross-sections of CdSe and CdTe NWs with diameters between 5 and 12 nm were compared to those of their QD counterparts, having the same sizes (i.e., diameters). Previously discovered linear relationships between CdSe/CdTe NW cross-sections and d2 [σabs,edge(CdSe) ) 2.33 × 10-14d2; σabs,edge(CdTe) ) 5.31 × 10-14d2 - 2.31 × 10-12] were used to calculate model σabs values for each material. Corresponding CdSe/CdTe QD crosssections were estimated using experimentally determined (M-1 cm-1) versus d relationships found by Peng and co-workers [CdSe ) 21145d2.0226 and CdTe ) 10043d2.12].9 Resulting values were subsequently converted to absorption cross-sections using eq 1. Next, obtained NW σabs values were scaled to account for volume differences between nanowires and QDs. Resulting volume-normalized NW cross-sections are then plotted in Figure 7 along with corresponding QD cross-sections. The figure shows that, even after accounting for volume disparities, scaled CdSe and CdTe NW cross-sections are larger than those of complementary colloidal QDs. Within the diameter range of interest, factors of ∼1-8 differences are readily
Figure 7. Scaled band edge cross-sections of (a) CdSe and (c) CdTe NWs compared to those of corresponding QDs (where, for example, 2.0e-14 represents 2.0 × 10-14). NW to QD cross-section ratios for (b) CdSe and (d) CdTe NWs. Analogous comparison at 350 nm for (e) CdSe NWs. Obtained cross-section ratio (f).
apparent (Figure 7b,d). The larger scaled NW cross-sections may originate from differences in the joint density of states between NWs and QDs. In the former, 1D subbands underlie the linear absorption when confinement effects are present. In the latter, discrete atomic-like transitions are responsible for the absorption. Cross-sectional variations between the two should therefore be prominent at the band edge where the joint density of states of each is sparse. This motivates the comparison in Figure 7. Differences between the two, however, should be less prominent further to the blue where both 0D and 1D systems experience a bulklike continuum of states. Indeed, an analogous comparison to CdSe QD cross-sections at 350 nm (3.54 eV) shows nearly identical NW/QD cross-sections (Figure 7e). To execute the comparison, a linear proportionality between σabs and d2 was found for CdSe NWs at 350 nm, σnw,abs ) 1.57 × 10-13d2, and is illustrated in more detail in the Supporting Information. Size-dependent QD cross-sections at 350 nm were then obtained from Leatherdale’s relation, σabs,350nm ) 5.501 × 105r3 where r is the nanocrystal radius in cm.11 Figure 7e shows that when scaled for volume, both NW and QD cross-sections are nearly identical with values within a factor of 2. Interestingly, the NW/QD cross-section ratio is found to be size-independent (Figure 7f). Despite apparent differences, NW/QD cross-section comparisons are actually more complicated since σabs is related to the product of an oscillator strength (fosc) and the underlying 0D or 1D joint density of states. Since QD oscillator strengths likely experience significant size-dependent changes, large variations of the QD cross-section at the band edge could compensate for the smaller volume and reduced density of states relative to NWs. A more detailed and in depth cross-section
CdSe and CdTe Nanowire Optical/Electrical Study comparison therefore awaits better knowledge of relevant QD and NW fosc values. Conclusions Frequency-dependent cross-sections and corresponding molar extinction coefficients for high-quality CdSe and CdTe NWs have been determined. The NWs were made via a recently developed solution-based synthesis using low melting bimetallic nanoparticle catalysts. Resulting CdSe (CdTe) wire diameters range from 6 to 42 nm (7.5-11.5 nm) and are within the intermediate confinement regime of either material. Lengths exceed 10 µm in some cases. Experimental techniques employed include the following: TEM measurements to determine NW diameters, size distributions, and lengths, ICP-AES measurements to obtain NW concentrations, and extinction spectroscopy to measure the linear absorption of the wires. Large absorption cross-sections are found at the band edge and in the blue. Specifically, for CdSe or CdTe, size-dependent σabs () values vary from ∼4 × 10-13 to 5 × 10-11 cm2 (∼1 × 108 to 1 × 1010 M-1 cm-1) at the band edge. At 488 nm, σabs () increases to values between ∼2 × 10-12 and 1 × 10-10 cm2 (between ∼5 × 108 and 2 × 1010 M-1 cm-1). These numbers agree well with prior theoretical estimates of NW absorption cross-sections at high energies where the joint density of states becomes bulklike. Furthermore, obtained σabs () values are nearly 5 orders of magnitude larger than those of corresponding colloidal CdSe and CdTe QDs, primarily due to volume differences. Even volume-normalized NW cross-sections appear larger by nearly a factor of 8, possibly due to fundamental differences in the underlying joint density of states. Implications of these conclusions abound. For example, in low dimensional photodetectors and photovoltaics, although both QDs and NWs span essentially the same spectral range and have absorption spectra which increase toward the blue, NW device efficiencies may be higher than those of comparable QD assemblies, especially in the red, due to their larger absorption cross-sections and intrinsic polarization sensitivities. Acknowledgment. We thank Jim Goebl, Istvan Robel, and Gennady Margolin for early assistance with these measurements, for assistance with the transient differential absorption experiments, and for help in using Matlab to simulate the NW scattering efficiencies. The authors thank the University of Notre Dame, the Notre Dame Faculty Research Program, and the ACS Petroleum Research Fund for financial support. M.K. thanks the Notre Dame Center for Environmental Science and Technology (CEST), the Notre Dame Radiation Laboratory, and the Office of Basic Energy Sciences of the U.S. Department of Energy for financial support and for the use of their facilities. M.K. also thanks the National Science Foundation for a NSF CAREER award in support of this work. M.K. is a Cottrell Scholar of Research Corporation. V.P. and D.B. contributed equally to this work. Supporting Information Available: Additional low-resolution TEM micrographs of CdTe NWs, table of CdSe and CdTe NW diameters, corresponding size distributions, and stoichiometries, a list of R11 m (θ) coefficients, comparison of exact versus Taylor-expanded expressions for the scattering crosssection, a graph illustrating χabs as a function of d2 at the band edge, 488 and 387 nm, derivation of χ| and χ⊥, and a graph illustrating σabs as a function of d2 for CdSe NWs at 350 nm. This material is available free of charge via the Internet at http:// pubs.acs.org.
J. Phys. Chem. B, Vol. 110, No. 50, 2006 25331 References and Notes (1) Peng, Z. A.; Peng, X. G. J. Am. Chem. Soc. 2001, 123, 183. (2) Murray, C. B.; Norris, D. J.; Bawendi, M. G. J. Am. Chem. Soc. 1993, 115, 8706. (3) Katari, J. E. B.; Colvin, V. L.; Alivisatos, A. P. J. Phys. Chem. 1994, 98, 4109. (4) Trentler, T. J.; Hickman, K. M.; Goel, S. C.; Viano, A. M.; Gibbons, P. C.; Buhro, W. E. Science 1995, 270, 1791. Yu, H.; Buhro, W. E. AdV. Mater. 2003, 15, 416. Yu, H.; Li, J. B.; Loomis, R. A.; Wang, L. W.; Buhro, W. E. Nat. Mater. 2003, 2, 517. Yu, H.; Loomis, R. A.; Gibbons, P. C.; Wang, L-W.; Buhro, W. E. J. Am. Chem. Soc. 2003, 125, 16168. (5) Grebinski, J. W.; Richter, K. L.; Zhang, J.; Kosel, T. H.; Kuno, M. J. Phys. Chem. B 2004, 108, 9745. (6) Grebinski, J. W.; Hull, K. L.; Zhang, J.; Kosel, T. H.; Kuno, M. Chem. Mater. 2004, 16, 5260. (7) Hull, K. L.; Grebinski, J. W.; Kosel, T. H.; Kuno, M. Chem. Mater. 2005, 17, 4416. (8) Kuno, M.; Ahmad, O.; Protasenko, V. Bacinello, D.; Kosel, T. H. Chem. Mater., in press. (9) Yu, W. W.; Qu, L.; Guo, W.; Peng, X. Chem. Mater. 2003, 15, 2854. Yu, W. W.; Qu, L.; Guo, W.; Peng, X. Chem. Mater. 2004, 16, 560. Note that corrections to the original study have necessitated a new expression relating to d for CdSe QDs. This expression can be found in the main text. (10) Jun, Y.-W.; Lee, J.-H.; Choi, J.-S.; Cheon, J. J. Phys. Chem. B 2005, 109, 14795. (11) Leatherdale, C. A.; Woo, W.-K.; Mikulec, F. V.; Bawendi, M. G. J. Phys. Chem. B 2002, 106, 7619. (12) Yu, P.; Beard, M. C.; Ellingson, R. J.; Ferrere, S.; Curtis, C.; Drexler, J.; Luiszer, F.; Nozik, A. J. J. Phys. Chem. B 2005, 109, 7084. (13) Schmelz, O.; Mews, A.; Basche, T.; Herrmann, A.; Mullen, K. Langmuir 2001, 17, 2861. (14) Rajh, T.; Micic, O. I.; Nozik, A. J. J. Phys. Chem. 1993, 97, 11999. (15) Shim, M.; Guyot-Sionnest, P. J. Chem. Phys. 1999, 111, 6955. (16) Vossmeyer, T.; Katsikas, L.; Giersig, M.; Popovic, I. G.; Diesner, K.; Chemseddine, A.; Eychmuller, A.; Weller, H. J. Phys. Chem. 1994, 98, 7665. (17) Jacobsohn, M.; Banin, U. J. Phys. Chem. B 2000, 104, 1. (18) Kuno, M.; Mikulec, F. V.; Bawendi, M. G. J. Chem. Phys.1997, 106, 9869. (19) Jiang, J.; Bosnik, K.; Maillard, M.; Brus, L. J. Phys. Chem. B 2003, 107, 9964. (20) Shi, Y.; McClain, W. M. J. Chem. Phys. 1993, 98, 1695. (21) Ricard, D.; Ghanassi, M.; Schanne-Klein, M. C. Opt. Commun. 1994, 108, 311. (22) Klimov, V. I. J. Phys. Chem. B 2000, 104, 6112. (23) Protasenko, V. V.; Hull, K. L.; Kuno, M. AdV. Mater. 2005, 17, 2942. (24) Robel, I.; Bunker, B. A.; Kamat, P. V.; Kuno, M. Nano Lett. 2006, 6, 1344. (25) Singh, A.; Li, X.; Protasenko, V.; Galantai, G.; Kuno, M.; Xing, H.; Jena, D. Manuscript in preparation. (26) Wei, S-H.; Zhang, S. B. Phys. ReV. B 2000, 62, 6944. (27) Yoffe, A. D. AdV. Phys. 2002, 51, 799. (28) Zhou, R.; Chang, H.-C.; Protasenko, V.; Kuno, M.; Singh, A. K.; Jena, D, H. Xing. Submitted for publication. (29) Protasenko, V.; Hull, K. L.; Kuno, M. Chem. Educ. 2005, 10, 1. (30) Bohren, C. F.; Huffman, D. R. Absorption and Scattering of Light by Small Particles; John Wiley & Sons: New York, 1983; p 66. (31) Erdelyi, A. Higher Transcendental Functions; McGraw-Hill: New York, 1953; Vol. 2. In MatLab the hypergeometric function 2F2(a,b;c,d;z) is called “hypergeom([a b],[c d],z)”. In Mathemetica the same function has the name “HypergeometricPFQ[{a,b},{c,d},z]”. (32) Ninomiya, S.; Adachi, S. J. Appl. Phys. 1995, 78, 4681. (33) Marple, D. T. F.; Ehrenreich, H. Phys. ReV. Lett. 1962, 8, 87. (34) Debenham, M.; Dew, G. D. J. Phys. E: Sci. Instrum. 1981, 14, 544. (35) Ruck, G. T.; Barrick, D. E.; Stuart, W. D.; Krichbaum, C. K. Radar Cross Section Handbook; Plenum Press: New York, 1970; Vol. 1. (36) Reference 35, eqs 4.1-47 (Taylor-expanded) and 4.1-7, 4.1-8, and 4.1-42 (exact). (37) Muljarov, E. A.; Zhukov, E. A.; Dneprovskii, V. S.; Masumoto, Y. Phys. ReV. B 2000, 62, 7420. (38) Gudiksen, M. S.; Wang, J. F.; Lieber, C. M. J. Phys. Chem. B 2002, 106, 4036.
