Understanding the Plasmon Resonance in Ensembles of

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Understanding the Plasmon Resonance in Ensembles of Degenerately Doped Semiconductor Nanocrystals Rueben J. Mendelsberg,†,§ Guillermo Garcia,† Hongbo Li,‡ Liberato Manna,‡ and Delia J. Milliron*,† †

The Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States Istituto Italiano di Tecnologia, Via Morego 30, 16163 Genova, Italy § Plasma Applications Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States ‡

S Supporting Information *

ABSTRACT: Inevitable variations in size and composition within nanocrystal ensembles affect their optical absorbance as revealed by effective medium theory calculations. We critically analyzed the effects of such inhomogeneity and of the surface ligands on the localized surface plasmon resonance absorption of In2O3:Sn and Cu2−xSe nanocrystal dispersions. Modeling the absorbance line shape readily provides valuable and quantitative insight into the structural, electrical, and optical properties of colloidal nanocrystals.

T

the Mott limit, which puts the Fermi level within the conduction band (n type) or the valence band (p type). As in metals, the (degenerate) free carriers undergo collective oscillations when driven by photons within the resonant frequency range. Their contribution to the frequency (ω)dependent complex dielectric function is given by the wellknown Drude equation3,5,10,17−20

he observation of localized surface plasmon resonance (LSPR) in impurity-1−4 or vacancy-5−8 doped semiconductor nanostructures has opened doors to new phenomena with potential application to sensing9 and smart windows.3 In contrast to metals, tuning the LSPR frequency (ωsp) in semiconductors is possible since their carrier concentrations can be precisely controlled during synthesis2,7,10 and dynamically and reversibly modulated.3,11−13 However, this tunability creates a challenge when analyzing optical absorbance data from semiconductor nanostructures since the absorbance depends on the electrical properties, which can vary from sample to sample and from nanostructure to nanostructure. Applying a rigorous approach to the analysis of absorbance spectra allows for efficient and quantitative assessment of the electronic properties (carrier concentration and mobility), characteristics that are not easily accessed in nanostructures by transport measurements. A detailed understanding of LSPR absorbance in metal or semiconductor nanocrystal dispersions requires the use of the Mie scattering theory6,10,14−17 or, as we show, an effective medium approximation (EMA). In this work, absorbance spectra of some purely theoretical dispersions were calculated to evaluate the effects of inhomogeneities among an ensemble. Then, we account for the presence of organic ligand shells, which generally accompany inorganic nanocrystals fabricated by colloidal chemistry. We utilized these refined EMA theories to fit the absorbance spectra of real Cu2−xSe and indium tin oxide (ITO) nanocrystals, uncovering the complexity of free carrier generation in these new materials. Notably, the relationship between the free carrier concentration and the composition differs dramatically between these two prototypical cases. Outside the range of the bandgap or other interband or impurity absorption features, the optical response of degenerate semiconductors is governed by the free charge carriers. In degenerately doped semiconductors, the doping level is above © 2012 American Chemical Society

εD(ω) = ε∞ −

ωP2 2

ω + i ωΓ(ω)

(1)

where ε∞ is the high frequency dielectric constant, ωP is the bulk plasma frequency, and Γ is the damping function. ωP is related to the free carrier concentration (n) by ω2P = ne2/m*ε0 where e is the elementary charge, m* is the carrier effective mass, and ε0 is the permittivity of free space. In bulk crystals, the collective oscillations of free carriers manifest as longitudinal plasmons19 with a frequency ω2BP = (ω2P/ε∞) − Γ2. For nanocrystals dispersed in a host medium, the plasmonic oscillations are confined by the particle and become LSPRs. Thus, optical absorbance from ensembles of such nanocrystals shows a clear peak centered on the LSPR frequency (ωsp). Despite the spatial confinement provided by the nanocrystal, the electromagnetic fields generated by LSPRs extend into the surroundings, making ωsp a function of the host dielectric constant18,21 (εH) as well as the particle shape.10,15−17 Formal analysis of absorbance spectra of plasmonic nanocrystals is typically carried out within Mie scattering theory.6,11,14,22,23 In the quasistatic limit (where the photon wavelength is much larger than the particle size), the Mie absorption cross-section for spherical particles is given by6 Received: March 21, 2012 Revised: April 28, 2012 Published: May 25, 2012 12226

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The MG-EMA approach is readily modified to quantitatively account for the inhomogeneities in size and composition that are inevitably present in real nanocrystal dispersions. Although it is known that the dopant concentration varies between nanocrystals within an ensemble,2 leading to a distribution of ωP, these effects have not been previously accounted for when analyzing plasmonic absorption spectra. This inhomogeniety can be treated by extending the MG equation to include multiple constituents, such as25 εeff − εH ε − εH = ∑ fi i εeff + 2εH ε i + 2εH (4) i

(2)

where k = 2π(εH) ω/c, εP is the particle dielectric function, R is the particle radius, and c is the speed of light. The Mie absorbance is then calculated from the Beer−Lambert law as AMie = NσAL/ln(10) where N = 3f V/4πR3 is the particle number density, f V is the particle volume fraction, and L is the optical path length. As such, AMie is independent of the particle radius, but it is only valid when the particles are sufficiently separated such that they do not experience far field interactions. In the dilute limit, an EMA is equivalent to Mie theory; yet, EMAs are more versatile and can be applied to a wider range of experimental conditions including concentrated dispersions and thin films.20 The effective dielectric function (εeff) of a dispersion of spherical nanocrystals can be described by the Maxwell-Garnett (MG)-EMA14,24,25 1/2

εeff − εH ε (ω) − εH = fV P εeff + 2εH εP(ω) + 2εH

where f i is the volume fraction of the i-th component with dielectric function εi. Likewise, the Mie approach can be extended to several constituents by summing the cross-sections of each component: AMie =

(3)

3L 4π ln 10

∑ i

fi σi R i3

(5)

which results in identical spectra as eq 4 for ∑f i < 0.01. In the ideal case, statistical models predict a Poisson distribution of carrier-generating defects within an ensemble of doped semiconductor nanocrystals.26 This leads to a distribution of ωP that can be well approximated by a narrow Gaussian (Supporting Information, Figure S2). Putting Gaussian ωP distributions with a range of standard deviations (σ) into eq 4 results in the calculated absorbance spectra shown in Figure 2. In this calculation, the mean plasma frequency was

which accounts for far field interactions but neglects multipole coupling between adjacent nanocrystals.14 The MG absorbance (AMG) is then easily calculated from the imaginary part of the dielectric function [(εeff)1/2] using the Beer−Lambert law. The relationship between the Mie and the MG-EMA approaches was explored by calculating the absorbance of model nanocrystal dispersions for a range of f V (Figure 1).

Figure 1. Comparison of the absorbance calculated with Mie theory and MG-EMA for a range of volume fractions, f V. For all calculations, the total volume of absorbing material (f V × L) was kept constant. AMie is independent of f V and is shown as the black dotted line. The two results start to noticeably deviate when f V is between 10−2 and 10−1.

Figure 2. Calculated absorbance of a compositionally inhomogeneous nanocrystal dispersion using a Gaussian distribution of ωP with various half widths σ. In each case, ∑f i = f V = 2.5 × 10−5. The dashed green lines are the best fit when assuming a homogeneous ensemble with a single ωP.

set to ⟨ωP⟩ = 20000 cm−1, and a noticeable blue shift and broadening of the LSPR are observed with increasing σ. The LSPR also becomes slightly asymmetric; both the blue shift and the asymmetry are due to the larger extinction coefficient of nanocrystals with higher-than-average ωP. Thus, if these spectra are fit assuming a single plasma frequency (dashed green lines in Figure 2), the extracted ωP is higher than ⟨ωP⟩ and Γ is overestimated (Supporting Information, Figure S3). The impact of compositional inhomogenity on quantitative analysis was assessed by considering a Poisson distribution of dopant concentrations within an ensemble of ITO-like nanocrystals. This model ensemble has ⟨ωP⟩ = 20000 cm−1, and a Gaussian ωP distribution with σ ≈ 250 cm−1 ≈ ⟨ωP⟩/90 is an excellent approximation. Such a narrow distribution has a negligible effect on the calculated absorbance spectrum. However, real nanocrystal dispersions likely have a significantly

Equation 1 was used for εP along with parameter values that would be reasonable for ITO nanocrystals dispersed in tetrachloroethylene (TCE), that is, εH = 2.26, ωP = 20000 cm−1, Γ = 2500 cm−1, ε∞ = 4, and f V × L = 2.5 × 10−6 cm. The product f V × L was fixed to ensure the same amount of material was contributing to each calculated spectrum. Below f V = 0.1, AMie and AMG are identical to several decimal places. When f V ≈ 0.1, AMie and AMG begin to deviate, and the difference depends to some extent on ωP (see the Supporting Information, Figure S1). Hence, Mie theory is valid at least up to f V = 0.01 for nanocrystals showing Drude behavior, above which an EMA should be used for full confidence in the results. Dilute nanocrystal dispersions prepared for absorbance measurements typically have f V ≈ 10−6 to 10−4, where the Mie and MG-EMA approaches are equivalent. 12227

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Figure 3. Discrete Γ(R) distributions (a) and calculated absorbance (b) of theoretical nanocrystal dispersions with radii following log-normal distributions. In each case, ∑f i = f V = 2.5 × 10−5.

Figure 4. (a) LSPR frequency (ωsp) calculated for a theoretical ensemble of 4.5 nm nanocrystals with ligand shells (εs = 2.13) of various thickness (ds). Other parameters are the same as in Figure 1, and the yellow dotted line shows the curve for 13.5 nm nanocrystals with ds = 2 nm. The black dotted line is eq 7, which is commonly used to relate ωsp to ωP and εH but contains intrinsic error. (b) The error in the position of ωsp as calculated from the simple LSPR equation (eq 7) for ligand-free nanocrystals for several values of ωP.

broader distribution.2 Still, increasing the width up to σ = 3000 cm−1 (≈ ⟨ωP⟩/7) blue shifts the extracted (homogeneous) ωP by only 3% as compared to ⟨ωP⟩, while Γ is overestimated by over 45% (Supporting Information, Figure S3). Thus, accounting for compositional inhomogeneities may not be important when the goal is extracting a free carrier concentration, but it certainly should be considered during a detailed analysis of plasmon damping in ensembles of nanostructures. Besides variations in composition, inhomogeneities in nanocrystal size within an ensemble can present a challenge when the material properties of interest are strongly size dependent. For small nanocrystals, surface scattering of free carriers is known to cause a size-dependent Drude damping parameter given by14 Γ(R) = Γ0 + CvF/R where Γ0 is the bulk damping constant, vF is the Fermi velocity, and C is a theorydependent constant (Supporting Information, Figure S4). This 1/R dependence can be combined with log-normal approximations27 for the nanocrystal R distribution (Supporting Information, Figure S5) to give the resulting Γ(R) distributions and absorbance spectra (as calculated by eq 4), which are shown in Figure 3. Using modern synthetic techniques, the R distribution of colloidal semiconductor nanocrystals can be narrow, with a half width at half max of 5−15%. The realistic Γ(R) distribution shown in Figure 3 was derived from TEM measurements of ≈5 nm ITO nanocrystals (Supporting Information, Figure S5) and represents the upper end of this range. An ensemble with such a size distribution has a nearly identical absorbance spectrum to a perfectly uniform one, indicating that inhomogeneities in size can be neglected for spherical nanocrystals with moderate polydispersity. If the width of the realistic R distribution is doubled, a narrowing of the absorbance can be observed

(Figure 3). Such counterintuitive narrowing will always accompany a nearly or fully symmetric R distribution of any finite width since the larger-than-average particles take up more volume and have smaller damping, Γ, than the most abundant particles. These calculations also show that ωP and ωsp do not depend on the width of the R distribution, which is fortunate since they are commonly the quantities of greatest interest. Semiconductor nanocrystals are typically capped with a shell of organic ligands allowing good dispersion in various solvents. These shells influence the LSPR18,21 and can be accounted for by extending eq 3 to a core−shell geometry25 εeff − εH (ε − εH)(εP + 2εs) + w(εP − εs)(εH + 2εs) = fV s εeff + 2εH (εs + 2εH)(εP + 2εs) + 2w(εP − εs)(εs − εH)

(6)

where εs is the dielectric constant of the shell and w = (b/a)3 with a being the whole particle radius and b the radius of the core. In this case, f V represents the total volume fraction of the entire particle with the total volume fraction of the core being f V × w. Core−shell particles can also be treated by Mie theory using a similar form as eq 6 for the absorption cross-section.21 Absorbance calculated with eq 6 using eq 1 shows that the ligand shell partially shields the LSPR from the host dielectric environment (Supporting Information, Figure S6), in agreement with what has been observed for metal nanoparticles.21,28 As such, a smaller shift in ωsp when changing εH is observed for nanocrystals with thick ligand shells, as shown in Figure 4a for 4.5 and 13.5 diameter nanocrystals with ligand shell thickness (ds) between 0 and 2 nm. For plasmonic sensing of refractive index changes, the ligand shell will significantly reduce the sensitivity, which can become a problem for small nanocrystals (2R ≈ 5 nm) coated with thick ligand shells. 12228

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Figure 5. Measured and fitted absorbance and TEM images of the nanocrystals. The solid yellow line shows the fit assuming a perfectly uniform ensemble with no ligands (eq 3), the dotted red line is the fit to the core shell EMA (eq 6) with the ligand shell thickness (εs = 2.13) fixed at 1.8 nm, and the dashed red line is the fit assuming no ligands but an inhomogeneous dopant concentration in the ensemble. A frequency dependent Γ was used to fit the ITO data to account for the asymmetric line shape (on an energy scale) caused by the ionized impurities. A few fits shown in Table 1 were omitted from this figure for clarity as they lie on top of the fits shown.

Table 1. Best Fit Parameters for the Two Nanocrystal Dispersions Using a Variety of EMAsa material Cu1.85Se

ITOg

model MG (eq 3) ωP-dist. (eq 4) core−shell (eq 6) LSPR equation (eq 7) MG (eq 3) ωP-dist. (eq 4) core−shell (eq 6) core−shell + ωP-dist.e LSPR equation (eq 7)

ωP (cm−1)

n (and p) (cm−3)

μ (cm2/V s)

f V (10−6)

other

36100 36100b 35900 39200 14900 14600b 14700 14400b 16800

× × × × × × × × ×

10.8 10.8 10.8 11.5 7.3 7.6 7.4 7.6 11.0

3.6 3.6c 3.8d

σ = 0 cm−1 ds = 1.8 nmh

5.3 4.9c 5.5d 5.14f

σ = 1, 500 cm−1 ds = 1.8 nmh σ = 1, 500 cm−1

3.06 3.06 3.03 3.59 9.92 9.47 9.68 9.20 1.26

21

10 1021 1021 1021 1020 1020 1020 1020 1021

Effective carrier mass of 0.4me and 0.21me was used in the n and μ calculations for the ITO and Cu1.85Se, respectively. bThis is the mean plasma frequency ⟨ωP⟩. cThis is the total volume fraction ∑f i = f V. dThis is the total core volume fraction f V × w. eLigand shell thickness was fixed at ds = 1.8 nm. fThis is the total core volume fraction ∑f i × w. gA frequency-dependent Γ was used for the fitting. hFixed parameter. a

EMA models assuming a Gaussian distribution of ωP and the presence of organic ligands are shown in Figure 5 with the fit parameters reported in Table 1. In each case, ε∞ must be fixed (10 for Cu1.85Se6 and 4 for ITO19) to arrive at a unique fit, since it is strongly correlated with ωP for nanocrystal dispersions. Only that data outside the range of the bandgap absorption were used for the fits. For ITO nanocrystals and some other wide bandgap n type semiconductors,19,20,29 ionized impurity scattering causes a frequency dependence in Γ, which is modeled here using an empirical equation (see the Supporting Information and ref 20). For Cu2−xSe and other chalcogenides, the high value of ε∞ effectively screens the free carriers from the ionized impurities,30 leading to a symmetric absorbance line shape and a frequency independent Γ. For both the Cu1.85Se and the ITO, very similar fits and electrical properties were obtained for each of the models shown in Table 1. For the Cu1.85Se, no inhomogeneity in ωP within the ensemble could be detected and fixing σ to anything above zero degraded the fit quality, suggesting a highly uniform composition for the C1.85Se ensemble. On the other hand, σ reliably converged to a value near 1500 cm−1 for the ensemble of ITO nanocrystals. Free electrons in ITO are generated by incorporation of Sn onto In sites, with Poisson-like statistics governing the Sn incorporation during growth. Here, σ ≈ ⟨ωP⟩/9, which means the model shows the width of the dopant concentration distribution in the ITO ensemble is about 10 times the theoretical minimum. This suggests further investigation into free carrier distribution in nanocrystal ensembles is warranted.

When trying to exploit the connection between ωsp and ωP to extract the electrical properties, it is tempting to approximate their relationship by5 ωsp =

ωP2 − Γ2 ε∞ + 2εH

(7)

and simply take ωsp as the absorbance peak position and Γ as the width. (For noble metal nanocrystals, eq 7 is sometimes further approximated5 by substituting a 1 in the place of ε∞, but this introduces significant error for degenerate semiconductors.) However, this simple equation, derived by setting the real part of εD = −2εH, gives rise to inherit error even when no ligands are present (Figure 4a). The error decreases significantly as ωP increases in the absence of ligands (Figure 4b), which explains why this type of simple approximation works well for metal nanocrystals.28 The error also increases as ε∞ increases (Supporting Information, Figure S7), meaning eq 7 is not reliable for materials with a large ε∞ and/or a small ωP. The error in eq 7 can be rationalized by noting that the absorbance is determined by Im{(εeff)1/2} [or Im{(εD − εH)/ (εD + 2εH)} using Mie theory], which is strongly influenced by the imaginary part of εD. Now that the framework has been laid out for dealing with nanocrystal inhomogeneity and ligand shells, attention is turned to the measured absorbance of two very different (degenerate) semiconducting nanocrystal dispersions: p type 13.5 nm Cu1.85Se nanocrystals with oleylamine ligands in TCE6,11 (εH = 2.27) and n type 4.5 nm ITO nanocrystals capped with oleic acid and also in TCE (Figure 5).3 Least-squares fits to MG12229

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Accounting for the ligands slightly red-shifted the extracted ωP as expected, but there is only a small difference as compared to ωP extracted when assuming bare nanocrystals. Fixing the ligand length to ds = 1.8 nm results in a shift in ωP of only 200 cm−1, which changes the extracted carrier concentration (n) by only 2% for the ITO ensemble. Because the ligands have a similar dielectric constant as the solvent (εs = 2.13 and εH = 2.27), a large error in ωP when neglecting them would have been surprising. The same is true when using a model that includes both a core−shell structure as well as an ω P distribution (Table 1). However, if εH is much different from εs or if εH is changing, the ligand shells must be taken into account, especially for accurate measurements when using LSPRs for refractive index sensing. The above procedure for using absorbance to measure free carrier density in doped nanocrystals is a powerful technique, but it is best combined with other measurements when trying to explore the connection between structural, optical, and electrical properties. For example, comparing the extracted carrier concentration in the ITO nanocrystals (using an effective mass of19 m* = 0.4me) to the measured 14 at % Sn content shows that only 11% of the Sn contributes free carriers. Sn incorporated onto In sites should be ionized since the dopant concentration is above the Mott limit. The discrepancy between Sn content and carrier concentration implies that a large number of compensating electron traps must be present in the ITO nanocrystals. Moreover, Sn clusters and/or SnO2 inclusions that do not create free carriers could be also be forming. This quantitative approach shows that impurity concentration does not have a simple correspondence to free carrier concentration. In the case of Cu2−xSe, the effective mass is highly dependent on the carrier concentration due to the nonparabolicity of the valence band,31 which makes it challenging to derive the electrical properties from the absorbance measurement. Elemental analysis on these Cu2−xSe particles indicates that x ≈ 0.15, which gives an expected vacancy concentration of 3.07 × 1021 cm−3 (assuming that the Cu2Se has a density32 of 7.0 g/ cm3). Excellent agreement between the vacancy and the free hole concentrations is achieved with m* = 0.21me for all of the models (Table 1), which is in agreement with m* = 0.2−0.25me expected31 for Cu2Se with free hole concentrations >1021 cm−3. Thus, caution must be taken when calculating optically derived carrier concentrations and ωP, and the value of the carrier effective mass used in the calculation should always be quoted. In summary, analysis of absorbance line shapes using the Drude theory coupled with an EMA is an efficient alternative to Mie theory. EMAs are readily refined to account for the effects of nanocrystal inhomgeneites in composition and size as well as the presence of a ligand shell. Fitting measured absorbance of two drastically different degenerate semiconductors revealed that these inhomogeneities do not cause major error in the extracted Drude parameters and electrical properties in real semiconductor nanocrystal dispersions. Calculations showed that ligand shells will cause a shift in the LSPR and reduce the sensitivity of the LSPR to the host dielectric environment. Furthermore, line shape analysis is much more accurate than estimating ωP using eq 7 and is crucial for drawing quantitative conclusions. In this study, the nanocrystals were all assumed to be spherical, but EMAs exist for many geometries and the theoretical approach developed here would apply well to understanding real-world effects of shape anisotropy.4,10,16

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ASSOCIATED CONTENT

S Supporting Information *

Sections on comparison of Mie and MG-EMA results, dopant inhomogeneity, size dependence of Γ, core−shell EMA, error in simple LSPR frequency approximation, real dispersions, and references. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge R. Buonsanti, A. Llordes, and E. Runnerstrom for supporting this work. Research was supported by the LDRD Program of Lawrence Berkeley National Laboratory (Dr. Mendelsberg) and by an Office of Basic Energy Sciences Early Career Research Program grant (Dr. Milliron, G. Garcia) under U.S. Department of Energy Contract No. DE-AC0205CH11231. Portions of this work were performed as a user project at the LBNL Molecular Foundry, which is supported by the Office of Science, Office of Basic Energy Sciences, under the same contract. Additional support was provided by European Union through the FP7 starting ERC Grant NANO-ARCH (contract number 240111) as well as the Italian FIRB grant (contract #RBAP115AYN) (Dr. Manna and Dr. Li).

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