Analytical modeling of localized surface plasmon resonance in heterostructure copper sulfide nanocrystals Andrew H. Caldwell, Don-Hyung Ha, Xiaoyue Ding, and Richard D. Robinson Citation: The Journal of Chemical Physics 141, 164125 (2014); doi: 10.1063/1.4897635 View online: http://dx.doi.org/10.1063/1.4897635 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The effect of a semiconductor-metal interface on localized terahertz plasmons Appl. Phys. Lett. 98, 111106 (2011); 10.1063/1.3565247 Dependence of the localized surface plasmon resonance of noble metal quasispherical nanoparticles on their crystallinity-related morphologies J. Chem. Phys. 134, 024507 (2011); 10.1063/1.3523645 Controlling plasmonic resonances in binary metallic nanostructures J. Appl. Phys. 107, 114313 (2010); 10.1063/1.3407527 Surface plasmon resonance in nanostructured metal films under the Kretschmann configuration J. Appl. Phys. 106, 124314 (2009); 10.1063/1.3273359 Enhancement in middle-ultraviolet emission in a surface-plasmon-assisted coaxial nanocavity Appl. Phys. Lett. 93, 091902 (2008); 10.1063/1.2973159

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THE JOURNAL OF CHEMICAL PHYSICS 141, 164125 (2014)

Analytical modeling of localized surface plasmon resonance in heterostructure copper sulfide nanocrystals Andrew H. Caldwell,1 Don-Hyung Ha,1 Xiaoyue Ding,2 and Richard D. Robinson1,a) 1 2

Department of Materials Science and Engineering, Cornell University, Ithaca, New York 14853, USA School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA

(Received 1 August 2014; accepted 25 September 2014; published online 30 October 2014) Localized surface plasmon resonance (LSPR) in semiconductor nanocrystals is a relatively new field of investigation that promises greater tunability of plasmonic properties compared to metal nanoparticles. A novel process by which the LSPR in semiconductor nanocrystals can be altered is through heterostructure formation arising from solution-based cation exchange. Herein, we describe the development of an analytical model of LSPR in heterostructure copper sulfide-zinc sulfide nanocrystals synthesized via a cation exchange reaction between copper sulfide (Cu1.81 S) nanocrystals and Zn ions. The cation exchange reaction produces dual-interface, heterostructure nanocrystals in which the geometry of the copper sulfide phase can be tuned from a sphere to a thin disk separating symmetrically-grown sulfide (ZnS) grains. Drude model electronic conduction and Mie-Gans theory are applied to describe how the LSPR wavelength changes during cation exchange, taking into account the morphology evolution and changes to the local permittivity. The results of the modeling indicate that the presence of the ZnS grains has a significant effect on the out-of-plane LSPR mode. By comparing the results of the model to previous studies on solid-solid phase transformations of copper sulfide in these nanocrystals during cation exchange, we show that the carrier concentration is independent of the copper vacancy concentration dictated by its atomic phase. The evolution of the effective carrier concentration calculated from the model suggests that the out-of-plane resonance mode is dominant. The classical model was compared to a simplified quantum mechanical model which suggested that quantum mechanical effects become significant when the characteristic size is less than ∼8 nm. Overall, we find that the analytical models are not accurate for these heterostructured semiconductor nanocrystals, indicating the need for new model development for this emerging field. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4897635] I. INTRODUCTION

Plasmon resonance is of considerable technological interest because it can amplify the electric field of optical signals at wavelengths characteristic of electrons in conductors. Devices exploiting this combination of electric field amplification and sub-diffraction limit localization of light signals promise the electronic control of optical signals at the nanometer scale and are of great relevance for photonics, sensors, and photovoltaics.1 At nanoscale dimensions a surface plasmon resonance mode is localized and therefore called “localized surface plasmon resonance” (LSPR). Conductive nanocrystals exhibit LSPR when the size of the nanocrystals is smaller than the wavelength of incident light such that plasmons resonate as stationary surface modes.2 Metal nanocrystals with LSPR can be used as light trapping components in solar cells, as biological sensors, and as nanometer-scale waveguides.1, 3 While much work has been done with LSPR in metal nanocrystals,4–11 the phenomenon of LSPR in semiconducting nanocrystals is a relatively new field of inquiry. Few semiconducting materials have the charge carrier densities necessary to sustain plasmons. It is for this reason that current a) Author to whom correspondence should be addressed. Electronic mail:

[email protected].

0021-9606/2014/141(16)/164125/8/$30.00

research on LSPR in semiconducting nanocrystals focuses on semiconducting materials that can be heavily doped or are self-doping, such as transition metal chalcogenides.12–19 Our investigation focuses on the use of copper sulfide (Cu2−x S) nanocrystals. The copper sulfide system has several copper-deficient phases: high chalcocite (hexagonal Cu2.0−1.94 S at T > 103.5 ◦ C), low chalcocite (monoclinic Cu2.0−1.997 S), djurleite (Cu1.97 S–Cu1.93 S), roxbyite (Cu1.81 S), digenite (Cu1.8 S), and anilite (Cu1.75 S).12, 20 These copper sulfide phases exhibit p-type electronic character arising from copper vacancy doping. The hole concentration in Cu2−x S is generally understood to be stoichiometry-dependent and increases with the number of copper vacancies, x.12, 21 Reported carrier densities for the Cu2−x S system are typically on the order of 1021 cm−3 .12, 17 Semiconductor nanocrystals have an important advantage over metal nanocrystals in that the LSPR frequency can be controlled by solution-based processes that modify the carrier concentration, such as doping and both chemical and thermal oxidation.12, 13, 16, 21 Luther et al.,12 Kriegel et al.,16 and Hsu et al.21 have reported tunability of the LSPR frequency in Cu2−x S nanocrystals through exposure to air, which introduces copper vacancies.16, 22 Other methods to control the LSPR frequency in semiconductor nanocrystals focus on synthetic routes or chemical treatments to selectively modify the

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geometry and size of the LSPR-supporting phase. Targeted phosphorus doping of Si nanowires has been shown to produce segmented nanowires consisting of alternating insulating and conducting regions.23, 24 Hsu et al.21, 25 synthesized Cu2−x S (x ∼ 0.04) nanodisks exhibiting two extinction spectra peaks corresponding to “in-plane” and “out-of-plane” resonances, as predicted from theories of classical electrostatics for high-aspect ratio particles of similar geometry.2 Nanoscale cation exchange provides control over the spatial arrangement of different phases within nanocrystals, allowing for the synthesis of heterostructures with tunable properties.26–28 Cation exchange therefore can be an important tool for controlling the plasmonic properties of semiconductor nanocrystals that combines the spatial resolution of top-down fabrication techniques with the high throughput of solution-based syntheses. The use of cation exchange to modify the optical properties of metal chalcogenides nanocrystals remains a novel field of inquiry. Dilena et al.29 investigated the plasmonic properties of Cu2−x (Sy Se1−y ) nanocrystals converted to Cd(Sy Se1−y ) upon reacting with Cd ions; however, heterostructure formation was not reported. Huang et al.30 studied the plasmonic absorbance spectra of Cu1.94 S–ZnS “sandwich-like” heterostructure nanocrystals. Our group31 recently reported a novel solution-based method of tuning plasmon resonance in Cu1.81 S (roxbyite) nanocrystals using cation exchange-induced heterostructure formation to control the geometry of the semiconducting Cu1.81 S phase. This cation exchange reaction between spherical Cu1.81 S nanocrystals and Zn ions produced dual-interface heterostructure nanocrystals in which ZnS grains symmetrically nucleate on opposite ends of the Cu1.81 S nanocrystal and grow towards the center of the nanocrystal, as shown in the transmission electron microscope (TEM) images in Fig. 1 (top).31 Precise control of the degree of cation exchange was exercised by controlling the reaction parameters, allowing for exact confinement of the copper sulfide layer between the two ZnS grains. However, a narrow window of conditions, including particle size, permits these uniform sandwiched structures, which places limitations on the degrees of freedom of the heterostructure. Although the thickness of the copper sulfide layer is well-controlled, it is coupled to the volume fraction of ZnS such that the copper sulfide layer thickness cannot be studied independently from the size of the ZnS caps. The Cu2−x S–ZnS heterostructure nanocrystals exhibit LSPR wavelength tunability in the infrared spectrum, from 1300 nm to 2100 nm, as shown in the absorbance spectra in Fig. 1. The growth of the ZnS grains during the cation exchange reaction results in a monotonic LSPR red-shift. As shown in previous studies,18, 21, 23–25 LSPR is sensitive to the geometry of the doped semiconducting phase, and thus one would expect the heterostructure formation described here to directly modify the plasmonic behavior of the nanocrystals by tuning the aspect ratio of the confined Cu2−x S layer. However, the exact mechanisms responsible for the LSPR red-shift are unclear, as several other parameters may influence LSPR during cation exchange: size effects,17, 19 changes in the local dielectric medium cause by the increasing volume fraction of ZnS,2 and solid-solid phase transitions in the copper sulfide layer induced by heterostructure formation.31

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FIG. 1. (Top) TEM images of Cu2−x S–ZnS heterostructure nanocrystals. (Bottom) UV-Vis-NIR absorbance spectra for Cu2−x S–ZnS nanocrystals dispersions. h is the thickness of the copper sulfide layer, and r is the aspect ratio of the copper sulfide layer, defined as the ratio of the diameter (∼22 nm) to h.

Here, an analytical model describing the dependence of the LSPR wavelength on the geometry of the copper sulfide layer and the volume fraction of ZnS was developed to determine how the aforementioned parameters can influence the LSPR and to, in a more general sense, elucidate how heterostructure formation can influence LSPR in doped semiconductor nanocrystals. Other studies of semiconductor nanocrystal LSPR have incorporated modeling based on classical (Drude) or semi-classical electrostatics to extract an understanding of nanocrystal plasmonic behavior, such as the relative dominance of different resonance modes,17 LSPR geometry-dependence,21 carrier densities12, 13, 17, 24, 25 and divergence from the Drude model.18, 19 Unique to the analytical model discussed herein, however, is the description of a heterostructure architecture that significantly influences the plasmonic behavior. We find that the presence of the ZnS capping regions in the heterostructure nanocrystal drastically changes the predicted evolution of the LSPR wavelength during cation exchange, depending on the resonance mode. In describing the carrier density dependence of LSPR, the model allows us to make observations comparing the structural evolution of copper sulfide during cation exchange31 to changes in the carrier density. Specifically, we find that the carrier concentration is not entirely dependent on the copper sulfide phase, suggesting that electronic defects alone do not control the carrier density. By fitting the model to experimental data, we determine that classical electrostatics provides an adequate description of LSPR in the nanocrystals up to a characteristic size of the copper sulfide region (∼8 nm), at which point quantum mechanical effects may become significant, as

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suggested by a simplified quantum mechanical model of the nanocrystals.

II. METHODS A. Nanocrystal synthesis and cation exchange

The Cu1.81 S (roxbyite) nanocrystals used in the cation exchange reaction were synthesized according to, with minor variations, the method reported by Li et al.32 This method follows the principles of the hot-injection method for synthesizing colloidal semiconductor nanocrystals.33, 34 The cation exchange reaction was performed on the as-synthesized Cu1.81 S nanocrystals, in a procedure reported previously.31 Briefly, the Cu1.81 S nanocrystals were dispersed in trioctylphosphine (TOP), an organic solvent that coordinates strongly to the nanocrystals and is vital to mediating the cation exchange process.31, 35 A solution of Zn ions was prepared in a separate reaction vessel by solvating zinc (II) chloride in oleylamine under nitrogen atmosphere at 100 ◦ C. The zinc solution was added to a three neck flask along with toluene, an organic solvent. At 50 ◦ C the Cu1.81 S/TOP solution was injected into the reaction vessel containing the Zn ion and toluene solution, initiating the cation exchange reaction.26, 27 Aliquots of the solution were extracted at several time intervals following the injection of Cu1.81 S/TOP.

B. Absorbance measurements

Absorbance measurements of Cu2−x S–ZnS nanocrystal dispersions were taken to determine how the LSPR wavelength evolved during cation exchange (see supplementary material36 ). During the cation exchange reaction, ZnS grains grow on opposing faces of the initially spherical copper sulfide (roxbyite) nanocrystals. The thickness of the copper sulfide layer (“h”) decreases with increasing reaction time such that the copper sulfide layer can be confined to an atomically thick, two-dimensional plate at the center of the nanocrystal. The absorbance spectra for Cu2−x S–ZnS nanocrystal dispersions, from the initial Cu1.81 S (roxbyite) nanocrystals to the fully-transformed ZnS nanocrystals, are shown in Fig. 1 (bottom). The LSPR wavelength for each sample was taken to be the wavelength at the absorbance maximum. The LSPR data indicate that the resonance wavelength of the surface plasmons in the copper sulfide phase can be tuned over a wide range of wavelengths, from ∼1300 nm for the initial Cu1.81 S nanocrystals to ∼2100 nm for nanocrystals with a thin (∼2 nm) copper sulfide layer. Previous reports of LSPR wavelengths for vacancy-doped copper sulfide nanocrystals lie within this range of infrared wavelengths.12, 13, 16 For our heterostructures, as the thickness of the copper sulfide layer decreases the LSPR wavelength increases and the absorbance peak broadens (Fig. 1), which is likely a consequence of (a) the decrease in the total volume of copper sulfide, and (b) the surface scattering of free carriers for small particle sizes.37 The second of these effects can be treated by modifying the equation for the collision frequency, addressed later.12, 37

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C. Carrier concentration

Cu2−x S (0 < x < 0.25) has p-type electronic character arising from intrinsic vacancy doping, and this selfdoping gives rise to “quasi-metallic” behavior that lends itself to adequate modeling by classical theories of electronic conduction.16, 38 The carrier concentration for the initial Cu1.81 S (roxbyite) nanocrystals can be calculated following the procedure described by Luther et al. in which the LSPR wavelength, taken from absorbance measurements, can be used to solve for the carrier concentration assuming Drude model electronic conduction and Mie-Gans theory (see supplementary material36 ).12 The initial roxbyite nanocrystals exhibit surface plasmon resonance at approximately 1360 nm with a FWHM of roughly 650 nm. Using a solvent (tetrachloroethylene) dielectric constant of 2.25,12 and a hole effective mass of 0.8me ,12, 25 where me is the free electron mass, the carrier concentration is calculated to be 3.3 × 1021 cm−3 . This value is in agreement with other carrier concentrations calculated for Cu2−x S nanocrystals.12, 25 III. ANALYTICAL MODELING

An analytical model of the LSPR wavelength of the heterostructure nanocrystals was developed using a classical approach following Mie-Gans theory and similar treatments in the literature (see supplementary material36 ).12, 21, 23 During cation exchange, growth of the ZnS grains confines the copper sulfide region into a disk-like layer. Treating the copper sulfide layer as a flat disk, the degree of cation exchange can be quantified by the aspect ratio of the copper sulfide “disk,” which is, importantly, a quantity that can be measured using TEM. Thus, the model is an analytical expression for the LSPR wavelength as a function of the aspect ratio of the copper sulfide layer, denoted as r (Fig. 2(b)). X-ray diffraction (XRD) and high-resolution TEM analyses confirm the existence of a spatially well-defined heterostructure architecture with distinct phase regions exhibiting no

(a)

(b)

(c)

FIG. 2. Schematic diagrams illustrating: (a) geometrical deconstruction of a heterostructure Cu2−x S–ZnS nanocrystal into ZnS spherical caps and a disklike Cu2−x S layer; (b) oblate spheroid approximation of the Cu2−x S disk; (c) core-shell approximation of the heterostructure nanocrystal.

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discernable alloying, allowing us to treat the copper sulfide layer and ZnS capping regions as separate geometrical entities.31 Cation exchange results in a change in the geometry of the copper sulfide layer, from a sphere to a thin disk (Fig. 2), and thus it is not appropriate to model the geometry as a sphere. In Mie-Gans theory, the extinction from an oblate spheroid can be determined analytically, and so, following a similar analysis by Hsu et al.,21 we have chosen to approximate the disk-like copper sulfide layer as an oblate spheroid (Fig. 2) for the convenience of developing an analytical model. However, an additional complication arises from the growth of the ZnS grains, which change the permittivity of the local environment of the copper sulfide layer and thus strongly affect the plasmonic behavior.2 To account for the growth of this second phase, the heterostructure

nanocrystals were approximated as having a core-shell structure (Fig. 2(c)), effectively treating the copper sulfide layer as a “coated ellipsoid,” thereby allowing an analytical solution to be developed.2 In this core-shell approximation, the “core” is copper sulfide, and the “shell” is ZnS. Returning to the simple geometric picture of the copper sulfide “disk,” one can then treat the ZnS grains as spherical caps, the volume of which can be readily calculated as a function of r, the aspect ratio of the copper sulfide layer (Figs. 2(a) and 2(b)), thus allowing the ZnS shell volume fraction to be a function of r. Using both the oblate spheroid approximation and the core-shell approximation, an analytical model for the LSPR wavelength as a function of the copper sulfide layer aspect ratio was developed as follows. The polarizability, α i , of a core-shell ellipsoid is given in Eq. (1),2

 (2)  ] + f ε2 (ε1 − ε2 )) v((ε2 − εm )[ε2 + (ε1 − ε2 ) L(1) i − f Li , αi  (1) (2)  (2) ([ε2 + (ε1 − ε2 ) L1 − f Li ][εm + (ε2 − εm )L(2) i ] + f Li ε2 (ε1 − ε2 ))

ωsp

   = 

ωp2 ε∞ −

(1) (2) (1) ε2 ((ε2 −εm )L(2) i ((Li −x)+Li (x−1))−εm (1−Li ))

− γ 2,

(1)

(2)

(1) (2) (1) (ε2 −εm )L(2) i ((1+Li −x)+Li (x−1))+εm Li

where v the volume of the particle, f is the volume fraction of the core ellipsoid, Li (1) and Li (2) (i = 1, 2, 3) are the geometrical factors for the inner (core) and outer (shell) ellipsoids, respectively, ε1 is the real part of the permittivity of the core, ε2 is the real part of the permittivity of the shell, and εm is the real part of the permittivity of the medium. The geometrical factors Li (j) (i = 1, 2, 3 and j = 1, 2) (see Eq. (S6) of the supplementary material36 ) have the property that L1 (j) + L2 (j) + L3 (j) = 1. Since the copper sulfide layer is being treated as an oblate spheroid in which a < b = c (where a, b, c are the semi-major axes), only two axes are unique, rendering only two unique geometrical L factors: L1 (j) corresponding to the shorter axis and L2 (j) = L3 (j) corresponding to the longer axes. The geometrical factors L1 and L2 define the out-of-plane and the in-plane resonance modes, respectively, for oblate spheroid particles. One can then develop an expression for the LSPR wavelength (λsp = 2π c/ωsp ) as a function of the known dielectric constants, the volume fraction of the ZnS shell, and the geometrical factors, as given in Eq. (2) (see supplementary material36 for derivation).

A. Influence of ZnS caps on LSPR wavelength

The results of the model are shown in Fig. 3 along with the experimentally-determined LSPR wavelengths (Fig. 3(a), black open circles). It should be noted that only the LSPR wavelengths that are clearly discernable from the absorbance spectrum (Fig. 1) are included in the analytical modeling

analysis. The absorbance peaks for r > 4.4 (h < 5 nm) are broad and extend near the wavelength limit for the instrument, but we note that the plasmonic properties of nanocrystal samples with r > 4.4 deviate significantly from the classical model developed here, suggesting the influence of quantum confinement—this will be addressed subsequent to the analytical model. Therefore, absorbance data for nanocrystal dispersions with r > 4.4 (h < 5 nm) are not included in Figs. 3 and 4. The LSPR wavelength was taken to be the wavelength at the absorbance maximum. The aspect ratio, r, for each experimental data point was calculated from TEM analysis of the constant diameter (d) and variable height (h) of the copper sulfide “disk.” Two resonance modes exist for nanoparticles with disk-like morphologies: an out-of-plane resonance mode (red) and an in-plane resonance mode (blue). The dashed curves in Fig. 3(a) describe the plasmonic behavior of the nanocrystals treated only as Cu1.81 S oblate spheroids in a manner following the analysis of Hsu et al.21 The solid curves describe the more accurate coreshell model that includes the dielectric contribution of the ZnS grain “capping regions.” An important difference exists between these two models of the wavelength shift as a function of aspect ratio. This difference in wavelength shift is particularly prominent for the out-of-plane LSPR mode, which is predicted to red-shift during cation exchange for the model that includes the ZnS capping regions (Fig. 3(a), red solid curve), rather than blue-shift, as predicted for the simple geometrical model considering only the copper sulfide domain (Fig. 3(a), red dashed curve). The presence of the ZnS caps

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(a)

(b)

FIG. 3. Plot of LSPR wavelength (λsp ) analytical modeling as a function of the aspect ratio of the copper sulfide layer, r, for in-plane and out-of-plane modes. (a) Model results with and without the core-shell approximation. (b) Model results accounting for the size-dependence of the collision frequency, γ , i.e., the “extended” Drude model (dashed red and blue lines). Solid lines are core-shell model as shown in (a). For both (a) and (b), experimentally-determined data points (black) were taken from optical absorbance measurements. Aspect ratios for the experimental data were calculated from TEM analysis. Error bars are derived from the variance in disk height “h” determined from TEM analysis.

changes the permittivity of the local environment of the copper sulfide layer and induces the LSPR redshift observed experimentally. Thus, the blue-shift resulting from the increasing aspect ratio of the copper sulfide domain and the redshift corresponding to the presence of the ZnS caps are competing effects, yet it is clear that the influence of the ZnS caps dominates the plasmonic properties, as the core-shell model predicts a continuous out-of-plane LSPR redshift with increasing aspect ratio. This result is in agreement with the findings of Chou et al.,39 whose discrete dipole approximation modeling of LSPR modes in Si nanowires shows the same competing effects between a blue-shift with increasing aspect ratio of the LSPR-supporting domain and a red-shift with the presence of neighboring dielectric domains. Although Chou et al. did not report on how the out-of-plane LSPR redshift scales with the relative volume of the dielectric domains, our modeling shows that this red-shift does scale with the size of the ZnS caps.

(a)

Out of Plane Mode core-shell model core-shell model with fitted Nh Expt. data

B. Dominance of the out-of-plane resonance mode

Additionally, the results of our model suggest the presence of two absorbance peaks, one corresponding to each of the resonance modes. The asymmetry evident in the absorbance peaks shown in the plot in Fig. 1 suggests both inplane and out-of-plane resonance mode contributions (also see energy scale plot in Fig. S6 of the supplementary material36 ). Deconvolution of the absorbance peaks into inplane and out-of-plane modes is problematic, however, as the initial spherical Cu1.81 S nanocrystal dispersion also has an asymmetric absorbance peak, and thus it is unclear to what extent the asymmetry can be attributed to the relative intensity of the two resonance modes as opposed to an artifact of the nanocrystal dispersion, such as aggregation, which can result in asymmetric absorbance peaks. Despite this, an argument can be made that this LSPR is dominated by an outof-plane resonance mode due to the existence of the ZnS

(b) In Plane Mode core-shell model core-shell model with fitted Nh Expt. data

FIG. 4. Plots of the LSPR wavelength (λsp ) as a function of the aspect ratio of the copper sulfide layer, r, comparing the original core-shell model with an extended model incorporating XANES structural data for the out-of-plane mode (a) and in-plane mode (b). The solid lines correspond to the core-shell model as shown in Fig. 3. The red (a) and blue (b) open circles correspond to the LSPR wavelengths calculated from the extended model incorporating XANES-derived carrier concentration data. The dashed lines are polynomial fits and are merely guides for the eye. Experimental LSPR data are shown in black.

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“capping regions” that should act to damp the in-plane LSPR. In this damping process, plasmon resonance at the copper sulfide interface establishes an image dipole in the adjacent, high-dielectric medium that screens the in-plane LSPR in a manner similar to that described by Chou and Filler24 for LSPR in doped Si nanowires. However, the experimental data follow the theoretical in-plane LSPR response quite well for the nanocrystals with r ≤ 2.44 (first four data points). For the final data point (r = 4.4), the theoretical in-plane LSPR deviates significantly from the experimental data. Ignoring any in-plane LSPR damping from the ZnS capping regions, a dispersion of randomly-oriented, perfectly monodisperse oblate spheroid particles would indeed show two absorbance peaks. However, a distribution of copper sulfide layer thicknesses and shapes exist within each dispersion. It is not surprising, therefore, to observe only one distinct absorbance peak due to polydispersity,2 even without consideration of in-plane LSPR damping. Future studies of these systems should seek to measure the plasmons of solitary nanoparticles to gain clarity on mode assignments. Our assignment of the out-of-plane LSPR mode as the dominant mode would appear to contradict the report by Chou et al.39 which suggests that the absorption cross-section of the out-of-plane LSPR mode should increase as the ZnS caps begin to form, serving to increase the absorbance of the nanocrystal dispersions as the ZnS grains develop. Our experimental absorbance spectrum (Fig. 1), however, shows a monotonic decrease in the absorbance with increasing aspect ratio (i.e., the size of the ZnS caps). This discrepancy can be explained by the decreasing volume fraction of copper sulfide as the aspect ratio increases. The number of carriers that can support LSPR decreases as the copper sulfide phase is replaced with ZnS. Furthermore, from the report by Chou et al. it is unclear how the increase in the absorption cross-section of the out-of-plane LSPR mode scales with the relative volume of the dielectric domains. It cannot be presumed that the reported absorption cross-section enhancement will increase continuously as the ZnS caps grow such that its improvement of the absorbance outweighs the absorbance decrease that arises from the shrinking copper sulfide volume fraction. C. Size effects

A limitation to the analytical model thus far described is the absence of size-dependent effects. The Drude model can be extended to account for nanocrystal size by including a size-dependent term such that the collision frequency, γ , increases as nanocrystal size decreases.40 The modified equation for the collision frequency is shown in Eq. (3) and is written such that size effects exist for all nanocrystals smaller than the initial Cu1.81 S nanocrystals, AvF (r − 1). (3) d0 In Eq. (3), γ bulk is the bulk collision frequency, vF is the Fermi velocity of copper sulfide (5.3 × 105 m/s, treating Cu1.81 S as a free-electron gas), r is the aspect ratio of the copper sulfide “disk,” d0 is the initial diameter of the nanocrystals (22 nm), and A is a slope parameter equal to unity for a sphere with isotropic scattering. Here, we let A = 1.40 The results γ (r) = γbulk +

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of the core-shell model incorporating a size-dependent collision frequency are shown in Fig. 3(b) for both out-of-plane (red dashed line) and in-plane (blue dashed line) resonance modes. (Also shown are the core-shell modeling results from Fig. 3(a), in solid lines.) This “extended” Drude model for the out-of-plane LSPR mode more adequately captures the magnitude of the red-shift experimentally observed, although it fails to describe the decrease in the rate of the LSPR red-shift during cation exchange. D. Structure-dependence of the carrier concentration

As reported by Ha et al.,31 X-ray absorption spectroscopy (XAS) analysis of the heterostructure nanocrystals indicates a change in the copper sulfide structure during cation exchange. The X-ray absorption near-edge structure (XANES) profile of each sample was fit to a linear combination of an experimental XANES profile of Cu1.81 S and Cu2 S, thus quantifying the amount of Cu1.81 S phase present relative to a copper-rich Cu2 S-like phase. The onset of the transformation (h = 20 nm, i.e., ZnS caps are each 1 nm) is marked by an immediate solid-solid phase transition to a Cu2 S-like phase.31 As the cation exchange reaction progresses, the volume fraction of Cu1.81 S increases and then decreases slightly as the thickness of the copper sulfide layer becomes less than ∼5 nm. This unusual change in the copper sulfide stoichiometry was also corroborated with XRD. As the copper sulfide carrier concentration is generally considered phasedependent,12, 38 the solid-solid phase transformation indicated from XANES and XRD should lead to changes in the carrier density during cation exchange that will influence the LSPR wavelength. This phase transformation can be reflected in the model by pegging the carrier concentration to the experimentally determined phases of the nanocrystal and adjusting the carrier concentration according to the volume fraction of the two phases. To incorporate this phase transformation in the model, we used a linear combination of the carrier concentrations for roxbyite (Cu1.81 S, Nh = 3.3 × 1021 cm−3 ) and copper-rich djurleite (Cu1.97−1.94 S, Nh ∼ 1.0 × 1021 cm−3 ).12, 25 The plots in Figs. 4(a) and 4(b) show the results of this extended model for the out-of-plane (red open circles) and in-plane (blue open circles) LSPR resonance modes, respectively. It is clear that the LSPR shift expected from the change in the carrier concentration resulting from the solid-solid phase transformation does not match the monotonic LSPR red-shift observed experimentally. It can be concluded, then, that the carrier concentration is not controlled by the solid-solid phase transformation, i.e., the carrier concentration is independent of the phase of the material and its stoichiometrically-associated copper vacancy concentration. This result is not entirely unexpected. Xie et al.17 have shown that LSPR in copper sulfide (CuS) nanocrystals is not dependent on vacancy self-doping and that carrier concentration need not have a stoichiometric relationship with the cation vacancy concentration. E. Deviation from the analytical model

As a measure of the validity of the analytical models, the theoretical carrier concentration required to produce the

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(a)

(b) Quantum model Drude model (out-of-plane)

Extended Drude model (out-of-plane)

FIG. 5. (a) Plot of LSPR energy vs. nanocrystal diameter generated from the quantum mechanics model. (b) Plot of LSPR energy versus nanoparticle size comparing the quantum mechanics model to both the Drude model and the extended Drude model (size-dependent collision frequency). The out-of-plane LSPR energy is plotted for both the Drude model and the extended Drude model. Quantum mechanical considerations become significant for the nanocrystals with diameter less than ∼8 nm corresponding to r > 2.75 for our system.

experimentally measured LSPR data was calculated using both the core-shell model and the extended Drude coreshell model (see Fig. S5 of the supplementary material36 ). A change in the copper sulfide stoichiometry is expected to eventually accompany the solid-solid phase transformation to a more copper-rich phase;31 thus, as r increases, one can expect the carrier concentration to decrease and approach that of copper-rich phases such as djurleite (Cu1.97−1.94 S), which is approximately three times less than that of roxbyite.12 For both variants of the model, the out-of-plane LSPR mode for 1 < r < 4 results in the predicted decrease in Nh (Fig. S5). The trend suggested by the out-of-plane modes is therefore more reasonable than the rapid increase in Nh predicted for the in-plane LSPR mode, suggesting that the out-of-plane LSPR mode is dominant. The roxbyite (Cu1.81 S) carrier concentration for our nanoparticles is 3.3 × 1021 cm−3 , and from Fig. S5 we can see that beyond some critical value of r, the size-dependent coreshell model predicts a marked increase in the carrier concentration above that of roxbyite. This un-physical deviation in the carrier concentration occurs for the out-of-plane mode for r > 4 and for the in-plane mode for r > 1.5. The deviation indicates that a LSPR blue-shift—one that is not predicted from the size-dependent core-shell model—is required to match the trend of the experimental data as the r value increases beyond some critical dimension of the copper sulfide layer. This inferred blue-shift is indicative of deviations from the classical, size-dependent core-shell model treated here that may suggest the influence of quantum effects, as described previously by Schimpf et al.19 and Scholl et al.41 for ZnO and Ag nanocrystals, respectively. Following the method described by Scholl et al.,41 an approximate quantum mechanical description of LSPR in the copper sulfide nanocrystals was developed (see the supplementary material36 ). This quantum mechanical model only considers the copper sulfide material properties and ignores geometric considerations. A plot of nanocrystal diameter versus LSPR energy is shown in Fig. 5(a), which shows a LSPR blue-shift similar to that calculated from previous studies.19, 41 Figure 5(b) compares the theoretical size dependent plasmonic behaviors between the quantum mechanics model, classical Drude model, and the extended Drude model. The plot in Fig. 5(b) indicates that quantum me-

chanical effects should become significant when the characteristic diameter of a spherical nanocrystal is less than ∼8 nm, which translates to r > 2.75 for our system, at which point the LSPR red-shift predicted from the Drude Model fails to describe the experimental trend in the LSPR energy. We predict that this quantum mechanical blue-shift affects our r = 4.4 LSPR data point, providing an explanation for the decrease in the red-shift that is experimentally-observed as r > 2.75. IV. CONCLUSIONS

From the classical core-shell model of the heterostructure nanocrystals described above, we can conclude that the hole concentration is not entirely dependent on the structure of the copper sulfide phase, indicating that electronic defects arising from cation vacancy formation are only partly responsible for the total carrier density, as suggested by Xie et al.17 The model also indicates that the presence of the ZnS capping regions in the heterostructure nanocrystals have a pronounced effect on the plasmonic properties, particularly for the out-of-plane LSPR mode. Assuming a dominant out-ofplane LSPR mode, an extended core-shell model with a sizedependent collision frequency term was developed that captures the experimentally observed LSPR red-shift. The effective carrier concentration, as determined from the modeling, was plotted as a function of r (Fig. S5), and the large increase in the carrier concentration expected with increasing r for inplane LSPR suggests that out-of-plane LSPR is the dominant resonance mode. Assuming a dominant out-of-plane mode, the inadequacies of the extended core-shell model to describe the trend in the experimental data are apparent for r > 2.75. To account for the discrepancy between the classical model and the experimental data at large values of r, a quantum mechanical model was introduced which suggested that the LSPR blue-shift with increasing r may become significant for r > 2.75, or when the characteristic dimension of the nanoparticle is less than 8 nm. The indication that quantum mechanical affects become significant for copper sulfide nanocrystals with characteristic dimensions less than 8 nm substantiates the use of a model based on classical electrostatics to describe LSPR in these nanocrystals for r < 2.75 (when the characteristic dimension is >8 nm). In addition, the quantum

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plasmonic effect may explain why the rate of the experimental LSPR red-shift decreases with increasing r, a trend opposite that predicted from the extended Drude model. Overall, however, our results show that plasmonic behavior in capped semiconductor nanoparticles does not follow the generally accepted analytical models, and thus more advanced theoretical treatments may be necessary to understand these new systems. ACKNOWLEDGMENTS

This work made use of the Cornell Center for Materials Research Shared Facilities which are supported through the NSF MRSEC program (DMR-1120296). The work was supported in part by the National Science Foundation under Award No. CHE-1152922. D-H.H. and R.D.R. were supported as part of the Energy Materials Center at Cornell (EMC2 ), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Science under Award No. DE-SC0001086. 1 J.

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