Comment pubs.acs.org/JPCC
Comment on “HgS and HgS/CdS Colloidal Quantum Dots with Infrared Intraband Transitions and Emergence of a Surface Plasmon” Niket Thakkar,† Alina M. Schimpf,‡,§ Carolyn E. Gunthardt,‡ Daniel R. Gamelin,*,‡ and David J. Masiello*,‡,† †
Department of Applied Mathematics and ‡Department of Chemistry, University of Washington, Seattle, Washington 98195, United States
J. Phys. Chem. C 2016, 120 (21), pp 11744−11753. DOI: 10.1021/acs.jpcc.6b04014 J. Phys. Chem. C 2016, 120. DOI: 10.1021/acs.jpcc.6b10286
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implementations of both papers, many of which were already discussed in ref 2, such a small discrepancy is not significant, and consequently it changes none of the conclusions drawn in ref 2. Ref 1 also raises the following issues with the way in which the ZnO dipolar LSPR frequencies are evaluated in ref 2: (1) the above approximation introduces an unphysical oscillatorstrength threshold that does not allow a smooth evolution from independent to collective electronic motion, (2) the above approximation generates a root in the region of anomalous dispersion, and (3) the use of the term “quantum plasmon.” Each of these points is now addressed: (1) The approximate way in which the roots of the polynomial Re ε(ω) + 2εm = 0 are determined does lead to an oscillator-strength threshold that must be overcome for a single- (or many-degenerate) electronic transition(s) to contribute to collective motion, although not the threshold reported in ref 1. Equation 6 in ref 1 suffers from a typographical error that would indeed lead to drastic physical consequences if correct. Instead, the threshold is
n ref 1, a model describing the surface plasmon resonances of semiconductor nanocrystals possessing extra band-like charge carriers is presented and contrasted with the model previously applied in ref 2. The latter is portrayed as having significant deficiencies that lead to erroneous interpretations. The two models were not adequately compared in ref 1, however, or in the work of ref 3 that is highlighted in ref 1 (and which was also published after ref 2). Upon comparison, we find that the two models are in fact equivalent, contain exactly the same physics, and have consequently led to the same conclusions about the experimental phenomenon under discussion. To determine the nanocrystal’s dipolar localized surface plasmon resonances (LSPRs), ref 2 considered the poles of the induced Green’s function associated with the quasistatic solution of Maxwell’s equations for a dielectric sphere. Contained within the sphere are many noninteracting quantum-mechanical electrons bound by an infinite sphericalwell potential of the same radius. Equivalently, ref 1 considered the poles of the induced polarization for the same problem but in the case of just one electron. Note that the induced polarization is derived from the induced Green’s function and shares the same poles; e.g., the sphere’s dipolar LSPR occurs at the frequency where ε(ω) + 2εm = 0. What is different between refs 1 and 2 is how these poles are calculated in practice. This is not a dif ference in model but rather a dif ference in implementation. Because of the mathematical simplicity of the one-electron case, ref 1 applied the condition ε(ω) + 2εm = 0 analytically exactly. This condition is equivalent to Re ε(ω) + 2εm = 0 together with Im ε(ω) = 0. In ref 2, many resonances were found to be significant, and the approximation Re ε(ω) + 2εm = 0 together with Im ε(ω) ≈ 0 was used. This well-established approximation has been invoked previously on numerous occasions; refs 4−7 represent just a few. This approximation was invoked because the exact high-order polynomial defining the nanocrystal’s dipolar LSPR was impossible to factor analytically. Instead, its roots were determined numerically using a graphical method to find the frequencies where Re ε(ω) crossed −2εm while simultaneously requiring Im ε(ω) ≈ 0. The only difference between the two implementations thus lies in the practical significance of this approximation. Analysis shows that the difference between using Im ε(ω) ≈ 0 and the exact Im ε(ω) = 0 results in variations of ≲1% over the full range of modeling results reported in ref 2. In light of the many idealizations and approximations inherent to the theoretical © XXXX American Chemical Society
2πA > ω0γ εIB + 2εm
(1)
which, when applied as presented in ref 2, is of no significance because the transitions that contribute to collective electronic motion carry enough oscillator strength to exceed eq 1 in all of the scenarios described. This result is due to the existence of multiple degenerate electronic transitions stacking up to produce the plasmon. Consequently, as shown in Figure 4b of ref 2, a continuous evolution of intraband transitions (white circles) and plasmon resonances (black circles) is predicted as a function of ZnO nanocrystal radius, spanning from a distinct, nearly classical plasmon at large radii (and large number of electrons) to convergence of the two features at small radii (and small number of electrons). This figure is a testament to the fact that the oscillator strength threshold was never encountered (because we would not have been able to locate the LSPR frequencies denoted by the black bullets if it were). Figure 1 reproduces Figure 4b of ref 2 schematically, and highlights both of these limits as well as the quantum plasmon regime bridging the two that was identified in our study. Given Received: October 9, 2016 Revised: November 9, 2016 Published: November 21, 2016 A
DOI: 10.1021/acs.jpcc.6b10200 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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equivalent model is the definition of the LSPR frequency. Reference 2 defines it as the poles of the induced Green’s function. This is the same definition used for LSPRs in the coinage metals.10 In the limit of one intraband transition, this pole occurs exactly at3,11 Ω=
ω02 + Ω 20 − (γ /2)2
(2)
Alternatively, ref 1 proposes to define the transition as a “surface plasmon resonance” when the Drude frequency Ω0 exceeds the single-electron transition frequency ω0 and as an “intraband transition” otherwise, acknowledging that there exists no true physical threshold separating the two. This attempt to partition the transitions into two distinct groups contradicts the notion of a continuous evolution from one type to the other. In ref 2, nonclassical plasmons on both sides of this artificial divide were termed “quantum plasmons”, a name that we maintain is valuable for describing the intermediate regime (Figure 1) bridging the classical LSPR and quantum single-particle excitation limits in the electronic spectra of n- or p-doped semiconductors precisely because of this smooth evolution from one limit to the other. Finally, we draw the reader’s attention to the fact that ref 2 marks the first identification of such nonclassical (“quantum”) surface plasmon resonances in any colloidal semiconductor nanocrystals. This work built upon many earlier assignments of LSPRs in colloidal semiconductor nanocrystals (cited in ref 2) that had assumed classical Drude-like behavior. In ref 2, a clear deviation from classical LSPR behavior was demonstrated experimentally using ZnO quantum dots, and its origins were correctly identified and explained theoretically. Reference 1 does not dispute these points. Reference 2 concluded that identification of this quantum plasmon regime helps to “reconcile conflicting evidence for interpretation of the IR bands of doped semiconductor nanocrystals as plasmon resonances versus single-electron excitations”. We are therefore gratified to see that the major conclusions of ref 2 have been confirmed and built upon in ref 1 using a different material system (HgS), including the conclusion again that this new understanding indeed “reconciles differing interpretations of the intraband absorption as single-electron transitions vs plasmon resonances”.
Figure 1. Illustration of the evolution of the dipolar LSPR energy (upper curve) and intraband transition energy (lower curve) as a function of nanocrystal radius at constant carrier density, adapted from ref 2. The LSPR energy merges with the intraband transition energy ℏω0 in the small radius (few-carrier) limit and extrapolates to the classical Drude LSPR energy ℏ Ω 20 − (γ /2)2 familiar to the coinage metals in the large radius (many-carrier) limit. The intermediate region, denoted by the gray box, is the quantum plasmon regime. Overall, the energy of the quantum plasmon continuously evolves away from ℏω0 to the value ℏ ω02 + Ω 20 − (γ /2)2 as the plasmon inherits increased free-electron gas character with increasing nanocrystal radius at fixed carrier density.
that this smooth convergence of the intraband and collective electronic excitations at small radii (few carriers) was illustrated in Figure 4b and discussed multiple times in ref 2 (e.g., page 1068: “As the radius decreases, the LSPR and lowest-energy singleelectron transition energies converge. A similar convergence occurs for f ixed radius as Ne is reduced.” See also pages 1069, 1070, and the caption of Figure S9. The Reply denies the existence of this text if ref 2), it is unclear why ref 1 emphasizes the expectation of a similar smooth evolution between intraband and collective electronic excitations as a significantly different result. We note that the Reply does not use the parameters from our model to make its Figure 1, and the use of different parameters is required to generate the discontinuity shown. Application of the implementation from ref 1 to avoid this discontinuity then merely retrieves the same conclusion already drawn in ref 2 using our original parameters and implementation, namely, that the two types of transitions converge in the low-carrier-density limit. Therefore, no new physical insight is achieved. (2) The graphical method employed in ref 2 does produce roots in the regions of both normal and anomalous dispersion. Reference 2 describes that the latter were identified and rejected, and the former were adopted as the physical solutions. The justification for this selection lies in comparing the magnitudes of Im ε(ω) at the normal and anomalous roots, ΩN and ΩA. Only ΩN, which is blue-shifted from ω0 and lies in the (normal) dispersive tail of Re ε(ω), has Im ε(ΩN) ≈ 0, with this approximation improving as the ZnO nanocrystal radius increases. (3) The term quantum plasmon is used in ref 2 because the dipolar LSPRs are distinctly not classical, i.e., non-Drude. This name has been used in a variety of settings where quantum mechanics must be invoked to describe the physics of the plasmon. Examples include the excitation of multiple-quanta LSPRs in metallic nanorods8 as well as the observation of distinctly nonclassical LSPR behavior in small metallic nanoparticles,9 both using electron spectroscopy techniques. More important than the discussion of how a particular constraint was enforced within two implementations of an
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AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected]. ORCID
David J. Masiello: 0000-0002-1187-0920 Present Address §
Department of Chemistry and Biochemistry, University of California San Diego, La Jolla, CA 92093, United States Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the U.S. National Science Foundation (Graduate Research Fellowship DGE-1256082 to A.M.S. and N.T., CAREER Award CHE-1253775 to D.J.M., and DMR-1206221, DMR-1505901 to D.R.G.). B
DOI: 10.1021/acs.jpcc.6b10200 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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REFERENCES
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