Comment pubs.acs.org/JPCC
Reply to “Comment on ‘HgS and HgS/CdS Colloidal Quantum Dots with Infrared Intraband Transitions and Emergence of a Surface Plasmon’” Guohua Shen and Philippe Guyot-Sionnest* James Franck Institute, The University of Chicago, 929 East 57th Street, Chicago, Illinois 60637, United States
J. Phys. Chem. C 2016, 120 (21), 11744−11753. DOI: 10.1021/acs.jpcc.6b04014 J. Phys. Chem. C 2016, 120. DOI: 10.1021/acs.jpcc.6b10200
T
his is a response to a comment on our paper,1 written by the authors of ref 2.2 Some of the comment is reviewed positively by us. (1) Equation 1 in the comment corrects a typo that was made in eq 6 of our paper. (2) The comment proposes qualitative distinctions between intraband transition, quantum plasmon, and plasmon regimes. This is an improvement from ref 2. (3) The comment correctly identifies the sphere dipole resonance as given by the poles of 1/(ε + 2εm) rather than by solving Re(ε) = −2εm as was presented in ref 2. Other parts of the comment merit discussion. The sentence “The latter is portrayed as having significant deficiencies that lead to erroneous interpretations” is itself an interpretation of the statement in our paper: “Solving for Re(ε) = −2εm has two consequences: first an unphysical result of two resonances arising when there should be only one and, second, an oscillation strength threshold”. The equation of ref 2, Re(ε) = −2εm,2 indeed raised questions. It implied that there must be conditions where the intraband transition at low electron density is too weak given the size of the nanoparticles such that the real part of the dielectric constant does not become negative enough. In this condition, the threshold is not met, and the resonance should remain at its frequency. It is also clear that there must be a number of electrons at which the threshold is reached and Re(ε) = −2εm, and this would then lead to a sudden jump in the resonant frequency. These strange consequences were not discussed in ref 2, although the comment inaccurately claims that “the smooth convergence of the intraband and collective electronic excitation was··· discussed multiple times in ref 2”. This seemed an issue to us: the frequency jump should be detectable; the dielectric constant should be measurable; and the model should be testable in the limit of small electron number. Alternatively, something more profound, such as correlations, was at play as the number of electrons increased. In thinking about these unphysical consequences, we came up with a simple model based on the local field. This gave a physically reasonable gradual change of the resonance frequency with oscillator strength. We then found that this result was in fact included in prior derivations by Kreibig and co-workers,3 by Dionne and co-workers,4 and by Jain.5 As correctly written in the comment, our treatment was a special case of the more general approach which is that the dipole resonances of a sphere are given by the poles of 1/(ε + 2εm). However, ref 2 considered only the approximation Re(ε) = −2εm. © XXXX American Chemical Society
This approximation fails at low electron number. Figure 1 shows the difference for a Lorentzian intraband transition at 0.5
Figure 1. Resonance as a function of the oscillator strength. The black line is using the result from ref 1 or solving for the pole of 1/(ε + 2εm) when ε has a single resonance at 0.5 eV with a damping of 0.165 eV. The red line is calculated with the condition Re(ε) = −2εm following ref 2. The threshold is at ∼25. The green line is the discarded resonance in the anomalous region. The other parameters, the nanocrystal radius of 1.75 nm, the dielectric constants for ZnO, εib= 3.72, and the medium, εm = 2.25, are from ref 2.
eV with a 0.165 eV damping parameter. The parameters apply to the smaller ZnO nanocrystals of 1.75 nm reported in ref 2. The blue shifts for the resonance calculated using 1/(ε + 2εm) (black line) or Re(ε) = −2εm (red and green lines) converge at large oscillator strength but they differ at low oscillator strength with the threshold at an oscillator strength of ∼25 for this particular set of parameters. Twenty-five is the strength of the resonance required before the condition Re(ε) = −2εm is met. At this dot size, ref 2 reports that there are three electrons. Assuming 1S−1P and 1P−1D to be degenerate electron intraband transitions, one may estimate the maximum collective oscillator strength of the three noninteracting electrons as 3/ m* ∼ 10 when using m* = 0.28,2 and this oscillator strength is well below the threshold of 25. Therefore, it is indeed important to not use the approximation of ref 2 for low electron numbers. Figure 1 in the comment introduces qualitative regions for “intraband” transitions, “quantum plasmon”, and “Drude Received: October 11, 2016 Revised: November 11, 2016 Published: November 21, 2016 A
DOI: 10.1021/acs.jpcc.6b10286 J. Phys. Chem. C XXXX, XXX, XXX−XXX
Comment
The Journal of Physical Chemistry C
(4) Scholl, J. A.; Koh, A. L.; Dionne, J. A. Quantum Plasmon Resonances of Individual Metallic Nanoparticles. Nature 2012, 483, 421−427. (5) Jain, P. K. Plasmon-in-a-Box: On the Physical Nature of FewCarrier Plasmon Resonances. J. Phys. Chem. Lett. 2014, 5, 3112−3119. (6) Faucheaux, J. A.; Jain, P. K. Plasmons in Photocharged ZnO Nanocrystals Revealing the Nature of Charge Dynamics. J. Phys. Chem. Lett. 2013, 4, 3024−3030. (7) Buonsanti, R.; Llordes, A.; Aloni, S.; Helms, B. A.; Milliron, D. J. Tunable Infrared Absorption and Visible Transparency of Colloidal Aluminum-Doped Zinc Oxide Nanocrystals. Nano Lett. 2011, 11, 4706−4710. (8) Zhang, H.; Kulkarni, V.; Prodan, E.; Nordlander, P.; Govorov, A. O. Theory of Quantum Plasmon Resonances in Doped Semiconductor Nanocrystals. J. Phys. Chem. C 2014, 118, 16035−16042.
plasmon”. This is an improvement over ref 2 which stated that “once interpreted as intraband single electron transitions, the infrared absorption of doped semiconductor nanocrystals is now commonly attributed to localized surface plasmon resonances”. We agree that there should be a discussion about how the intraband transitions should be called. Our paper proposed an arbitrary but quantitative measure of whether a system is in the plasmon regime or intraband regime, as did Jain earlier.5 The authors of the comment could similarly propose a quantitative measure of the different regions. It may be that the community will use the catchy words “quantum plasmon” liberally. We had noted that the term “quantum plasmon” is confusing because the local field effect is purely classical, while it is the intraband transition that is quantum. One concern is that the same physics of local field corrections applies broadly to dipole-allowed excitations including vibrational, electronic, intraband, and interband transitions, of molecules, quantum dots, wires, wells, everything with a dielectric boundary. One should then think carefully about choosing to rename any dipole transition in a dielectric object a “quantum plasmon”. Regarding the final paragraph of the comment, we agree that ref 2 was one of the first detailed studies of how the collective effects of multiple single electron resonances emerge in colloidal nanocrystals. An important insight, together with a preceding work by Fauchaux and Jain,6 was the assignment of the ZnO infrared resonance to a plasmon resonance for the highly charged ZnO nanocrystals. We note that both built upon the prior observation of ZnO nanocrystal plasmon resonance with Al doping by the Milleron group.7 Our work investigated a different and novel system: HgS with ambient doping. One of our experimental results was that there is a similar evolution to a plasmon resonance at large sizes and large electron number as reported in ref 2, and this was properly referenced. We pointed out and corrected an approximation in the model of ref 2, such that the evolution from the intraband single electron transition with one electron to a collective oscillation with many electrons should be a smooth process. This was supported by prior models3−5 and microscopic calculations.8 We think that this comment and response stress the fact that there is now agreement about the smooth evolution in the frequency, as shown in both figures of the comment and response, and future investigations can focus on the possibly more subtle effects of electron interactions, such as line width and lifetime of the collective excitation.
■
AUTHOR INFORMATION
ORCID
Philippe Guyot-Sionnest: 0000-0003-0178-6255 Notes
The authors declare no competing financial interest.
■
REFERENCES
(1) Shen, G.; Guyot-Sionnest, P. Hgs and Hgs/Cds Colloidal Quantum Dots with Infrared Intraband Transitions and Emergence of a Surface Plasmon. J. Phys. Chem. C 2016, 120, 11744. (2) Schimpf, A. M.; Thakkar, N.; Gunthardt, C. E.; Masiello, D. J.; Gamelin, D. R. Charge-Tunable Quantum Plasmons in Colloidal Semiconductor Nanocrystals. ACS Nano 2014, 8, 1065−1072. (3) Genzel, L.; Martin, T. P.; Kreibig, U. Dielectric Function and Plasma Resonances of Small Metal Particles. Z. Phys. B: Condens. Matter Quanta 1975, 21, 339−346. B
DOI: 10.1021/acs.jpcc.6b10286 J. Phys. Chem. C XXXX, XXX, XXX−XXX
