Single-electron induced surface plasmons on a topological nanoparticle

It is rarely the case that a single electron affects the behaviour of several hundred thousands of atoms. Here we demonstrate a phenomenon where this happens. The key role is played by topological insulators—materials that have surface states protected by time-reversal symmetry. Such states are delocalized over the surface and are immune to its imperfections in contrast to ordinary insulators. For topological insulators, the effects of these surface states will be more strongly pronounced in the case of nanoparticles. Here we show that under the influence of light a single electron in a topologically protected surface state creates a surface charge density similar to a plasmon in a metallic nanoparticle. Such an electron can act as a screening layer, which suppresses absorption inside the particle. In addition, it can couple phonons and light, giving rise to a previously unreported topological particle polariton mode. These effects may be useful in the areas of plasmonics, cavity electrodynamics and quantum information.

I think the method is acceptable for low intensity incident wave and small particle sizes. The methodology is summarised below. The authors used a set of fitting parameters (from the experimental in Ref. 19) for the dielectric function of the bulk Bi2Se3 (see Eq. 2). The authors have also used the parameter A/R in Ref. 2 of the supplementary materials (which was obtained from a k-dot-p perturbation theory) to evaluate the eigenstates confined on a spherical surface (Eqs. 5 to 10 in Supplementary Materials. Then, within the linear approximation, the timedependent surface charge density due to an oscillating driving electric-field is obtained by the transition rates (Eq. 14). The surface charge density is further used to evaluate the total dipole moment of the whole particle including the surface and bulk contributions in quasi-static approximation. The dipole moment is then used to find the absorption cross section of the particle.
The results could be useful for comparisons with experiments in the future and therefore should be published in some good journals. However, the presentation in this manuscript should be improved to better illustrate the physics behind. For example, the meaning of "Quantum Plasmons" in the title is not defined. It looks like that the "Quantum" part is about the discrete electronic states due to finite surface. It will be nice if the author can use better term to emphasize on the difference from quantum dots and quantization of plasmons.
Reviewer #2 (Remarks to the Author) The manuscript under review predicts a novel light absorption resonance in topological nanoparticles. In a nutshell, it is argued that a nanoparticle made of an ordinary insulator will only display two (light) absorption resonances while one made of topological insulator will display an intermediate one due to the topological surface states.
I believe this theoretical effect is predicted in an oversimplified context and has very little chances to be observed experimentally. For this reason I cannot recommend the article for publication.
1. The theoretical model starts by placing a (single) Dirac metal on a sphere. In reality, the nanoparticle will be etched out of crystal, hence one will have to deal with the surface states on different surface terminations. It is known that Dirac cones are sensitive to surface direction and surface termination, hence the picture of a single Dirac metal on the surface of the sphere is, the least, an oversimplification. 4. At a more in-depth level, the metallic character is generally guarantied on flat surfaces where the back-scattering is prohibited, but on curved surfaces this is no longer the case. For this reason, I believe the Dirac metal will rather be a trivial Anderson insulator. 5. Given all the above, I believe only a first principle calculation will be convincingly enough on proving or disproving the effect. Perhaps, in a first phase one could investigate discrete lattice models terminated in various ways to give a nanoparticle.
I have other minor comments if the authors want to consider in the future. I think it is one of our main duties, as a community, to give proper acknowledgement to the works who initiated the fields mentioned in the introduction, such as In this work the authors present a model study on the interaction of topological insulator nanostructures with the electromagnetic field. The results are novel and interesting and the paper is well written. The topic is timely and can potentially attract the interest of theoretical and experimental physicists in the fields of plasmonics and quantum condensed matter physics.
Before I can recommend publication on Nature Communications, the authors need to address the comments below: i) According to the title this is a study on quantum plasmons. However quantum plasmonics is generally regarded as the field of research that involves the study of the quantum properties of light and its interaction with matter at the nanoscale (see e.g. Nature Physics 9, 329-340 (2013)). All the results presented in this paper have been derived within a semiclassical model where the TI is treated within a quantum model and the electromagnetic field is treated classsically. Hence the title and also some description in the abstract and the text can be misleading and/or inappropriate.
ii) In the abstract, and in the text in different points the authors point out that the proposed phenomenon may pave the way for quantum behaviour to be observed at room temperature. The authors should justify better this statement. For example also exciton absorption lines in organic thin films or in molecular aggregates can be considered as quantum phenomena which can be observed at room temperature.
Most important, all the calculations presented by the authors have been performed at zero temperature. The room temperature behaviour could be very different. The authors should make the effort to present some result as a function of temperature. I think that it is not sufficient to claim that the surface states have shown very little temperature dependence. This is a key point, since the interest of experimentalists and the possibility for interesting applications strongly depend on the behaviours at nonzero and room temperatures.

Response to the reviewers
We are grateful to the reviewers for reading our manuscript carefully and providing such meaningful reviews. Their comments and criticisms have been helpful in revising the manuscript and improving its presentation.

Reviewer 1
The reviewer gives very positive evaluation of our work. Their only suggestion is to choose a more descriptive title which we have done in the revised manuscript.
The results could be useful for comparisons with experiments in the future and therefore should be published in some good journals. However, the presentation in this manuscript should be improved to better illustrate the physics behind. For example, the meaning of "Quantum Plasmons" in the title is not defined. It looks like that the "Quantum" part is about the discrete electronic states due to finite surface. It will be nice if the author can use better term to emphasize on the difference from quantum dots and quantization of plasmons.
We are very grateful to the reviewer for reading the manuscript and thank them for positive feedback. We fully agree with their comment and have changed the title to "Topological nanoparticle: single-electron induced surface plasmons" We have better explained the motivation behind it stressing the similarities with plasmons in metallic nanoparticles. A sentence contrasting the behaviour with that of non-topological quantum dots has been added.

Reviewer 2
The reviewer states that the predicted phenomenon is novel however they doubt that our model is rigorous and can be realised in practice proposing further calculations. Following up on the suggestion, we have performed the suggested calculations which show good agreement with our analytical model. We have fully addressed the reviewer's concerns adding numerous supporting evidence and mentioning available experimental techniques to produce topological nanoparticles.
I believe this theoretical effect is predicted in an oversimplified context and has very little chances to be observed experimentally. For this reason I cannot recommend the article for publication.
We understand the reviewer's point of view and have expanded the manuscript including the arguments for why our model is robust in prediction of phenomenon and experimentally accessible relying on the strong basis which exists in the literature. This is detailed below where we carefully address the reviewer's comments.
The theoretical model starts by placing a (single) Dirac metal on a sphere. In reality, the nanoparticle will be etched out of crystal, hence one will have to deal with the surface states on different surface terminations. It is known that Dirac cones are sensitive to surface direction and surface termination, hence the picture of a single Dirac metal on the surface of the sphere is, the least, an oversimplification...At a more in-depth level, the metallic character is generally guarantied on flat surfaces where the back-scattering is prohibited, but on curved surfaces this is no longer the case. For this reason, I believe the Dirac metal will rather be a trivial Anderson insulator.
We agree that deviation from ideal spherical shape will modify the surface spectrum of the nanoparticle however it cannot remove the effect. The existence of the surface states is guaranteed by the non-trivial Z 2 invariant of the material. On a flat surface they form a Dirac cone. As noted by Haldane [1] (and references therein), the Dirac cone covers area A k which is a small fraction of the 2D 1BZ. This implies that in real space their wavefunctions cannot be localised over areas smaller that ≈ (2π) 2 /A k rendering them insensitive to atomic size details of the surface. This is even more so for the nanoparticle surface where few discrete states available cannot produce fine features in real space. On a general surface, the surface states are influenced by surface curvature and surface termination. The curvature enters the employed model explicitly through Berry phase [2]. Also, the Dirac cone is only moderately distorted on different surfaces as has been shown in the work of Zhang, Kane and Mele [3] whose results we present in Fig. 1 below for convenience. More generally, the new resonance is not restricted to spheres in the same way as ordinary plasmon resonance occurs in spherical metallic particles but also particles of other shapes [4]. The advantage of spherical geometry is that it allows analytic solution and provides physical insight, the effect however will remain for nanoparticles of other shapes. The information above has also been added in the revised manuscript. The effect is relevant for nanoparticles of radius less than 100nm. The authors fail to say how such a nanoparticle will be fabricated out of Bi2Se3 which has a very large unit cell... Recall that the metallic character of the surface states are protected by the bulk; if there is not enough "bulk," there cannot be any protection, hence the main assumption of topological protection is under question.
We completely agree with the reviewer that producing spherical nanoparticles is challenging but as described above the reported phenomenon relies on the nanoscale size of the particle rather than its shape. Though Bi 2 Se 3 has a large unit cell, the resolution lies in its hexagonal layered structure (c=2.864 nm and a=0.414 nm [5]). Along the c-axis it consists of three identical quintuple layers (≈ 1 nm) shifted relative to one another (ABC-stacking). Thus a sphere with 10 nm radius is 20 quintuple layers high and up to 50 lattice parameters wide which should provide enough 'bulk' for protection. Looking from another point of view, the protection occurs for particle sizes much larger than the decay length of surface states into the bulk. Decay length of 0.5 nm for the model employed (Eq. 15 in [2]) agrees well with experimental evidence that states on opposite surfaces of thin films noticeably interact only for thicknesses below 5 nm [6]. The above suggests that the radius of 10 nm used for all absorption calculations is a reasonable limit (this information has been added in the revised manuscript). In practice, such nanoparticles can be produced with two techniques: • Vapour-liquid-solid growth technique on gold nanoparticles yields Bi 2 Se 3 nanowires of thicknesses down to 25 nm [7]. Stopping the growth early would enable one to obtain nanoparticles.
• Electron-beam lithography technique allows resolution of < 10 nm [8]. It is being used extensively in nanophotonics and has already been applied to fabricate Bi 2 Se 3 microribbons. [9].
Given all the above, I believe only a first principle calculation will be convincingly enough on proving or disproving the effect. Perhaps, in a first phase one could investigate discrete lattice models terminated in various ways to give a nanoparticle.
We certainly agree that first principles calculations would strengthen our case. Such calculations exist for the case of topological insulator cylinder [10] and are shown in Fig. 2 (Left) for convenience. In addition, we have applied a tight-binding model [2] to the spherical nanoparticle. The results presented in Fig. 2 (Right) show that analytical model adequately describes states close to the Dirac point especially for large radii. Furthermore, the dispersion is weakly modified if one instead considers a cubic particle as expected from the topological nature of the states. These results have been added to the revised manuscript with details described in supporting information. Unfortunately, even the nanoparticles of smallest sizes considered would consist of thousands of atoms and are too costly to investigate with density functional theory. We certainly agree that our model has limitations but this is the best that can be done currently. Further work employing ambitious density functional theory calculations using dedicated large-scale simulation methods (such as ONETEP) would indeed be interesting, but ultimately the validation can only from experiment, so there is value in making this result available to the community at this stage. We agree and have added references to the papers mentioned.

Reviewer 3
The reviewer rated the results and their presentation positively. They suggest publishing the manuscript on condition of improving the title and adding finite temperature simulations. Following up on the suggestion we have performed the simulations and have fully addressed all of the reviewer's concerns.
The results are novel and interesting and the paper is well written. The topic is timely and can potentially attract the interest of theoretical and experimental physicists in the fields of plasmonics and quantum condensed matter physics.
We thank the reviewer for high evaluation of the results and are very grateful for their comments which have been fully addressed as detailed below.
i) According to the title this is a study on quantum plasmons. However quantum plasmonics is generally regarded as the field of research that involves the study of the quantum properties of light and its interaction with matter at the nanoscale (see e.g. Nature Physics 9, 329-340 (2013))... Hence the title and also some description in the abstract and the text can be misleading and/or inappropriate.
We agree and have changed the title to "Topological nanoparticle: single-electron induced surface plasmons" We have also modified abstract and relevant text stressing the similarities with plasmons in metallic nanoparticles. The classical treatment of light is made more explicit and we have added a reference to the mentioned paper.
ii) In the abstract, and in the text in different points the authors point out that the proposed phenomenon may pave the way for quantum behaviour to be observed at room temperature. The authors should justify better this statement... Most important, all the calculations presented by the authors have been performed at zero temperature. The room temperature behaviour could be very different. The authors should make the effort to present some result as a function of temperature.
We agree and have expanded the relevant part of the manuscript adding a figure with finitetemperature simulations in the supporting information. These modifications are outlined below.  Fig. 1 in [12].
Presence of Dirac surface states at room temperature is confirmed in ARPES studies as presented in Fig. 3 for convenience. States close to the Dirac point can only scatter into other surface states due to small energy range of phonons in Bi 2 Se 3 [13] as shown in Fig. 3(a). Early studies found that the surface states couple to the surface phonon developed from bulk α (2.0 THz) [14,15]. The coupling to a single optical phonon at ≈ 8 meV (1.9 THz) was confirmed with: • Time-resolved ARPES on (001) surface at 40 K [16] • Transport measurement on elongated crystals at 30 K [17] Scattering cannot happen if the energy spacing of the surface states is larger than the energy of available phonons. To model this we assume that the surface states with energy spacing ≥8 meV are unaffected by finite temperature while those with less are completely smeared out. This is a very strict approximation which also acknowledges the possibility of scattering by phonons of lower energy (e.g. acoustic). Finite temperature enters our model by multiplying δ R (the surface term) with a smooth step function centered at R T = 37.5 nm (surface level spacing A/R T = 8 meV). We use the bulk dielectric function, in measured at 300 K from a different source [9]. The resulting absorption cross section presented in Fig. 4 is valid for T ≤40 K. This is a conservative estimate because small number of the surface states available in nanoparticle compared to bulk samples reduces scattering. In addition, the states far from the Dirac point have bulk-like character and participate less in surface scattering [1]. Thus the possibility of observing the effect at room temperature cannot be ruled out, however, without direct experimental evidence we have relaxed our claims in the revised manuscript.