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Topological-Insulator Nanocylinders

by Michele Governale, Fabio Taddei

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Submission summary

Authors (as registered SciPost users): Michele Governale
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-02-13
Date submitted: 2022-11-23 11:27
Submitted by: Governale, Michele
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


Nanostructures, such a quantum dots or nanoparticles, made of three-dimensional topological insulators (3DTIs) have been recently attracting increasing interest, especially for their optical properties. In this paper we calculate the energy spectrum, the surface states and the dipole matrix elements for optical transitions with in-plane polarization of 3DTI nanocylinders of finite height L and radius R. We first derive an effective 2D Hamiltonian by exploiting the cylindrical symmetry of the problem. We develop two approaches: the first one is an exact numerical tight-binding model obtained by discretising the Hamiltonian. The second one, which allows us to obtain analytical results, is an approximated model based on a large-R expansion and on an effective boundary condition to account for the finite height of the nanocylinder. We find that the agreement between the two models, as far as eigenenergies and eigenfunctions are concerned, is excellent for the lowest absolute value of the longitudinal component of the angular momentum. Finally, we derive analytical expressions for the dipole matrix elements by first considering the lateral surface alone and the bases alone, and then for the whole nanocylinder. In particular, we focus on the two limiting cases of tall and squat nanocylinders. The latter case is compared with the numerical results finding a good agreement.

Author comments upon resubmission

The Referee raises two points of criticisms, which we address and rebut in detail below.

The Referee's report has been of help to improve the manuscript and we are grateful for this. Following their comments we have extensively modified the introduction and conclusions sections and we have added a new figure (Fig. 7) in the manuscript.

Finally, we wish thank the Referee for bringing to our attention a relevant reference, which we have now included.

Detailed Answers

1) We agree with the Referee that our work is an extension of our previous work, although not a trivial one. The previous work dealt with infinitely-long cylinders. In the present manuscript we focus on finite-height cylinders, which are of current experimental interest (see reply to point 3). This is done with a two-pronged approach: a) we derive analytically a quantisation-condition for the longitudinal momentum; b) we use a finite-difference numerical model which is applicable to cylinders of any aspect ratio.

We wish to stress here that the quantisation condition that we find (which is fully validated by the comparison with the numerical results) differs from the one of Kundu et al.

We indeed overlooked the work mentioned by the Referee. However, our approach and the one of Kundu et al. do differ and as a consequence we find different results, as detailed in point 4.

2) We agree with the Referee that we did not elaborate sufficiently on the motivations for our work.
Obviously experimental realisations of topological-insulator nanocylinders are of finite height. And one possible way to characterise their topologically-protected surface states is by optical spectroscopy. Two important mechanisms affecting the optical absorption of a finite cylinder are neglected in the infinite-height limit considered in our previous paper Ref. 54. First, the finite height changes the energies of the surface states and therefore the frequencies of the absorption lines. Second, the wavefunction on the sides of the cylinder is affected by the finite height and moreover the bases also need to be accounted for. The fact that the wavefunction is different in the case of finite-height cylinders implies that the dipole matrix elements, which determine the strength of the absorption, are different than the one computed in the infinite-height limit.

Finally, there is the very interesting theoretical point of finding the correct quantisation condition for the longitudinal momentum entering the wavefunction of the lateral side surface of the cylinder. Indeed, the side surface state of the cylinder, when it approaches the top/bottom bases, bends and merges with the surface states on the bases and hard-wall boundary conditions cannot be imposed.

In the revised version of the paper we have emphasised these points.

3) We strongly disagree with the Referee on this point. The dipole matrix elements determine the absorption amplitudes in semiconducting low-dimensional structures [See H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th ed. (World Scientific, Singapore, 2004) and I. Pelant and J. Valenta, Luminescence Spectroscopy of Semiconductors (Oxford University Press, Oxford, UK, 2012).

In the case of transitions between surface states of topological-insulator cylinders of realistic radii the frequencies of the absorption lines are in the THz range.

We wish to mention that in the last months we have been providing data to the experimental group led by Stephanie Law who are measuring the absorption of THz radiation in nanocylinders made of Bi$_2$Se$_3$. Our calculations are very valuable in order to predict the frequencies and intensities of the absorptions lines.

The matrix elements plotted in Fig. 6 provide the absorption amplitude for THz radiation with circular polarisation. In particular, the dependence on the radius $R$ is an important prediction in order to maximise the optical response: the absorption amplitude is roughly linear with $R$ with a slope that depends on $n$.

In the revised version of the paper we have added a figure (Fig. 7) showing an example of the absorption spectrum.

4) We are grateful to the Referee for pointing this paper out, we had indeed overlooked the work of Kundu et al., which is definitely very relevant. However, we remark that the only overlap is the numerical solution for the eigenenergies and eigenfunctions. In the following we list the novel contributions of our paper:

a) The analytical approach in Kundu et al. was based on the Dirac-fermion (surface) theory, where only the linear terms are retained in the Hamiltonian. They have found as a quantisation condition for the longitudinal momentum the standard one, namely $k L=n\pi$. On the contrary, we use the full (bulk) Hamiltonian for the wavefunctions of the lateral side of the cylinder and propose a different quantisation condition. We find that such a quantisation condition yields the correct dependence of the eigenenergies on the radius $R$ of the cylinder. On the contrary, the standard quantisation condition of Kundu et al. does not predict the correct dependence on $R$ as can be easily shown by comparing Eq. (26) with $k_z=n\pi/L$ (the result of Kundu et al.) with our tight-binding results (Fig 1 of our manuscript) and our analytical results given by Eqs. (33) with $k_z$ given by Eq. (32). While we do not provide a figure with this comparison in the paper, we are happy to send it to the editor if required.

b) We calculate the probability density for the four different components of the spinor wavefunction, emphasizing peculiar behaviours close to the surfaces of the nanocylinder (for example the fact that ``the density on the two bases appears to be small in the region around their centres."). Moreover, we find that the four components of the wavefunction obtained analytically are a very good approximation for the ones obtained numerically, at least not too close to the bases.

c) We calculate the dipole matrix elements for optical transitions both numerically and analytically. These results are pertinent to current experimental studies. The optical properties of the cylinder are not addressed in Kundu et al. This is the added value of our paper, which we have now stressed in the revised version of the paper in the introduction, in the conclusions and in the results sections. On the contrary, the issue addressed by Kundu in their paper is mainly the spin and parity structure of the eigenstates. Their main finding is that the spin direction in a topologically protected surface mode is not locked to the surface.

List of changes

1) Extensive changes in the introduction;
2) Conclusions completely rewritten;
3) Small changes at the beginning of Section 4;
3) Added Fig. (7) and text describing it in Section 4;
4) Added 3 new References including to the paper by Kundu et al suggested by the Referee.

Published as SciPost Phys. Core 6, 032 (2023)

Reports on this Submission

Anonymous Report 1 on 2023-1-18 (Invited Report)


I thank the authors for their constructive response and the significant work in rewriting parts of the manuscript. I do believe the paper will be of more benefit to readers in the current form. The motivation to carry out the calculation is clear, the mentioned works are discussed appropriately, and the physical response associated to the dipole matrix elements is presented explicitly. I have no further comments and I believe the work is now suitable for SciPost Physics Core.

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