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Assessing the role of interatomic position matrix elements in tight-binding calculations of optical properties

by Julen Ibañez-Azpiroz, Fernando de Juan, Ivo Souza

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Submission summary

Authors (as registered SciPost users): Julen Ibanez · Ivo Souza · Fernando de Juan
Submission information
Preprint Link:  (pdf)
Date submitted: 2021-12-02 10:39
Submitted by: Ibanez, Julen
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


We study the role of hopping matrix elements of the position operator $\mathbf{\hat{r}}$ in tight-binding calculations of linear and nonlinear optical properties of solids. Our analysis relies on a Wannier-interpolation scheme based on \textit{ab initio} calculations, which automatically includes matrix elements of $\mathbf{\hat{r}}$ between different Wannier orbitals. A common approximation, both in empirical tight-binding and in Wannier-interpolation calculations, is to discard those matrix elements, in which case the optical response only depends on the on-site energies, Hamiltonian hoppings, and orbital centers. We find that interatomic $\mathbf{\hat{r}}$-hopping terms make a sizeable contribution to the shift photocurrent in monolayer BC$_2$N, a covalent acentric crystal. If a minimal basis of $p_z$ orbitals on the carbon atoms is used to model the band-edge response, even the dielectric function becomes strongly dependent on those terms.

Current status:
Has been resubmitted

Author comments upon resubmission

We thank the editor in charge for the helpful processing of our manuscript. We also wish to thank both referees for their thoughtful reports. We have modified the manuscript in order to clarify the points raised by the referees, which we believe has resulted in an improved version.

List of changes

- The Abstract now highlights the importance of interatomic position matrix elements.
- The introduction now comments also on Hamiltonian on-site and hopping matrix elements, and draws analogies between Hamiltonian and position hoppings.
- Section 2 of previous version has been restructured into Sections 2 and 3.
- The main results are now discussed in a different order; Section 5.2 is devoted to the basis set composed of 4 Wannier functions, while Section 5.3 focuses on the set composed of 2 Wannier functions, including discussion of the k.p model.
- The discussion section focuses on hopping terms of the position operator and their relevance.
- Figures and Table: Table 1, Fig. 5 and Fig. 6 of previous version are not present in the modified version, while Fig. 5 and Fig. 8 of the modified version are new. The rest of figures have been slightly modified to address the comments of the referees.

Reports on this Submission

Anonymous Report 2 on 2021-12-13 (Invited Report)


The response of the authors to my comments and those of the other referee is satisfactory, and I believe that the paper is a meaningful advance in the understanding of the microscopic origin of optical response in materials.


The manuscript meets the standards of SciPost Physics in its current form and can be published as is, in my opinion.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
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Report 1 by Jae-Mo Lihm on 2021-12-6 (Invited Report)

  • Cite as: Jae-Mo Lihm, Report on arXiv:1910.06172v3, delivered 2021-12-06, doi: 10.21468/SciPost.Report.4002


The authors have responded to my comments adequately. I believe this work on the interatomic position matrix elements will encourage a push to study and include its effect on the calculation of various linear and nonlinear response properties in real materials and thus recommend publication in SciPost Physics.

I have only three minor comments.

1. Regarding the authors' reply on RC1, Ref. [14] (Ref. [10] in the previous version) defines $\omega_{mn} = \omega_m-\omega_n$ (Below Eq. (32), p. 5341), which is different from the authors' definition of $\hbar \omega_{nm} = E_m-E_n$ (below Eq. (4) of the current version). But the form of Eq. (57) of Ref. [14] is identical to Eq. (8) of the submitted manuscript, which means that the delta function reads $\delta(\omega_m - \omega_n - \omega)$ in Ref. [14] and $\delta(\omega_n - \omega_m - \omega)$ in the submitted manuscript. Could the authors check that this is correct?

2. In Section 5.3.2, the coefficients f_i, f_ia, and f_iab are used in Eqs. (20, 21) but not defined in the main text. It will be easier to follow if these coefficients as well as the k dot p Hamiltonian are defined in the main text, rather than in the appendix.

3. Typo in p.16: “sectot” -> “sector”

  • validity: top
  • significance: high
  • originality: top
  • clarity: top
  • formatting: perfect
  • grammar: perfect

Author:  Julen Ibanez  on 2021-12-16  [id 2033]

(in reply to Report 1 by Jae-Mo Lihm on 2021-12-06)

We thank referee Jae-Mo Lihm for his comments, which we have taken into account for the revised version of the manuscript. Here is our reply:

1- Following Appendix A of Ref 22, it can be seen that the order of the nm indexes inside the delta function does not alter the result of the shift current. However, we do agree with the referee that the convention $\omega_{nm}=\omega_{m}-\omega_{n}$, which was also used in Ref. 22, is somewhat anti-intuitive, hence we decided to switch to the more common convention $\omega_{nm}=\omega_{n}-\omega_{m}$.

2- We now define the k.p Hamiltonian and the expansion coefficients in the main text.

3- We have fixed the typo.

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