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Assessing the role of interatomic position matrix elements in tightbinding calculations of optical properties
by Julen IbañezAzpiroz, 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:  https://arxiv.org/abs/1910.06172v3 (pdf) 
Date submitted:  20211202 10:39 
Submitted by:  Ibanez, Julen 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We study the role of hopping matrix elements of the position operator $\mathbf{\hat{r}}$ in tightbinding calculations of linear and nonlinear optical properties of solids. Our analysis relies on a Wannierinterpolation 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 tightbinding and in Wannierinterpolation calculations, is to discard those matrix elements, in which case the optical response only depends on the onsite 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 bandedge response, even the dielectric function becomes strongly dependent on those terms.
Current status:
Author comments upon resubmission
List of changes
 The Abstract now highlights the importance of interatomic position matrix elements.
 The introduction now comments also on Hamiltonian onsite 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.
Submission & Refereeing History
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Strengths
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.
Report
The manuscript meets the standards of SciPost Physics in its current form and can be published as is, in my opinion.
Report 1 by JaeMo Lihm on 2021126 (Invited Report)
 Cite as: JaeMo Lihm, Report on arXiv:1910.06172v3, delivered 20211206, doi: 10.21468/SciPost.Report.4002
Report
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 $\hbar \omega_{nm} = E_mE_n$ (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”
Author: Julen Ibanez on 20211216 [id 2033]
(in reply to Report 1 by JaeMo Lihm on 20211206)We thank referee JaeMo 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 antiintuitive, 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.