# Conductivity Sum Rule in the Nearly Free Two-Dimensional Electron Gas in an Uniaxial Potential

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Multiband Conductivity Tensor

**q**→0 limit in the Equation (2) and dividing it in the intraband (s = s′) and interband (s ≠ s′) channel and assuming that the intraband phenomenological relaxation constants are equal to γ we get

## 3. Two-Dimensional UniAxNFE Model

## 4. Charge Transport Concentrations of the 2D UniAxNFE Model

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Little, W.A.; Holcomb, M.J. Oscillator-Strength Sum Rule in the Cuprates. J. Supercond.
**2004**, 17, 551–558. [Google Scholar] [CrossRef] - Benfatto, L.; Sharapov, S.G.; Andrenacci, N.; Beck, H. Ward identity and optical conductivity sum rule in the d-density wave state. Phy. Rev. B
**2005**, 71, 104511. [Google Scholar] [CrossRef] [Green Version] - Kupčić, I. General theory of intraband relaxation processes in heavily doped graphene. Phys. Rev. B
**2015**, 91, 205428. [Google Scholar] [CrossRef] [Green Version] - Ashcroft, N.W.; Mermin, N. Solid State Physics; Saunders Collage: Rochester, NY, USA, 1976. [Google Scholar]
- Dressel, M.; Grüner, G. Electrodynamics of Solids: Optical Properties of Electrons in Matter; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]
- Millis, A.J. Optical conductivity and correlated electron physics. In Strong Interactions in Low Dimensions. Physics and Chemistry of Materials with Low-Dimens; Baeriswyl, D., Degiorgi, L., Eds.; Springer: Dordrecht, The Netherlands, 2004. [Google Scholar]
- Rukelj, Z.; Radić, D. DC and optical signatures of the reconstructed Fermi surface for electrons with parabolic band. New J. Phys.
**2022**, 24, 053024. [Google Scholar] [CrossRef] - Rukelj, Z.; Akrap, A. Carrier concentrations and optical conductivity of a band-inverted semimetal in two and three dimensions. Phys. Rev. B
**2021**, 104, 075108. [Google Scholar] [CrossRef] - Rukelj, Z. Dynamical conductivity of lithium-intercalated hexagonal boron nitride films: A memory function approach. Phys. Rev. B
**2020**, 102, 205108. [Google Scholar] [CrossRef] - Kupčić, I.; Rukelj, Z.; Barišić, S. Quantum transport equations for low-dimensional multiband electronic systems: I. J. Phys. Condens. Matter
**2013**, 25, 145602. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rukelj, Z.; Radić, D. Topological Properties of the 2D 2-Band System with Generalized W-Shaped Band Inversion. Quantum Rep.
**2022**, 4, 476–485. [Google Scholar] [CrossRef] - Kupčić, I. Damping effects in doped graphene: The relaxation-time approximation. Phys. Rev. B
**2014**, 90, 205426. [Google Scholar] [CrossRef] [Green Version] - Carbotte, J.P.; Schachinger, E. Optical Sum Rule in Finite Bands. J. Low Temp. Phys.
**2006**, 144, 61–120. [Google Scholar] [CrossRef] - Mahan, G.D. Many-Particle Physics; Plenum Press: New York, NY, USA, 1990. [Google Scholar]
- Gusynin, V.P.; Sharapov, S.G.; Carbotte, J.P. Sum rules for the optical and Hall conductivity in graphene. Phys. Rev. B
**2007**, 75, 165407. [Google Scholar] [CrossRef] [Green Version] - Ashby, P.E.; Carbote, J.P. Chiral anomaly and optical absorption in Weyl semimetals. Phys. Rev. B
**2014**, 89, 245121. [Google Scholar] [CrossRef] [Green Version] - Sabio, J.; Nilsson, J.; Castro Neto, A.H. f-sum rule and unconventional spectral weight transfer in graphene. Phys. Rev. B
**2008**, 78, 075410. [Google Scholar] [CrossRef] [Green Version] - Kadigobov, A.M.; Bjeliš, A.; Radić, D. Topological instability of two-dimensional conductors. Phys. Rev. B
**2018**, 97, 235439. [Google Scholar] [CrossRef] [Green Version] - Kadigobov, A.M.; Radić, D.; Bjeliš, A. Density wave and topological reconstruction of an isotropic two-dimensional electron band in external magnetic field. Phys. Rev. B
**2019**, 100, 115108. [Google Scholar] [CrossRef] [Green Version] - Spaić, M.; Radić, D. Onset of pseudogap and density wave in a system with a closed Fermi surface. Phys. Rev. B
**2021**, 103, 075133. [Google Scholar] [CrossRef]

**Figure 1.**Diagrammatic representation of the fully dressed irreducible current-dipole correlation function, or the conductivity tensor [3,9]. The wiggly line represents the incoming photon with impulse $\mathbf{q}$ and frequency $\omega $. Double lines represent the fully dressed electron and hole propagators. The orange and green circle represent the current and the dipole matrix elements respectively. The red part is the vertex function which represents all contributions to the electron-hole scattering originating from the various scattering mechanism like the electron-phonon, electron-impurity etc.. In a simplified expression, the dressed electron and hole propagators are approximated with the bare ones, and the vertex part is approximated by a phenomenological relaxation constant.

**Figure 2.**(

**Left**) The electron energies (15) as functions of ${k}_{\Vert}$ and ${k}_{\perp}$ scaled with ${\epsilon}_{Q}$. The positions of four characteristic energy points (16) and (17) are indicated with yellow circles. The pseudogap region is located between ${\epsilon}_{L}$ and ${\epsilon}_{U}$. (

**Right**) Various charge concentrations as functions of scaled Fermi energy ${\epsilon}_{F}$ in units of concentration ${n}_{0}$. For both figures, the dimensionless gap parameter $\eta =0.2$.

**Figure 3.**(

**Left**) Real part of the interband conductivity component parallel to $\mathbf{Q}$ in units of ${\sigma}_{0}$, as a function of scaled incoming photon energy $\Omega $ plotted for several values of ${\epsilon}_{F}$. The extent of the $\Omega $-domain, where the optical conductivity is finite, is indicated by yellow arrows for the dimensionless gap parameter $\eta =0.2$. The area under the blue curve for ${\epsilon}_{F}=0.7$ is proportional to ${n}_{\Vert}^{inter}({\epsilon}_{F}=0.7)$ shown in Figure 2. (

**Right**) Positions of the three values of ${\epsilon}_{F}$ are indicated by colored arrows. Yellow lines indicate the with of the pseudogap and the maximal energy extent of the bands within the Brillouin zone, consistent with the extent of the allowed $\Omega $-values.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rukelj, Z.; Radić, D.
Conductivity Sum Rule in the Nearly Free Two-Dimensional Electron Gas in an Uniaxial Potential. *Condens. Matter* **2023**, *8*, 1.
https://doi.org/10.3390/condmat8010001

**AMA Style**

Rukelj Z, Radić D.
Conductivity Sum Rule in the Nearly Free Two-Dimensional Electron Gas in an Uniaxial Potential. *Condensed Matter*. 2023; 8(1):1.
https://doi.org/10.3390/condmat8010001

**Chicago/Turabian Style**

Rukelj, Zoran, and Danko Radić.
2023. "Conductivity Sum Rule in the Nearly Free Two-Dimensional Electron Gas in an Uniaxial Potential" *Condensed Matter* 8, no. 1: 1.
https://doi.org/10.3390/condmat8010001