# Collapse of the electron gas from three to two dimensions in Kohn-Sham density functional theory

## Abstract

Here under pressure, a quasi-2D electron gas can collapse toward the true 2D limit. In this limit, the exact exchange-correlation energy per electron has a known finite limit, but general-purpose semilocal approximate density functionals, like the local density approximation (LDA) and Perdew-Burke-Ernzerhof generalized gradient approximation (PBE GGA), are known to diverge to minus infinity. Here we consider a model density for a non-interacting electron gas confined to a thickness $L$ by infinite-barrier walls, with a fixed 2D density $$1/(\pi (r_s^{\text{2D}})^2)$$, and $$r_s^{\text{2D}}=4$$ Bohr. We estimate that LDA, PBE, and the strongly constrained and appropriately normed (SCAN) meta-GGA are accurate for the exchange-correlation energy over a wide quasi-2D range, $$1.5< L/r_s^{\text{2D}} < 3.85$$, but not for smaller $L$. Of these functionals, only SCAN tends to a finite limit when $L$ tends to 0. Since the non-interacting kinetic energy, treated exactly in Kohn-Sham theory, dominates in this limit within a deformable jellium model, all of the general-purpose functionals can estimate the pressure required to achieve any thickness (with SCAN and LDA better than PBE). This pressure vanishes around $$L/r_s^{\text{2D}} = 3.85$$, where the 3D electron density is roughly that of the valence electrons in metallic potassium, and reaches about 20 GPa at $$L/r_s^{\text{2D}} = 1.5$$ and 400 GPa at $$L/r_s^{\text{2D}} = 0.6$$.

- Authors:

- Temple Univ., Philadelphia, PA (United States)

- Publication Date:

- Research Org.:
- Energy Frontier Research Centers (EFRC) (United States). Center for Complex Materials from First Principles (CCM); Temple Univ., Philadelphia, PA (United States). Dept. of Physics

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1467076

- Alternate Identifier(s):
- OSTI ID: 1467139

- Grant/Contract Number:
- SC0012575

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review B

- Additional Journal Information:
- Journal Volume: 98; Journal Issue: 8; Journal ID: ISSN 2469-9950

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

### Citation Formats

```
Kaplan, Aaron D., Wagle, Kamal, and Perdew, John P. Collapse of the electron gas from three to two dimensions in Kohn-Sham density functional theory. United States: N. p., 2018.
Web. doi:10.1103/PhysRevB.98.085147.
```

```
Kaplan, Aaron D., Wagle, Kamal, & Perdew, John P. Collapse of the electron gas from three to two dimensions in Kohn-Sham density functional theory. United States. doi:10.1103/PhysRevB.98.085147.
```

```
Kaplan, Aaron D., Wagle, Kamal, and Perdew, John P. Tue .
"Collapse of the electron gas from three to two dimensions in Kohn-Sham density functional theory". United States. doi:10.1103/PhysRevB.98.085147. https://www.osti.gov/servlets/purl/1467076.
```

```
@article{osti_1467076,
```

title = {Collapse of the electron gas from three to two dimensions in Kohn-Sham density functional theory},

author = {Kaplan, Aaron D. and Wagle, Kamal and Perdew, John P.},

abstractNote = {Here under pressure, a quasi-2D electron gas can collapse toward the true 2D limit. In this limit, the exact exchange-correlation energy per electron has a known finite limit, but general-purpose semilocal approximate density functionals, like the local density approximation (LDA) and Perdew-Burke-Ernzerhof generalized gradient approximation (PBE GGA), are known to diverge to minus infinity. Here we consider a model density for a non-interacting electron gas confined to a thickness $L$ by infinite-barrier walls, with a fixed 2D density $1/(\pi (r_s^{\text{2D}})^2)$, and $r_s^{\text{2D}}=4$ Bohr. We estimate that LDA, PBE, and the strongly constrained and appropriately normed (SCAN) meta-GGA are accurate for the exchange-correlation energy over a wide quasi-2D range, $1.5< L/r_s^{\text{2D}} < 3.85$, but not for smaller $L$. Of these functionals, only SCAN tends to a finite limit when $L$ tends to 0. Since the non-interacting kinetic energy, treated exactly in Kohn-Sham theory, dominates in this limit within a deformable jellium model, all of the general-purpose functionals can estimate the pressure required to achieve any thickness (with SCAN and LDA better than PBE). This pressure vanishes around $L/r_s^{\text{2D}} = 3.85$, where the 3D electron density is roughly that of the valence electrons in metallic potassium, and reaches about 20 GPa at $L/r_s^{\text{2D}} = 1.5$ and 400 GPa at $L/r_s^{\text{2D}} = 0.6$.},

doi = {10.1103/PhysRevB.98.085147},

journal = {Physical Review B},

number = 8,

volume = 98,

place = {United States},

year = {2018},

month = {8}

}

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