Flexoelectricity and transport properties of phosphorene nanoribbons under mechanical bending

T. Pandey, L. Covaci, M. V. Milošević, and F. M. Peeters

Phys. Rev. B 103, 235406 – Published 2 June 2021

Abstract

We examine from first principles the flexoelectric properties of phosphorene nanoribbons under mechanical bending along armchair and zigzag directions. In both cases we find that the radial polarization depends linearly on the strain gradient. The flexoelectricity along the armchair direction is over 40% larger than along the zigzag direction. The obtained flexoelectric coefficients of phosphorene are four orders of magnitude larger than those of graphene and comparable to transition metal dichalcogenides. Analysis of charge density shows that the flexoelectricity mainly arises from the $p_{z}$ orbitals of phosphorus atoms. The electron mobilities in bent phosphorene can be enhanced by over 60% along the armchair direction, which is significantly higher than previous reports of mobility tuned by uniaxial strain. Our results indicate phosphorene is a candidate for a two-dimensional material applicable in flexible-electronic devices.

Received 12 March 2021
Revised 21 May 2021
Accepted 21 May 2021

DOI:https://doi.org/10.1103/PhysRevB.103.235406

Physics Subject Headings (PhySH)

Electric polarization Electronic structure Flexoelectricity Piezoelectricity

Nanoribbon

Density functional calculations First-principles calculations

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

T. Pandey ^*, L. Covaci, M. V. Milošević, and F. M. Peeters

Department of Physics and NANOlab Center of Excellence, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium

^*tribhuwan.pandey@uantwerpen.be

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Vol. 103, Iss. 23 — 15 June 2021

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Images

Figure 1
(a) Top view of a phosphorene nanoribbon, where the unit cell of monolayer phosphorene is highlighted in gray. Armchair (AC) and zigzag (ZZ) directions are also shown. (b) Side view of bent AC and ZZ phosphorene nanoribbons at a bending curvature of $0.009 Å^{- 1}$ . The APNR is bent along the ZZ direction and vice versa. The bending direction is shown by the blue dashed line. (c) Induced out-of-plane polarization $P_{z}$ vs strain gradient for ZPNR (width = 92.50 Å) and APNR (width = 82.45 Å). Symbols are the calculated values, and lines show the linear fit. The transverse flexoelectric coefficient is calculated from the slope of the linear fit. (d) Transverse flexoelectric coefficient $F_{T}$ as a function of ribbon width.
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Figure 2
Electron-density response to the bending at strain gradient $κ = 0.01 Å^{- 1}$ along the $z$ direction for ZPNR, shown with respect to a flat ( $κ = 0.0 Å^{- 1}$ ) nanoribbon. (b) Contours of the electron density difference between the flat and bent ( $κ = 0.01 Å^{- 1}$ ) ZPNR. The contours are plotted in the $y z$ plane (about 10 Å from the middle of the nanoribbons. (c) The Bader charge transfer due to bending with respect to flat ZPNR on a section of nanoribbons. The values are marked near each atom, and the color indicates the extent of charge transfer. (d)–(f) same as (a)–(c), but for APNR. For APNR the electron density is projected on the $x z$ plane. A relatively larger charge redistribution (hence polarization) is seen for ZPNR bent along the AC direction.
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Figure 3
Bending effects on the electronic structure of (a) armchair and (b) zigzag phosphorene nanoribbons for the $κ$ = 0 (flat) and $κ = 0.0045 Å^{- 1}$ cases. The orbital projected eigenvalues are shown in different colors. (c) and (d) The top and side views of the APNR wave function corresponding to valence band maxima and conduction band minima at $κ = 0$ (flat) and $0.0045 Å^{- 1}$ , respectively. (e) and (f) same as (c) and (d), but for ZPNR. The wave functions are presented at an isosurface value of 0.001 eV/Å. These calculations are at widths of 39.57 Å ( $1 \times 12 \times 1$ ) and 46.25 Å ( $10 \times 1 \times 1$ ) for APNR and ZPNR, respectively.
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Figure 4
Change in the room temperature carrier mobility as a function of strain gradient for (a) armchair phosphorene nanoribbons and (b) zigzag phosphorene nanoribbons. Open and solid symbols give the hole and electron mobilities, respectively. These calculations are for widths of 40 Å ( $1 \times 12 \times 1$ ) and 43 Å ( $10 \times 1 \times 1$ ) for APNR and ZPNR, respectively.
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