Electronic and optical properties of phosphorene quantum dots under electric and magnetic fields
Longlong Li Department of Physics (CMT Group), University of Antwerp, Belgium Institute of Solid State Physics, Chinese Academy of Sciences, China Hefei
2 3 4 Contents
Introduction
Model and Theory
Main Results
Conclusions
5 Overview of Family of 2D Materials
Up to now, more than 140 2D materials are explored. P. Miro, M. Audiffred and T. Heine, Chem. Soc. Rev. 43, 6537 (2014) The number is still growing and new 2D materials are expected to arise. The most known are graphene, h-BN, TMD, silicene, germanene, etc. These 2D materials have their own unique properties and hold prospects for practical applications. Recently, a new 2D material, named phosphorene, has drawn a lot of attention.
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ABC of Phosphorene: Bulk and Nano
7 Allotropes of Phosphorus
8 From Black Phosphorus to Phosphorene
Research on BP dates back to 1914: P. M. Bridgman, JACS 36, 1344 (1914)
100 years
Phosphorene research starts in 2014: Liu et al., ACS Nano 8, 4033 (2014)
2D Materials Family Superstars
Graphene: 2004
Phosphorene: 2014
9 Lattice Structure of Phosphorene
a single layer of BP puckered honeycomb lattice covalent bonds of P atoms 3 𝑠𝑠 four𝑝𝑝 atoms in a unit cell
(a) 3D view (b) Top view (c) Side view Red: P atoms in lower layer Blue: P atoms in upper layer , : unit cell lengths
𝑎𝑎 𝑏𝑏 10 Band Structure (From DFT Calculations)
Direct band gap of 2 eV at point
Band anisotropy along∼ andΓ axes
Anisotropic effective masses:ΓX ΓY
= 0.17 ; = 1.12 𝑔𝑔 𝐸𝐸 ΓX ΓY 𝑚𝑚𝑒𝑒 = 0.15𝑚𝑚 0 ;𝑚𝑚 𝑒𝑒 = 6.35𝑚𝑚 0 ΓX ΓY 𝑚𝑚ℎ 𝑚𝑚0 𝑚𝑚ℎ 𝑚𝑚0 2D Mater. 1, 025001 (2014) Nat. Commun. 5, 4475 (2014)
11 Comparison with other 2D Materials (Selected)
Material Structure Bandgap Mobility Graphene Flat 0 > 10 cm2/V/s [1] Mono TMD Trilayer 1.3 1.9 eV [2] 2005 cm2/V/s [2] 1.5 10 2 Phosphorene Puckered ∼ eV [3] ∼ cm /V/s [4] 3 ∼ ∼ In addition, phosphorene is an anisotropic 2D material, showing optical and transport anisotropy [3], which is not typical for most 2D materials.
[1] A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183 (2007) [2] Q. H. Wang et al., Nat. Nanotech. 7, 699 (2012) [3] J. Qiao et al., Nat. Commun. 5, 4475 (2014) [4] L. Li et al., Nat. Nanotech. 9, 372 (2014)
12 Investigation of Phosphorene: Current Status
For Bulk Strain-engineered band structure: PRL 112, 176801 (2014) Superior mechanical flexibility: APL 104, 251915 (2014) Strong excitonic effect: Nat.Nano 10, 517 (2015) Nonlinear optical response: Opt. Express 23, 11183 (2015) Magneto-optical transport: PRB 92, 045420 (2015) Integer quantum Hall effect: Nat. Nano 11, 593 (2016) Anisotropic spin-orbit coupling: PRB 94, 155423 (2016) For Nanoribbons Effects of edge, size, strain, and external fields: . EPL 108, 47005 (2014); JAP 116, 144301 (2014) . PRB 89, 245407 (2014); NJP 16, 115004 (2014) . PRB 92, 035436 (2015); PRB 91, 085409 (2015)
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Quantum Dots in 2D BP: Experimental Fabrication
D: Lateral size H: Thickness D 4.9 ± 1.6 nm H 4 ± 2 layers ∼ ∼
X. Zhang et al., Angew. Chem. Int. Ed. 54, 3653 (2015). Z. Sun et al., Angew. Chem. 127, 11688 (2015).
14 Theoretical Aspects on 2D BP Quantum Dots
Mid-gap edge states (TB) Anomalous size dependence of optical emission gap (DFT)
2D Mater. 2, 045012 (2015). J. Phys. Chem. Lett. 7, 370 (2016).
15 Our Research Subject & Motivation
Research Subject: Investigating electronic and optical properties of phosphorene quantum dots in the presence of electric and magnetic fields. Research Motivation: Why phosphorene: Finite direct band gap Anisotropic material Why quantum dots: Strong quantum confinement Significant edge effects Why electric and magnetic fields: Efficient tuning of the properties
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Model System & Theoretical Approch
17 Model System
Rectangular phosphorene quantum dot Armchair ( -direction) and zigzag ( -direction) edges In-plane electric𝑥𝑥 field and perpendicular𝑦𝑦 magnetic field
: Side length along armchair direction Why rectangular shape: : Side length along zigzag direction 𝐿𝐿 Easy to realize by experiment : Perpendicular magnetic field 𝑊𝑊 Why AC and ZZ edges: : Electric field along armchair direction 𝐵𝐵 Proven to be chemically stable : Electric field along zigzag direction 𝐹𝐹𝑥𝑥
𝐹𝐹𝑦𝑦 18 Tight-binding Hamiltonian
(1) Without electric and magnetic fields: = + TB GW + , + 𝐻𝐻 � 𝜀𝜀𝑖𝑖𝑐𝑐𝑖𝑖 𝑐𝑐𝑖𝑖 � 𝑡𝑡𝑖𝑖𝑖𝑖𝑐𝑐𝑖𝑖 𝑐𝑐𝑗𝑗 , : 𝑖𝑖creation and𝑖𝑖 𝑗𝑗destruction operators +: on-site energy at site 𝑐𝑐𝑖𝑖 𝑐𝑐𝑗𝑗 : hopping energy between sites and 𝜀𝜀𝑖𝑖 𝑖𝑖 (2) In the presence of electric and magnetic fields: PRB 89, 201408: 𝑡𝑡𝑖𝑖𝑖𝑖 𝑖𝑖 𝑗𝑗 , = , , = ( , ) = 1.220 eV = 3.665 eV 𝑡𝑡1 − 𝜀𝜀𝑖𝑖 → 𝜀𝜀𝑖𝑖 −exp𝑒𝑒𝑭𝑭 ⋅i𝒓𝒓𝑖𝑖 𝑭𝑭 𝐹𝐹𝑥𝑥 𝐹𝐹𝑦𝑦 𝒓𝒓𝑖𝑖 𝑥𝑥𝑖𝑖 𝑦𝑦𝑖𝑖 𝒓𝒓𝑗𝑗 2 = 0.205 eV 2𝜋𝜋𝜋𝜋 𝑡𝑡 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 3 = 0.105 eV 𝑡𝑡 : magnetic→ 𝑡𝑡 vector potential�𝒓𝒓𝑖𝑖 𝑨𝑨 ⋅ 𝑑𝑑𝒍𝒍 𝑡𝑡 − ℎ 4 = 0.055 eV In the Landau gauge, = (0, , 0) 𝑡𝑡 − 𝑨𝑨 5 𝑡𝑡 − 19 𝑨𝑨 𝐵𝐵𝐵𝐵
Electronic States, DOS and Optical Absorption
Electronic States: = Pybinding package*
Density𝐻𝐻퐻 𝐸𝐸 of� States:⟹ = ( ) Pybinding package*
Optical𝐷𝐷 𝐸𝐸 Absorption:∑𝑛𝑛 𝛿𝛿 𝐸𝐸 − 𝐸𝐸 𝑛𝑛 ⟹ = Your own codes
𝐴𝐴 ℏ𝜔𝜔 =∑𝑖𝑖𝑖𝑖 𝐴𝐴𝑖𝑖𝑖𝑖 ℏ𝜔𝜔 ⟹ ( + ) 2 𝐴𝐴𝑖𝑖𝑖𝑖 ℏ𝜔𝜔= 𝐸𝐸 𝑗𝑗 − 𝐸𝐸𝑖𝑖 𝝐𝝐 ⋅ 𝑴𝑴𝑖𝑖𝑖𝑖 𝛿𝛿 𝐸𝐸𝑖𝑖 − 𝐸𝐸𝑗𝑗 ℏ𝜔𝜔 * D. Moldovan and F. M. Peeters, Pybinding v0.8.x: a python package 𝑴𝑴𝑖𝑖𝑖𝑖 ⟨𝑗𝑗 𝒓𝒓 𝑖𝑖⟩ for tight-binding calculations (doi: 10.5281/zenodo.56818).
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Main Results: Electronic and Optical Spectra
21 Well-defined Edge States
= 5.3 nm = 4.1 nm 𝐿𝐿 = = 0 𝑊𝑊 𝐹𝐹 𝐵𝐵 (a) Energy levels 𝐸𝐸𝑔𝑔 (b) DOS (c) Wave functions
22 Energy Spectrum: Electric- and Magnetic-field Dependence
= : flux : armchair dir
Φ 𝐵𝐵𝐵𝐵𝐵𝐵 𝐹𝐹
23 Optical Absorption: Electric- and Magnetic-field Effects
bb: bulk-to-bulk ee: edge-to-edge eb: edge-to-bulk
24 Robust Edge Absorption with Magnetic Field
Edge-to-edge Absorption
25 Polarization Sensitive Absorption
= 0 = 0.03 V/nm
𝐹𝐹 𝐹𝐹
26 Conclusions
The edge states are found within the bulk band gap of phosphorene quantum dot
The electric and magnetic fields have very different influences on the bulk and edge states
There are two (three) types of optical transitions observed in the presence of a magnetic (an electric) field.
The edge-to-edge absorption is robust with magnetic field
The bulk-to-bulk, edge-to-bulk, and edge-to-edge absorptions are all polarization sensitive.
27 Publications @ CMT Group, UA [1] L. L. Li, D. Moldovan, W. Xu, and F. M. Peeters, Electronic and optical properties of phosphorene quantum dots, Nanotechnology 28, 085702 (2016). [2] L. L. Li, M. Zarenia, W. Xu, H. M. Dong, and F. M. Peeters, Exciton states in a circular graphene quantum dot, Phys. Rev. B 95, 045409 (2017). [3] L. L. Li, D. Moldovan, P. Vasilopoulos, and F. M. Peeters, Aharonov-Bohm oscillations in phosphorene quantum rings, Accepted by Phys. Rev. B.
Future Plans and Considerations: Materials Systems: Phosphorene, Graphene or Other 2D Materials Physical Properties: Electronic, Optical and Transport Properties
28 Thank you for your attention
Thank you for your attention