Dislocation network structures in 2D bilayer system

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Dislocation network structures in 2D bilayer system Shuyang DAI School of Mathematics and Statistics Wuhan University Joint work with: Prof. Yang XIANG, HKUST Prof. David SROLOVITZ, UPENN S. Dai IMS Workshop, May 18 2018 1

Low-dimensional materials S. Dai IMS Workshop, May 18 2018 2 2D Monolayer materials: graphene, hexagonal boron nitride (hbn), transition-metal dichalcogenides (e.g., MoS 2 ),

Low-dimensional materials: Bilayers S. Dai IMS Workshop, May 18 2018 3 Monolayer materials: graphene, hbn, MoS2, van der Waals (vdw) interaction Bilayers: identical layers w/ or w/o a twist or different layers tunable electronic behavior Van der Waals Homo-/Hetero-structure [A. K. Geim et. al. Nature, (2013)]

Low-dimensional materials: Bilayers S. Dai IMS Workshop, May 18 2018 3 Monolayer materials: graphene, hbn, MoS2, van der Waals (vdw) interaction Bilayers: identical layers w/ or w/o a twist or different layers tunable electronic behavior Strong covalent bond Van der Waals Homo-/Hetero-structure Weak vdw [A. K. Geim et. al. Nature, (2013)]

Low-dimensional materials: Bilayers S. Dai IMS Workshop, May 18 2018 3 Monolayer materials: graphene, hbn, MoS2, van der Waals (vdw) interaction Bilayers: identical layers w/ or w/o a twist or different layers tunable electronic behavior Strong covalent bond Van der Waals Homo-/Hetero-structure Weak vdw Weak vdw interaction + Small bending rigidity Interlayer defect [A. K. Geim et. al. Nature, (2013)]

Low-dimensional materials: Bilayers S. Dai IMS Workshop, May 18 2018 3 Monolayer materials: graphene, hbn, MoS2, van der Waals (vdw) interaction Bilayers: identical layers w/ or w/o a twist or different layers tunable electronic behavior Strong covalent bond Weak vdw Weak vdw interaction + Small bending rigidity Interlayer defect [A. Kushima et al., Nano Lett., (2015)]

S. Dai IMS Workshop, May 18 2018 4 Defect types I Interlayer defects: Shift top layer w.r.t. bottom layer AB stacking Interlayer dislocation AC stacking [J. Alden, et. al., PNAS, (2013)] Interlayer dislocation: line defects that separate regions with different shifts

S. Dai IMS Workshop, May 18 2018 5 Defect types I Interlayer defects: Shift top layer w.r.t. bottom layer Dislocation:

S. Dai IMS Workshop, May 18 2018 5 Defect types I Interlayer defects: Shift top layer w.r.t. bottom layer Dislocation:

S. Dai IMS Workshop, May 18 2018 5 Defect types I Interlayer defects: Shift top layer w.r.t. bottom layer Dislocation:

S. Dai IMS Workshop, May 18 2018 5 Defect types I Interlayer defects: Shift top layer w.r.t. bottom layer Dislocation:

S. Dai IMS Workshop, May 18 2018 5 Defect types I Interlayer defects: Shift top layer w.r.t. bottom layer Dislocation:

S. Dai IMS Workshop, May 18 2018 5 Defect types I Interlayer defects: Shift top layer w.r.t. bottom layer Dislocation:

S. Dai IMS Workshop, May 18 2018 5 Defect types I Interlayer defects: Shift top layer w.r.t. bottom layer Dislocation:

S. Dai IMS Workshop, May 18 2018 5 Defect types I Interlayer defects: Shift top layer w.r.t. bottom layer Dislocation: Burgers vector b: characterize the dislocation

S. Dai IMS Workshop, May 18 2018 5 Defect types I Interlayer defects: Shift top layer w.r.t. bottom layer Dislocation: Burgers vector b: characterize the dislocation Dislocation energy b 2

S. Dai IMS Workshop, May 18 2018 5 Defect types I Interlayer defects: Shift top layer w.r.t. bottom layer Dislocation: Burgers vector b: characterize the dislocation Dislocation energy b 2 Angle between Ԧξ and b determines the type of dislocation: edge (90 o ), screw (0 o ), or mixed (0 o ~90 o )

S. Dai IMS Workshop, May 18 2018 5 Defect types I Interlayer defects: Shift top layer w.r.t. bottom layer Dislocation: Dislocations are main carriers of plastic deformation Glide motion Dislocation moves in response to a shear stress applied in a direction perpendicular to its line

S. Dai IMS Workshop, May 18 2018 5 Defect types I Interlayer defects: Shift top layer w.r.t. bottom layer Dislocation: Dislocations are main carriers of plastic deformation Dislocation in 2D structure: [B. Butz, et. al., Nature, (2014)]

Defect types II S. Dai IMS Workshop, May 18 2018 6 Interlayer defects: Moiré pattern rotate top layer w.r.t. bottom layer (twist) x 2 x 1 [J. Alden, et. al., PNAS, (2013)]

Defect types II S. Dai IMS Workshop, May 18 2018 6 Interlayer defects: Moiré pattern rotate top layer w.r.t. bottom layer (twist) AA AC AB AC AB AA x 2 AA AC AB x 1 [J. Alden, et. al., PNAS, (2013)] AA stacking

Defect types II S. Dai IMS Workshop, May 18 2018 6 Interlayer defects: Moiré pattern rotate top layer w.r.t. bottom layer (twist) AA AC AB AC AB AA x 2 AA AC AB x 1 [J. Alden, et. al., PNAS, (2013)] AA stacking Twisted bilayer: Interlayer dislocation network

S. Dai IMS Workshop, May 18 2018 7 Why multiscale Bilayer structure: local relaxation + bending Partial dissociation: Stacking fault [B. Butz, et. al., Nature, (2014)] Intralayer: strong chemical bond elastic sheet Interlayer: weak vdw interaction atomistic information Multiscale

General multiscale model S. Dai IMS Workshop, May 18 2018 8

General multiscale model S. Dai IMS Workshop, May 18 2018 8 Continuum Nonlinear interaction Γ Continuum

General multiscale model S. Dai IMS Workshop, May 18 2018 8 Continuum Nonlinear interaction Γ Continuum Total energy: E elastic : Continuum level E misfit : First-principle Equilibrium dislocation distribution:

Elastic contribution (Intralayer) S. Dai IMS Workshop, May 18 2018 9 Elastic deformation In-plane + Out-of-plane T. Zhang et. al., JMPS, 2014

Misfit contribution (Interlayer) S. Dai IMS Workshop, May 18 2018 10 AB G/G Energy Displacement

Misfit contribution (Interlayer) S. Dai IMS Workshop, May 18 2018 10 G/G AA Energy Displacement

Misfit contribution (Interlayer) S. Dai IMS Workshop, May 18 2018 10 G/G BA Energy Displacement

Misfit contribution (Interlayer) S. Dai IMS Workshop, May 18 2018 10 G/G SP Energy Displacement

Misfit contribution (Interlayer) S. Dai IMS Workshop, May 18 2018 10 G/G AB Energy Displacement

Misfit contribution (Interlayer) S. Dai IMS Workshop, May 18 2018 10 a b b SP a AA AB SP

S. Dai IMS Workshop, May 18 2018 10 Misfit contribution (Interlayer) 2D Generalized Stacking Fault Energy (GSFE) a b SP b a AB AA BA SP AB

S. Dai IMS Workshop, May 18 2018 11 General multiscale model Applied to twist Grain Boundaries φ and ψ for Al [Dai et. al., Acta Mat., 2013] Profiles along y = L y /4

S. Dai IMS Workshop, May 18 2018 11 General multiscale model Applied to twist Grain Boundaries φ and ψ for Cu [Dai et. al., Acta Mat., 2013] Profiles along y = L y /4

S. Dai IMS Workshop, May 18 2018 11 General multiscale model Applied to twist Grain Boundaries Hexagonal network Perfect screw Ideal-crystal [Dai et. al., Acta Mat., 2013] Stacking fault region Triangle network Partials

S. Dai IMS Workshop, May 18 2018 11 General multiscale model Applied to twist Grain Boundaries [Dai et. al., Acta Mat., 2013] Green dots: atoms within dislocation cores obtained in molecular simulation. Solid curves: dislocation network obtained in our model.

S. Dai IMS Workshop, May 18 2018 12 Misfit contribution (Interlayer) Interlayer separation effect

S. Dai Energy IMS Workshop, May 18 2018 12 Misfit contribution (Interlayer) Interlayer separation effect 0 Displacement

S. Dai IMS Workshop, May 18 2018 12 Misfit contribution (Interlayer) Interlayer separation effect 0Energy Displacement

S. Dai IMS Workshop, May 18 2018 12 Misfit contribution (Interlayer) Interlayer separation effect 0Energy Displacement

S. Dai IMS Workshop, May 18 2018 12 Misfit contribution (Interlayer) Interlayer separation effect 0Energy g coh Displacement

S. Dai IMS Workshop, May 18 2018 13 Misfit contribution (Interlayer) 3D Generalized Stacking Fault Energy (3D GSFE) δ is the interlayer spacing from Morse-potential convergence to long-range vdw interaction Γ should satisfies: Equivalent to the 2D GSFE additional condition related to elastic modulus Expressions for α, A, and B

S. Dai IMS Workshop, May 18 2018 14 Misfit contribution (Interlayer) 3D Generalized Stacking Fault Energy (3D GSFE) Interlayer shift Interlayer separation Calculation Fitting [S. Zhou et. al., PRB, 2015]

General multiscale model S. Dai IMS Workshop, May 18 2018 15 Generalized Peierls-Nabarro model + 3D GSFE Disregistry across slip plane: Continuum Nonlinear interaction Γ Continuum Total energy: E elastic : Continuum level E misfit : First-principle Equilibrium dislocation distribution: Γ = Γ(u 1, u 2, d)

S. Dai IMS Workshop, May 18 2018 16 General multiscale model Variables Displacements Upper (+): (u 1+ (x 1,x 2 ), u 2+ (x 1,x 2 ), f + (x 1,x 2 )) Lower ( ): (u 1 (x 1,x 2 ), u 2 (x 1,x 2 ), f (x 1,x 2 )) Flat state to Buckled state

S. Dai IMS Workshop, May 18 2018 16 General multiscale model Variables Displacements Upper (+): (u 1+ (x 1,x 2 ), u 2+ (x 1,x 2 ), f + (x 1,x 2 )) Lower ( ): (u 1 (x 1,x 2 ), u 2 (x 1,x 2 ), f (x 1,x 2 )) Total energy functional Flat state to Buckled state Like in the Peierls-Nabarro model for dislocations, we write the total energy as the sum of the elastic energy (due to the elastic deformation) and misfit energy (due to the interactions between layers) Elastic energy Misfit energy

General multiscale model Variables Displacements Upper (+): (u 1+ (x 1,x 2 ), u 2+ (x 1,x 2 ), f + (x 1,x 2 )) Lower ( ): (u 1 (x 1,x 2 ), u 2 (x 1,x 2 ), f (x 1,x 2 )) Total energy functional Flat state to Buckled state Like in the Peierls-Nabarro model for dislocations, we write the total energy as the sum of the elastic energy (due to the elastic deformation) and misfit energy (due to the interactions between layers) Elastic energy Misfit energy Relative displacements S. Dai IMS Workshop, May 18 2018 16

S. Dai IMS Workshop, May 18 2018 17 General multiscale model Relative Displacement 1. Buckling change the local coordinate system 2. The deformation of the neutral plane is approximated by the average of the displacements of the upper layer and the lower layer f u f(x) = ax u(x) = 0 Relative displacements between layers: Relative displacement due to buckling

S. Dai IMS Workshop, May 18 2018 18 General multiscale model Equilibrium structure: The equilibrium equations: Numerically, the minimum energy state can be found by iterating until the energy does not change (to numerical precision)

S. Dai IMS Workshop, May 18 2018 19 General multiscale model Inputs: Applied to bilayer graphene (BLG) 1. Elastic constants and bending rigidity [S. Chen et. al., PRB, (2011)] Elastic modulus: C 11 = 312.45J/m 2 ; C 12 = 91.65J/m 2 ; C 44 = 110.4J/m 2. Bending rigidity: κ = 22.08 10-20 J 2. 3D Generalized Stacking-Fault Energy (3D GSFE): calculated based on ACFDT-RPA [S. Zhou et. al., PRB, (2015)]

S. Dai IMS Workshop, May 18 2018 20 Interlayer dislocation in Bilayer Graphene (BLG) Edge dislocation: Generate an edge dislocation (only show upper layer) Define pre-existed strains:

S. Dai IMS Workshop, May 18 2018 21 Interlayer dislocation in Bilayer Graphene (BLG) Edge dislocation: θ b = b A + b B b b A b B θ=90 o 60 o 60 o

S. Dai IMS Workshop, May 18 2018 21 Interlayer dislocation in Bilayer Graphene (BLG) Edge dislocation: - breaks into 2 partial dislocation - agrees with MD results (dashed curves) θ b = b A + b B b b A b B θ=90 o 60 o 60 o Out-of-plane displacements (Bilayer shape) Relative displacements

S. Dai IMS Workshop, May 18 2018 21 Interlayer dislocation in Bilayer Graphene (BLG) Edge dislocation: - breaks into 2 partial dislocation - agrees with MD results (dashed curves) θ b = b A + b B b b A b B θ=90 o 60 o 60 o Out-of-plane displacements (Bilayer shape) Relative displacements b A b B

S. Dai IMS Workshop, May 18 2018 21 Interlayer dislocation in BLG General dislocations: Structural features θ θ 60 o 90 o 30 o 30 o 60 o 0 o 0 o 30 o 30 o

S. Dai IMS Workshop, May 18 2018 22 Interlayer dislocation in Bilayer Graphene (BLG) Core properties of partial dislocation: Type Edge 90 o Mixed 60 o Mixed 30 o Screw 0 o Width (nm) 1.5 (7.2) 2.4 (6.3) 3.7 (5.3) 4.5 (4.5) Energy ( 10-10 J) 0.318 0.508 0.905 1.081 Red: Core width in flat configuration (f + = f = 0) Buckled bilayer: Core width and energy as edge component Flat bilayer: Core width as edge component

S. Dai IMS Workshop, May 18 2018 22 Interlayer dislocation in Bilayer Graphene (BLG) Core properties of partial dislocation: Type Edge 90 o Mixed 60 o Mixed 30 o Screw 0 o Width (nm) 1.5 (7.2) 2.4 (6.3) 3.7 (5.3) 4.5 (4.5) Energy ( 10-10 J) 0.318 0.508 0.905 1.081 Red: Core width in flat configuration (f + = f = 0) Buckled bilayer: Core width and energy as edge component Flat bilayer: Core width as edge component Edge partial (90 o ) Screw partial (0 o )

S. Dai IMS Workshop, May 18 2018 22 Interlayer dislocation in Bilayer Graphene (BLG) Core properties of partial dislocation: Type Edge 90 o Mixed 60 o Mixed 30 o Screw 0 o Width (nm) 1.5 (7.2) 2.4 (6.3) 3.7 (5.3) 4.5 (4.5) Energy ( 10-10 J) 0.318 0.508 0.905 1.081 Red: Core width in flat configuration (f + = f = 0) Buckled bilayer: Core width and energy as edge component Flat bilayer: Core width as edge component Edge partial (90 o ) Screw partial (0 o )

Interlayer twist in BLG S. Dai IMS Workshop, May 18 2018 23 Counterclockwise twist AC AA AA AB AC AC AA AB AB Substructures in twisted BLG: AB/AC stacking Partial dislocation AA stacking (intersection of dislocations)

Interlayer twist in BLG S. Dai IMS Workshop, May 18 2018 23 Counterclockwise twist AC AA AA AB AC AC AA AB AB Substructures in twisted BLG: AB/AC stacking Partial dislocation AA stacking (intersection of dislocations)

Interlayer twist in BLG S. Dai IMS Workshop, May 18 2018 23 Counterclockwise twist AC AA AA AB AC AC AA AB AB Substructures in twisted BLG: AB/AC stacking Partial dislocation AA stacking (intersection of dislocations)

Interlayer twist in BLG S. Dai IMS Workshop, May 18 2018 23 Counterclockwise twist AC AA AA AB AC AC AA AB AB Substructures in twisted BLG: AB/AC stacking Partial dislocation AA stacking (intersection of dislocations) Question: How AA stacking relax?

Interlayer twist in BLG S. Dai IMS Workshop, May 18 2018 23 Counterclockwise twist AC AA AA AB AC AC AA AB AB Consider a toy configuration: Substructures in twisted BLG: AB/AC stacking Partial dislocation AA stacking (intersection of dislocations) Question: How AA stacking relax? Fixed to AA state

Interlayer twist in BLG S. Dai IMS Workshop, May 18 2018 23 Counterclockwise twist AC AA AA AB AC AC AA AB AB Consider a toy configuration: Substructures in twisted BLG: AB/AC stacking Partial dislocation AA stacking (intersection of dislocations) Question: How AA stacking relax? Fixed to AA state

Interlayer twist in BLG S. Dai IMS Workshop, May 18 2018 23 Counterclockwise twist AC AA AA AB AC AC AA AB AB Consider a toy configuration: Substructures in twisted BLG: AB/AC stacking Partial dislocation AA stacking (intersection of dislocations) Question: How AA stacking relax? Fixed to AA state Not possible for flat bilayer C 11 (312.45J/m 2 ) > C 44 (110.4J/m 2 ) But possible in the deformed bilayer

S. Dai IMS Workshop, May 18 2018 24 Interlayer twist in BLG Initial guess: bilayer is flat Case A Initial guess: bilayer is buckled a little at AA stacking Case B Case A Case B

S. Dai IMS Workshop, May 18 2018 24 Interlayer twist in BLG Initial guess: bilayer is flat Case A Initial guess: bilayer is buckled a little at AA stacking Case B Case A Case B

S. Dai IMS Workshop, May 18 2018 24 Interlayer twist in BLG Initial guess: bilayer is flat Case A Initial guess: bilayer is buckled a little at AA stacking Case B Case A 0.1 0 (Å) 0-0.1 Case B 2 1 0 (Å) 0-1 -2

Interlayer twist in BLG Initial guess: bilayer is flat Case A Initial guess: bilayer is buckled a little at AA stacking Case B Case A 0.1 0 0-0.1 (Å) Case B 2 1 0 (Å) Breathing mode S. Dai 0-1 -2 IMS Workshop, May 18 2018 Bending mode 24

S. Dai IMS Workshop, May 18 2018 25 Interlayer twist in BLG Breathing mode: At AA stacking, two layers buckles to different direction Bilayer shape Relative displacement f + f 0.2Å 0.25Å 0.1 0.2 0-0.1 0.15 0.1 0.05 5 0-5 -10 0 10-0.2 5 0-5 -10 0 10 0

x 2 (nm) Interlayer twist in BLG Breathing mode: At AA stacking, two layers buckles to different direction 3 sets of partial dislocations. Dislocations intersect at AA stacking with maximum buckling height. Bilayer shape 0.2Å 0.1 Relative displacement f + f 0.25Å 0.2 5 0-5 Dislocation network structure Upper 0-0.1 0.15 0.1 0.05 5 0-5 Lower 5 0-5 -10 0 10-0.2 5 0-5 -10 0 10 0-10 -5 0 5 x 1 (nm) 10 S. Dai IMS Workshop, May 18 2018 25

S. Dai IMS Workshop, May 18 2018 26 Interlayer twist in BLG Breathing mode: compare with MD (for free standing tblg) Upper layer out-of-plane displacement MD: [van Wijk et al. 2D Materials, 2015] θ = 1.2 o Intralayer: REBO Interlayer: registry dependent potential

S. Dai IMS Workshop, May 18 2018 26 Interlayer twist in BLG Breathing mode: compare with MD (for free standing tblg) Upper layer out-of-plane displacement MD: [van Wijk et al. 2D Materials, 2015] θ = 1.2 o Intralayer: REBO Interlayer: registry dependent potential

x 2 (nm) Interlayer twist in BLG Breathing mode: compare with MD (for free standing tblg) MD: [van Wijk et al. 2D Materials, 2015] θ = 1.2 o Intralayer: REBO Interlayer: registry dependent potential S. Dai Upper layer out-of-plane displacement Similar micro-structure Qualitative agreement IMS Workshop, May 18 2018 5 0-5 x 1 (nm) -10-5 0 5 θ 0.8 o 10 26 0.2Å 0.1 0-0.1-0.2

S. Dai IMS Workshop, May 18 2018 27 Interlayer twist in BLG Bending mode: Two layers buckles to same direction bulges Bilayer shape Relative displacement f + f 2Å 0.25Å 1 0.2 0-1 0.15 0.1 0.05 5 0-5 -10 0 10-2 5 0-5 -10 0 10 0

x 2 (nm) Interlayer twist in BLG Bending mode: Two layers buckles to same direction bulges 3 sets of partial dislocations. Dislocations intersect at AA stacking with maximum buckling height. Bilayer shape 2Å 1 Relative displacement f + f 0.25Å 0.2 5 0-5 Dislocation network structure Upper 0-1 0.15 0.1 0.05 5 0-5 Lower 5 0-5 -10 0 10-2 5 0-5 -10 0 10 0-10 -5 0 5 x 1 (nm) 10 S. Dai IMS Workshop, May 18 2018 27

x 2 (nm) Interlayer twist in BLG Bending mode: Two layers buckles to same direction bulges 3 sets of partial dislocations. Dislocations intersect at AA stacking with maximum buckling height. Dislocation twists: clockwise/counterclockwise for upper/lower bulge Bilayer shape 2Å 1 Relative displacement f + f 0.25Å 0.2 5 0-5 Dislocation network structure Upper 5 0-5 -10 0 10 0-1 -2 5 0-5 -10 0 10 0.15 0.1 0.05 0 5 0-5 twisted dislocations -10-5 0 5 x 1 (nm) Lower 10 S. Dai IMS Workshop, May 18 2018 27

Interlayer twist in BLG S. Dai IMS Workshop, May 18 2018 28 Perfect Moiré: small well-stacked domain, straight dislocations Breathing mode: large well-stacked domain, straight dislocations Bending mode: large well-stacked domain, spiral dislocations

Interlayer twist in BLG S. Dai IMS Workshop, May 18 2018 28 Perfect Moiré: small well-stacked domain, straight dislocations Breathing mode: large well-stacked domain, straight dislocations Bending mode: large well-stacked domain, spiral dislocations Intersection node comparison

Interlayer twist in BLG S. Dai IMS Workshop, May 18 2018 28 Perfect Moiré: small well-stacked domain, straight dislocations Breathing mode: large well-stacked domain, straight dislocations Bending mode: large well-stacked domain, spiral dislocations Node dislocation structure weakly depends on θ Intersection node comparison Node radius 4.5nm

Energy of twisted BLG S. Dai IMS Workshop, May 18 2018 29 Energy approaches to a constant value (Perfect Moiré, no relaxation)

Energy of twisted BLG S. Dai IMS Workshop, May 18 2018 29 Energy approaches to a constant value (Perfect Moiré, no relaxation) Bending mode is stable for small θ Breathing mode is stable when θ gets larger. θ c 1.6 o a 0 /(2r), r is radius of node Two nodes start to overlap

Conclusions S. Dai IMS Workshop, May 18 2018 30 Multiscale model to describe the deformation of bilayer system Applied to bilayer graphene: accurate description of interlayer defects: Single Dislocation 1. Edge component leads to buckling 2. Buckling reduces the core size and the energy Interlayer Twist Breathing mode vs. Bending mode (θ dependence) General approach for defects in multilayer systems

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