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Electronic Density of States for Incommensurate Layers Daniel Massatt 1 , Mitchell Luskin 1 , Christoph Ortner 2 Collaborators: Efthimios Kaxiras 3 , Stephen Carr 3 , Shiang Fang 3 , Eric Canc` es 4 , Paul Cazeaux 1 1 Department of Mathematics University of Minnesota -Twin Cities 3 Department of Physics Harvard University 2 Department of Mathematics University of Warwick 4 Department of Mathematics Ecole des Ponts May 18, 2017 DM, ML, CO (UMN) Electronic DoS May 18, 2017 1 / 16

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Electronic Density of States for Incommensurate Layers

Daniel Massatt 1, Mitchell Luskin 1, Christoph Ortner 2

Collaborators: Efthimios Kaxiras 3, Stephen Carr 3, Shiang Fang 3,Eric Cancès 4, Paul Cazeaux 1

1Department of MathematicsUniversity of Minnesota -Twin Cities

3Department of PhysicsHarvard University

2Department of MathematicsUniversity of Warwick

4Department of MathematicsEcole des Ponts

May 18, 2017

DM, ML, CO (UMN) Electronic DoS May 18, 2017 1 / 16

Incommensurate 2D Heterostructures

2D materials can be stacked androtated.

This leads to incommensuratesystems.

Our new method canapproximate the Density ofStates for incommensuratestructures that cannot beapproximated by supercells(such as low angle rotations).

[Geim, 2013]

DM, ML, CO (UMN) Electronic DoS May 18, 2017 2 / 16

Commensurate Approximation

Incommensurate system. Rotate blue lattice to makecommensurate.

DM, ML, CO (UMN) Electronic DoS May 18, 2017 3 / 16

[S. Carr]

DM, ML, CO (UMN) Electronic DoS May 18, 2017 4 / 16

Main Results

Rigorously define the Density of States (DoS) for incommensuratesystems.

Derive an efficient algorithm to calculate the DoS and Local Densityof States (LDoS) for incommensurate systems.

Derive error bounds, which allows controlling parameter selection tooptimize the accuracy and efficiency of the approximation.

DM, ML, CO (UMN) Electronic DoS May 18, 2017 5 / 16

Incommensurate System

Lattices defined by 2× 2invertible matrices Aj :

Rj = {Ajn : n ∈ Z2}.

We assume R1 and R2 areincommensurate, or for v ∈ R2,v +R1 ∪R2 = R1 ∪R2

⇔ v =(

00

).

We will be interested when thereciprocal lattices areincommensurate,

R∗j = {2πA−Tj n : n ∈ Z2}.

sheet 2sheet 1

Incommensurate rotated hexagonal bilayer, θ = 6◦.

DM, ML, CO (UMN) Electronic DoS May 18, 2017 6 / 16

Tight-Binding Model

Aj are orbital index sets.Ω = (R1 ×A1) ∪ (R2 ×A2).HRα,R′α′ = hαα′(R − R ′).Orbital interactions hαα′ areuniformly continuous on R2.They decay exponentially(r ∈ R2):|hαα′(r)| ≤ Ce−γ̃|r |.

DM, ML, CO (UMN) Electronic DoS May 18, 2017 7 / 16

Finite Matrix Approximation

Let Ω̃ ⊂ Ω be finite.The associated hamiltonian is H̃ = (Hij)i ,j∈Ω̃.

sheet 1sheet 2origin

DM, ML, CO (UMN) Electronic DoS May 18, 2017 8 / 16

Density of States (DoS)

For eigenvalues {�s}s∈Ω̃, the density of states is

DoS(�) =1

#Ω̃

∑s

δ(�− �s).

DoS can be defined weakly, g analytic:

D[H̃](g) = 1#Ω̃

Tr[g(H̃)] =

∫g(�)DoS(�)d�.

The thermodynamic limit is weakly defined by

D[H](g) = limΩ̃↑ΩD[H̃](g).

DoS can be broken into site contributions:

D[H̃](g) = 1#Ω̃

∑k∈Ω̃

[g(H̃)]kk .

DM, ML, CO (UMN) Electronic DoS May 18, 2017 9 / 16

Local Configuration Sampling

DM, ML, CO (UMN) Electronic DoS May 18, 2017 10 / 16

Parameterizing Local Configuration

The local geometry of site R1 ∈ R1 isdefined by

R1 ∪R2 − R1 = R1 ∪ (R2 − R1)= R1 ∪ (R2 −mod2(R1)).

mod2(R1) ∈ Γ2 := {A2α : α ∈ [0, 1)2}.

mod2(R1)

R1R2

sheet 1sheet 2

DM, ML, CO (UMN) Electronic DoS May 18, 2017 11 / 16

Equidistribution of Local Configurations

Theorem

Consider R1 and R2 such that their reciprocal lattices areincommensurate. Then for g ∈ Cper(Γ2), we have

1

#R1 ∩ Br

∑`∈R1∩Br

g(`)→ 1|Γ2|

∫Γ2

g(b)db. (1)

We will calculate site contribution to Density of States.

Integrate over Γj ’s for all site contributions.

DM, ML, CO (UMN) Electronic DoS May 18, 2017 12 / 16

DoS Approximation

Hr ,1(b) is defined over R1 ∩ Br and R2 ∩ Br + b, b ∈ Γ1.LDoS for sheet 2 can be defined as

Dα[H](b, g) = limr→∞

[g(Hr ,1(b))]0α,0α, α ∈ A2.

Theorem

D[H](g) = ν(∑α∈A1

∫Γ2

Dα[H](b, g)db +∑α∈A2

∫Γ1

Dα[H](b, g)db)

where

ν =1

|A2| · |Γ1|+ |A1| · |Γ2|.

DM, ML, CO (UMN) Electronic DoS May 18, 2017 13 / 16

The Approximation

For g�,η(ξ) =1√2πη

e−(ξ−�)2/2η2 , we have the final approximation

Dη(�) := ν

(∑α∈A1

∫Γ2

[g�,η(Hr ,2(b))]0α,0αdb+∑α∈A2

∫Γ1

[g�,η(Hr ,1(b))]0α,0αdb

).

For h smooth, we uniformly discretize the integrals.

DM, ML, CO (UMN) Electronic DoS May 18, 2017 14 / 16

Approximation Rate

Resolvent bounds yield an exponential decay rate:

Theorem

For α ∈ A1 and hαα′ exponentially decaying, we have

|Dα[H](b, g�,η)− [g�,η(Hr ,1(b))]0α,0α| . η−7e−γηr .

Then the error is optimized for r ∼ η−1 log(η−1).η is chosen small so g�,η ∼ δ(�− ·).

DM, ML, CO (UMN) Electronic DoS May 18, 2017 15 / 16

Numerical Methods

Kernel Polynomial Method to approximate g�,η(·) ∼ δ(�− ·).With more cost, can calculate DoS for multi-layers.

E-2 -1.5 -1 -0.5 0 0.5 1 1.5

DoS

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8bilayermonolayertest function

VHS

graphene bilayer with 6◦ twist DoS and monolayer graphene DoS.

DM, ML, CO (UMN) Electronic DoS May 18, 2017 16 / 16

D. Massatt, M. Luskin, C. OrtnerElectronic Density of States for Incommensurate Layers.Multiscale Model. Simul., to appear.arXiv preprint arxiv:1608.01968, Aug. 2016.

E. Cancès, P. Cazeaux, M. LuskinGeneralized Kubo Formulas for the Transport Properties ofIncommensurate 2D Atomic Heterostructures.arXiv preprint arxiv:1611.08043, Nov. 2016.

S. Carr, D. Massatt, S. Fang, P. Cazeaux, M. Luskin, E. Kaxiras.Twistronics: Manipulating the electronic properties of two-dimensionallayered structures through their twist angle.Phys. Rev. B, 95:075420, Feb 2017.

DM, ML, CO (UMN) Electronic DoS May 18, 2017 17 / 16