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 Cances 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

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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]

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Commensurate Approximation

Incommensurate system. Rotate blue lattice to makecommensurate.

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[S. Carr]

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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.

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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.

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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−γ|r |.

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Finite Matrix Approximation

Let Ω ⊂ Ω be finite.

The associated hamiltonian is H = (Hij)i ,j∈Ω.

sheet 1sheet 2origin

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Density of States (DoS)

For eigenvalues εss∈Ω, the density of states is

DoS(ε) =1

#Ω

∑s

δ(ε− εs).

DoS can be defined weakly, g analytic:

D[H](g) =1

#ΩTr[g(H)] =

∫g(ε)DoS(ε)dε.

The thermodynamic limit is weakly defined by

D[H](g) = limΩ↑ΩD[H](g).

DoS can be broken into site contributions:

D[H](g) =1

#Ω

∑k∈Ω

[g(H)]kk .

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Local Configuration Sampling

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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)

R1

R2

sheet 1sheet 2

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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.

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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|.

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The Approximation

For gε,η(ξ) = 1√2πη

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

Dη(ε) := ν

(∑α∈A1

∫Γ2

[gε,η(Hr ,2(b))]0α,0αdb+∑α∈A2

∫Γ1

[gε,η(Hr ,1(b))]0α,0αdb

).

For h smooth, we uniformly discretize the integrals.

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Approximation Rate

Resolvent bounds yield an exponential decay rate:

Theorem

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

|Dα[H](b, gε,η)− [gε,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 gε,η ∼ δ(ε− ·).

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Numerical Methods

Kernel Polynomial Method to approximate gε,η(·) ∼ δ(ε− ·).

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.

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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. Cances, 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.

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