Twistronics: Manipulating the Electronic Properties of Two-Dimensional Layered Structures Through Their Twist Angle

Twistronics: Manipulating the electronic properties of two-dimensional layered structures through their twist angle The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Carr, Stephen, Daniel Massatt, Shiang Fang, Paul Cazeaux, Mitchell Luskin, and Efthimios Kaxiras. 2017. “Twistronics: Manipulating the Electronic Properties of Two-Dimensional Layered Structures through Their Twist Angle.” Physical Review B 95 (7). https:// doi.org/10.1103/physrevb.95.075420. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:41384102 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#OAP Twistronics: Manipulating the Electronic Properties of Two-dimensional Layered Structures through their Twist Angle Stephen Carr,1 Daniel Massatt,2 Shiang Fang,1 Paul Cazeaux,2 Mitchell Luskin,2 and Efthimios Kaxiras1, 3 1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA. 2School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA. 3John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA. (Dated: November 4, 2016) The ability in experiments to control the relative twist angle between successive layers in two- dimensional (2D) materials offers a new approach to manipulating their electronic properties; we refer to this approach as \twistronics". A major challenge to theory is that, for arbitrary twist angles, the resulting structure involves incommensurate (aperiodic) 2D lattices. Here, we present a general method for the calculation of the electronic density of states of aperiodic 2D layered materials, using parameter-free hamiltonians derived from ab initio density-functional theory. We use graphene, a semimetal, and MoS2, a representative of the transition metal dichalcogenide (TMDC) family of 2D semiconductors, to illustrate the application of our method, which enables fast and efficient simulation of multi-layered stacks in the presence of local disorder and external fields. We comment on the interesting features of their Density of States (DoS) as a function of twist-angle and local configuration and on how these features can be experimentally observed. I. INTRODUCTION leaves important questions unaddressed: Are there distinct physical characteristics that distinguish the incommensurate from the commensurate case? Do the proper- A few short years after the experimental demonstration ties of commensurate systems approach the proper limit 1 of the existence of monolayer graphene , many other 2D of the incommensurate systems as the twist-angle is var- 2{6 materials, have been successfully fabricated . Although ied? single-layer 2D systems have intriguing physical prop- In the present work we introduce a robust framework erties, there has also been great interest in developing for the calculation of the properties of truly incommen- and understanding artificial heterostructures composed surate 2D heterostructures that can address such ques- of multiple atomic layers weakly bonded by van der Waals tions for situations involving arbitrary twists between 7 forces . Mechanical or chemical exfoliation and position- successive layers. Our method is inspired by previous ing of one layer on top of another allows for a relative mathematical works on disordered tight-binding models, twist between successive layers, which can destroy the which can be classified into two distinct concepts. First, alignment and thereby break the translational symme- an algebraic treatment of electronic transport in disor- 8,9 try in the combined system . The resulting structures dered systems14,15 that allows for a rigorous definition of may have commensurate stacking for special orientations, quantum-mechanical operators in a disordered material. but more generally are incommensurate. This allows Second, the fact that local tight-binding models create for interesting new behavior: studies of bilayer graphene exponentially localized observables, that is, they make it have found clear twist-dependent features in both the possible to controllably remove finite-size and edge effects 10,11 electronic density of states and the conductivity ; at from calculations16. We have already provided a rigorous very small twist-angles, a domain-wall phase appears, re- mathematical discussion of this method17, but here inves- 12 lated to the stacking configuration . Similar effects may tigate its implications and results for physical systems. occur in TMDC semiconductors, with their band-gaps Our modeling is based on effective tight-binding hamilto- affected by the substrate and the relative twist-angle nians without any adjustable parameters, obtained from 13 orientation . Incommensurate structures pose a great first-principles DFT results18,19. As a demonstration of challenge to theoretical studies since the standard de- the capabilities of the method, we study some proto- scription of solids with crystalline order, a periodic Bra- typical systems of 2D stacked layers, including bilayer arXiv:1611.00649v2 [cond-mat.mes-hall] 3 Nov 2016 vais lattice and the associated Bloch states of electrons, graphene, a semimetal, and bilayer MoS2, a representa- is entirely absent in the combined system although each tive semiconductor of the TMDC family. layer may still be a perfect 2D crystal. In the effort to capture the physics of incommensurate systems, a simple approximation is to consider II. FORMALISM large super-cells that can mimic the incommensurate system; in the case of first-principles calculations like den- The essence of our approach consists of the follow- sity functional theory (DFT), that can afford relatively ing ideas: A tight-binding model in d-dimensions is de- small cells, this approximation limits the physical system scribed by localized orbitals φi in a d-dimensional lat- rather severely to special values of the twist angle11. This tice, i 2 Zd, and the hopping matrix elements between 2 them labeled tij. To describe disorder in this model, we ality to Ω to represent all possible forms of disorder and consider the space of all possible defects and calculate applying them directly in each ! tight-binding model. physical properties for a carefully chosen subset of con- Our implementation of these ideas on a high- figurations. This is formulated by defining a configura- performance computing system are as follows: tion space Ω with specific local configurations ! 2 Ω with i) Creation of a heterostructure model out of layers that a probability distribution dP (!). Ω describes all possi- are disks of radius R; these disks are centered at a point ble environments that an atom in the infinite crystal can with \zero-shift", which is just one specific ! configura- experience, and we simulate physical observables by sam- tion. pling over this space of disordered configurations. This ii) Determination of all relevant hopping indices Hij in is in contrast to periodic approaches, which instead use the sparse hamiltonian by only looking for pairs of or- the Bloch wavenumber, ~k, as the sampling space. In in- bitals that are within the range of the hopping matrix commensurate systems translational symmetry has been elements tij. completely broken, and there is no Brillouin zone. Ω, iii) For each desired configuration !, displacement of one referred to as the \non-commutative Brillioun zone" for layer by the some amount with respect to the other layer, 14 ! this reason , is an alternative to this notion; neither Ω and computation of Hij for each non-zero hopping term; nor the Brillioun zone provide a diagonalized band struc- from this, we then calculate the local electronic density of ture with a finite number of eigenvalues at each point. states (LEDoS), or any other useful physical property like Viewing the interlayer interaction as a perturbative po- the conductivity. The LEDoS is derived from the global tential, the relative twist-angle can be interpreted as an EDoS, g(), by considering all eigenstates (indexed by s) aperiodic disorder field applied to the single-layer system. and orbitals (indexed by x): For a fixed twist angle, the location of the orbital φi in the field created by another layer varies. This variation N X 1 X X in location can be completely described by the offset, or g() = δ( − )jφ (x)j2 = g () (1) N s s x shift, between the two layers' unit cells, and thus Ω can x s=1 x be viewed as the compact two-dimensional space of all ! shifts. For each shift, we construct a system of finite iv) Application of the operator of interest to Hij radius which contributes a finite-size error. The error with a Kernel Polynomial Method (Chebyshev 31,32 decays exponentially with the radius, so it can be made polynomials) ; the Chebyshev polynomials Ti to approach zero in a controlable fashion. Our results form a complete basis for square integrable functions prove that this is a computationally feasible strategy. which take values in the range [−1; 1] and a linear In this picture, the difference between an incommen- combination of them can be chosen to approximate the surate and commensurate twist angle becomes trivial: a eigenspectrum of a tight-binding hamiltonian after a commensurate angle has a finite number of possible con- simple rescaling to ensure all eigenvalues lie in [−1; 1]. figurations because a periodic

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