Office: 509 Snow Hall

(785) 864-7116

405 Snow Hall

1460 Jayhawk Blvd.

University of Kansas

Lawrence, KS 66045, USA

Office: 509 Snow Hall

(785) 864-7116

405 Snow Hall

1460 Jayhawk Blvd.

University of Kansas

Lawrence, KS 66045, USA

arXiv:1806.10395 (29 pages)

**Abstract.** We derive and analyze a novel approach for modeling and computing the
mechanical relaxation of incommensurate 2D heterostructures. Our approach parametrizes
the relaxation pattern by the compact local configuration space rather than real space,
thus bypassing the need for the standard supercell approximation and giving a true
aperiodic atomistic configuration. Our model extends the computationally accessible
regime of weakly coupled bilayers with similar orientations or lattice spacing, for
example materials with a small relative twist where the widely studied large-scale moire
patterns arise. Our model also makes possible the simulation of multi-layers for which no
interlayer empirical atomistic potential exists, such as those composed of MoS2 layers,
and more generally makes possible the simulation of the relaxation of multi-layer
heterostructures for which a planar moire pattern does not exist.

Atomic reconstruction at van der Waals interface in twisted bilayer graphene.

(with H. Yoo, K. Zhang, R. Engelke, S.H. Sung, R. Hovden, A.W. Tsen, T. Taniguchi, K. Watanabe, G-C. Yi, M. Kim, M. Luskin, E.B. Tadmor and P. Kim)

(with H. Yoo, K. Zhang, R. Engelke, S.H. Sung, R. Hovden, A.W. Tsen, T. Taniguchi, K. Watanabe, G-C. Yi, M. Kim, M. Luskin, E.B. Tadmor and P. Kim)

arXiv:1804.03806 (15 pages)

**Abstract.** Interfaces between crystalline materials have been an essential engineering
platform for modern electronics. At the interfaces in two-dimensional (2D) van der Waals (vdW)
heterostructures, the twist-tunability offered by vdW crystals allows the construction of a
quasiperiodic moir\'e superlattice of tunable length scale, leading to unprecedented access
to exotic physical phenomena. However, these interfaces exhibit more intriguing structures
than the simple moir\'e pattern. The vdW interaction that favors interlayer commensurability
competes against the intralayer elastic lattice distortion, causing interfacial reconstruction
with significant modification to the electronic structure. Here we demonstrate engineered
atomic-scale reconstruction at the vdW interface between two graphene layers by controlling
the twist angle. Employing transmission electron microscopy (TEM), we find local commensuration
of Bernal stacked graphene within each domain, separated by incommensurate structural solitons.
We observe electronic transport along the triangular network of one-dimensional (1D) topological
channels as the electronic bands in the alternating domains are gapped out by a transverse
electric field. The atomic scale reconstruction in a twisted vdW interface further enables
engineering 2D heterostructures with continuous tunability.

Relaxation and Domain Formation in Incommensurate 2D Heterostructures. *Accepted for publication in Phys. Rev. B*

(S. Carr, D. Massatt, S.B. Torrisi, P.C, M. Luskin, E. Kaxiras)

(S. Carr, D. Massatt, S.B. Torrisi, P.C, M. Luskin, E. Kaxiras)

Accepted for publication in Physical Review B.
Preprint arXiv:1805.06972 (12 pages)

**Abstract.** We introduce configuration space as a natural representation for calculating
the mechanical relaxation patterns of incommensurate two-dimensional (2D) bilayers, bypassing
supercell approximations to encompass aperiodic relaxation patterns. The approach can be
applied to a wide variety of 2D materials through the use of a continuum model in combination
with a generalized stacking fault energy for interlayer interactions. We present computational
results for small-angle twisted bilayer graphene and molybdenum disulfide (MoS2), a representative
material of the transition metal dichalcogenide (TMDC) family of 2D semiconductors. We calculate
accurate relaxations for MoS2 even at small twist-angle values, enabled by the fact that our
approach does not rely on empirical atomistic potentials for interlayer coupling. The results
demonstrate the efficiency of the configuration space method by computing relaxations with
minimal computational cost for twist angles down to 0.05º, which is smaller than what can be
explored by any available real space techniques. We also outline a general explanation of domain
formation in 2D bilayers with nearly-aligned lattices, taking advantage of the relationship
between real space and configuration space.

Compression of Wannier functions into Gaussian-type orbitals, *Comp. Phys. Comm.*

(with A. Bakhta, E. Cancès, S. Fang, E. Kaxiras)

(with A. Bakhta, E. Cancès, S. Fang, E. Kaxiras)

Computer Physics Communications, **230**, 2018, pp 27-37.
doi.org/10.1016/j.cpc.2018.04.011 (12 pages)

**Abstract.** We propose a greedy algorithm for the compression of Wannier functions
into Gaussian-polynomials orbitals. The so-obtained compressed Wannier functions can be
stored in a very compact form, and can be used to efficiently parameterize effective
tight-binding Hamiltonians for multilayer 2D materials for instance. The compression method
preserves the symmetries (if any) of the original Wannier function. We provide algorithmic
details, and illustrate the performance of our implementation on several examples, including
graphene, hexagonal boron-nitride, single-layer FeSe, and bulk silicon in the diamond cubic
structure.

Quantum plasmons with optical-range frequencies in doped few-layer graphene, *Phys. Rev. B*

(S.N. Shirodkar, M. Mattheakis, P.C, P. Narang, M. Soljačić, E. Kaxiras)

(S.N. Shirodkar, M. Mattheakis, P.C, P. Narang, M. Soljačić, E. Kaxiras)

Physical Review B. **97**, 195435 (2018).
doi.org/10.1103/PhysRevB.97.195435 (6 pages)

**Abstract.** Although plasmon modes exist in doped graphene, the limited range of
doping achieved by gating restricts the plasmon frequencies to a range that does not
include the visible and infrared. Here we show, through the use of first-principles
calculations, that the high levels of doping achieved by lithium intercalation in bilayer
and trilayer graphene shift the plasmon frequencies into the visible range. To obtain
physically meaningful results, we introduce a correction of the effect of plasmon
interaction across the vacuum separating periodic images of the doped graphene layers,
consisting of transparent boundary conditions in the direction perpendicular to the layers;
this represents a significant improvement over the exact Coulomb cutoff technique employed
in earlier works. The resulting plasmon modes are due to local field effects and the
nonlocal response of the material to external electromagnetic fields, requiring a fully
quantum mechanical treatment. We describe the features of these quantum plasmons, including
the dispersion relation, losses, and field localization. Our findings point to a strategy
for fine-tuning the plasmon frequencies in graphene and other two-dimensional materials.

Cauchy-Born strain energy density for coupled incommensurate elastic chains, *ESAIM:M2AN*

(with M. Luskin)

(with M. Luskin)

ESAIM: Mathematical Modelling and Numerical Analysis **52(2)**, 2018, pp 729 - 749.
doi.org/10.1051/m2an/2017057 (21 pages)

**Abstract.** The recent fabrication of weakly interacting incommensurate two-dimensional
lattices requires an extension of the classical notion of the Cauchy-Born strain
energy density since these atomistic systems are not periodic. In this paper, we
rigorously formulate and analyze a Cauchy-Born strain energy density for weakly
interacting incommensurate one-dimensional lattices (chains) as a large body limit
and we give error estimates for its approximation by the popular supercell method.

Twistronics: Manipulating the Electronic Properties of Two-dimensional Layered Structures through their Twist Angle, *Phys. Rev. B*

(S. Carr, D. Massatt, S. Fang, P.C., M. Luskin and E. Kaxiras).

(S. Carr, D. Massatt, S. Fang, P.C., M. Luskin and E. Kaxiras).

Physical Review B **95**, 075420 (2017).
doi:10.1103/PhysRevB.95.075420 (6 pages)

**Abstract.** 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.

Generalized Kubo Formulas for the Transport Properties of Incommensurate 2D Atomic Heterostructures, *J. Math. Phys.*

(with E. Cancès and M. Luskin).

(with E. Cancès and M. Luskin).

Journal of Mathematical Physics **58**, 063502 (2017).
doi:10.1063/1.4984041 (23 pages)

**Abstract.** We give an exact formulation for the transport coefficients of incommensurate
two-dimensional atomic multilayer systems in the tight-binding approximation. This formulation
is based upon the C*-algebra framework introduced by Bellissard and collaborators to study
aperiodic solids (disordered crystals, quasicrystals, and amorphous materials), notably in the
presence of magnetic fields (quantum Hall effect). We also present numerical approximations and
test our methods on a one-dimensional incommensurate bilayer system.

Analysis of rippling in incommensurate one-dimensional coupled chains, *Multiscale. Model. and Simul.*

(with M. Luskin and E. Tadmor)

(with M. Luskin and E. Tadmor)

Multiscale Modeling and Simulation **15(1)**, 2017, pp. 56-73.
doi:10.1137/16M1076198
(18 pages)

**Abstract.** Graphene and other recently developed 2D materials exhibit exceptionally strong
in-plane stiffness. Relaxation of few-layer structures, either free-standing or on slightly
mismatched substrates occurs mostly through out-of-plane bending and the creation of large-scale
ripples. In this work, we present a novel double chain model, where we allow relaxation to occur
by bending of the incommensurate coupled system of chains. As we will see, this model can be seen
as a new application of the well-known Frenkel-Kontorova model for a one-dimensional atomic chain
lying in a periodic potential. We focus in particular on modeling and analyzing ripples occurring
in ground state configurations, as well as their numerical simulation.

Perturbation theory for weakly coupled two-dimensional layers, *J. Mater. Res.*

(G.Tritsaris, S. Shirodkar, T. Kaxiras, P. C., M. Luskin, P. Plechac and E. Cancès.)

(G.Tritsaris, S. Shirodkar, T. Kaxiras, P. C., M. Luskin, P. Plechac and E. Cancès.)

Journal of Material Research **31(7)**, 2016, pp. 959-966.
doi:10.1557/jmr.2016.99
(8 pages)

**Abstract.** A key issue in two-dimensional structures composed of atom-thick sheets of
electronic materials is the dependence of the properties of the combined system on the features
of its parts. Here, we introduce a simple framework for the study of the electronic structure of
layered assemblies based on perturbation theory. Within this framework, we calculate the band
structure of commensurate and twisted bilayers of graphene (Gr) and hexagonal boron nitride (h-BN),
and of a Gr/h-BN heterostructure, which we compare with reference full-scale density functional
theory calculations. This study presents a general methodology for computationally efficient
calculations of two-dimensional materials and also demonstrates that for relatively large twist
in the graphene bilayer, the perturbation of electronic states near the Fermi level is negligible.

Projective multiscale time-integration for electrostatic particle-in-cell methods,
*Accepted for publication in Comp. Phys. Comm.*

(with J. Hesthaven)

(with J. Hesthaven)

Accepted for publication in Computer Physics Communications.
doi.org/10.1016/j.cpc.2018.10.012 (26 pages)

**Abstract.** The simulation of problems in kinetic plasma physics are often challenging
due to strongly coupled phenomena across multiple scales. In this work, we propose a wavelet-based
coarse-grained numerical scheme, based on the framework of Equation-Free Projective Integration,
for a kinetic plasma system modeled by the Vlasov-Poisson equations. A kinetic particle-in-cell
(PIC) code is used to simulate the meso scale dynamics for short time intervals. This allows the
extrapolation over long time-steps of the behavior of a coarse wavelet-based discretization of
the system. To validate the approach and the underlying concepts, we perform two 1D1V numerical
experiments: nonlinear propagation and steepening of an ion wave, and the expansion of a plasma
slab in vacuum. The direct comparisons to resolved PIC simulations show good agreement. We show
that the speedup of the projective integration scheme over the full particle scheme scales linearly
with the system size, demonstrating efficiency while taking into account fully kinetic, non-Maxwellian
effects. This suggests that the approach is potentially interesting for kinetic plasma problems with
a large separation of scales.

Homogenization of a Multiscale Viscoelastic Model with Nonlocal Damping,
Application to the Human Lungs, *M3AS.*

(with C. Grandmont)

(with C. Grandmont)

Mathematical Models and Methods in Applied Sciences **25(6)**, 2015, pp 1125-1177.
doi:10.1142/S0218202515500293
(53 pages)

**Abstract.** We are interested in the mathematical modeling of the deformation of the
human lung tissue, called the lung parenchyma, during the respiration process. The parenchyma
is a foam-like elastic material containing millions of air-filled alveoli connected by a
tree-shaped network of airways. In this study, the parenchyma is governed by the linearized
elasticity equations and the air movement in the tree by the Poiseuille law in each airway.
The geometric arrangement of the alveoli is assumed to be periodic with a small period ε > 0.
We use the two-scale convergence theory to study the asymptotic behavior as ε goes to zero.
The effect of the network of airways is described by a nonlocal operator and we propose a
simple geometrical setting for which we show that this operator converges as ε goes to zero.
We identify in the limit the equations modeling the homogenized behavior under an abstract
convergence condition on this nonlocal operator. We derive some mechanical properties of the
limit material by studying the homogenized equations: the limit model is nonlocal both in space
and time if the parenchyma material is considered compressible, but only in space if it is
incompressible. Finally, we propose a numerical method to solve the homogenized equations and
we study numerically a few properties of the homogenized parenchyma model.

A fast boundary element method for the solution of periodic many-inclusion problems via
hierarchical matrix techniques, *ESAIM:Proc.*

(with O. Zahm)

(with O. Zahm)

ESAIM: Proceedings and Surveys, **48**, 2015, pp 156-168.
doi:10.1051/proc/201448006 13 pages)

**Abstract.** Our work is motivated by numerical homogenization of materials such as
concrete, modeled as composites structured as randomly distributed inclusions imbedded
in a matrix. In this paper, we propose a method for the approximation of the periodic
corrector problem based on boundary integral equations. The fully populated matrices obtained
by the discretization of the integral operators are successfully dealt with using the ℋ-matrix format.

Homogenization of a model for the propagation of sound in the lungs, *Multiscale. Model. Simul.*

(with C. Grandmont and Y. Maday)

(with C. Grandmont and Y. Maday)

Multiscale Modeling and Simulation **13(1)**, 2015, pp 43-71.
doi:10.1137/130916576 (29 pages)

**Abstract.** In this paper, we are interested in the mathematical modeling of the propagation
of sound waves in the lung parenchyma, which is a foam-like elastic material containing millions of
air-filled alveoli. In this study, the parenchyma is governed by the linearized elasticity equations,
and the air by the acoustic wave equations. The geometric arrangement of the alveoli is assumed to be
periodic with a small period $\varepsilon>0$. We consider the time-harmonic regime forced by
vibrations induced by volumic forces. We use the two-scale convergence theory to study the asymptotic
behavior as $\varepsilon$ goes to zero and prove the convergence of the solutions of the coupled
fluid-structure problem to the solution of a linear-elasticity boundary value problem.

Multiscale Modelling of sound propagation through the lungs' parenchyma, *ESAIM:M2AN.*

(with J. Hesthaven)

(with J. Hesthaven)

Mathematical Modelling and Numerical Analysis, **48(1)**, 2013, pp 27-52.
doi:10.1051/m2an/2013093 (23 pages)

**Abstract.** In this paper we develop and study numerically a model to describe some
aspects of sound propagation in the human lung, considered as a deformable and viscoelastic
porous medium (the parenchyma) with millions of alveoli filled with air. Transmission of sound
through the lung above 1 kHz is known to be highly frequency–dependent. We pursue the key idea
that the viscoelastic parenchyma structure is highly heterogeneous on the small scale $\varepsilon$
and use two–scale homogenization techniques to derive effective acoustic equations for
asymptotically small $\varepsilon$. This process turns out to introduce new memory effects. The effective
material parameters are determined from the solution of frequency–dependent micro–structure cell
problems. We propose a numerical approach to investigate the sound propagation in the homogenized
parenchyma using a Discontinuous Galerkin formulation. Numerical examples are presented.

Quelques modèles mathématiques homogénéisés appliqués
à la modélisation du parenchyme pulmonaire. PhD thesis.

The first part focuses on the coupling between parenchyma and bronchial tree. We begin by describing a model for the parenchyma deformation. We model (i) the parenchyma as a linear elastic material, (ii) the alveoli as periodically distributed cavities in the macroscopic parenchyma domain and (iii) the bronchial tree as a dyadic resistive tree. We write the equations of the model as a coupled fluid–structure system modeling the three–dimensional parenchyma’s displacement and depending on a parameter $\varepsilon$ which corresponds to the size of the periodicity cell. We study the two–scale convergence of the solutions of this system under an abstract hypothesis that describes the convergence of the action of the tree on the parenchyma. We obtain a macroscopic description of the parenchyma as a viscoelastic material where the tree induces a spatially non–local dissipation. In this part, we also study the abstract condition we have introduced. We propose two models for the irrigation of the domain by the tree inspired by the lung’s structure and for which the abstract condition can be verified. Finally, we describe a numerical method for the macroscopic problem and we illustrate the previous work by numerical simulations in two dimensions.

The second part focuses on the sound wave propagation in the parenchyma. We do not take into account the effect of the bronchial tree in this case. We homogenize in the frequency domain a first model coupling the linearized elasticity equations in the parenchyma and the acoustic equation in the air. We rigorously obtain the Rice model which describes sound propagation at low frequencies. We encounter a difficulty because the problem we investigate, of Helmholtz type, is not well–posed for all values of the frequency. To show the result, we use an argument by contradiction based on the Fredholm alternative. Then, we homogenize a second model which takes into account the viscoelastic and heterogeneous nature of the parenchyma at the microscopic level. The macroscopic viscoelastic coefficients depend on frequency. The material exhibits some new memory effects compared to its individual components. We propose a numerical method based on discontinuous Galerkin finite elements to solve the homogenized problem we have obtained. The numerical results obtained in a two–dimensional test case show that this model enables us to recover some physiological observations on the propagation of low–frequency ultrasound.

Autour de la modélisation du poumon. (in French) Master's thesis.

Ces problèmes ont donné lieu à un certain nombre de modèles simplifiés, visant à prendre en compte tel ou tel aspect, ou tel ou tel paramètre, jusqu'à des modèles tridimensionnels en géométrie réelle. Pour les expliquer, nous allons d’abord présenter rapidement la physiologie du poumon. Le modèle expérimental décrit est celui présenté par Weibel [1] .