# Publications

2021

Julian Klein, Thang Pham, Joachim Dahl Thomsen, Jonathan B. Curtis, Michael Lorke, Matthias Florian, Alexander Steinhoff, Ren A. Wiscons, Jan Luxa, Zdenek Sofer, Frank Jahnke, Prineha Narang, and Frances M. Ross. 6/30/2021. “Atomistic spin textures on-demand in the van der Waals layered magnet CrSBr.” arXiv. Publisher's VersionAbstract

Controlling magnetism in low dimensional materials is essential for designing devices that have feature sizes comparable to several critical length scales that exploit functional spin textures, allowing the realization of low-power spintronic and magneto-electric hardware.[1] Unlike conventional covalently-bonded bulk materials, van der Waals (vdW)-bonded layered magnets[2-4] offer exceptional degrees of freedom for engineering spin textures. However, their structural instability has hindered microscopic studies and manipulations. Here, we demonstrate nanoscale structural control in the layered magnet CrSBr creating novel spin textures down to the atomic scale. We show that it is possible to drive a local structural phase transformation using an electron beam that locally exchanges the bondings in different directions, effectively creating regions that have vertical vdW layers embedded within the horizontally vdW bonded exfoliated flakes. We calculate that the newly formed 2D structure is ferromagnetically ordered in-plane with an energy gap in the visible spectrum, and weak antiferromagnetism between the planes. Our study lays the groundwork for designing and studying novel spin textures and related quantum magnetic phases down to single-atom sensitivity, potentially to create on-demand spin Hamiltonians probing fundamental concepts in physics,[5-9] and for realizing high-performance spintronic, magneto-electric and topological devices with nanometer feature sizes.[10,11]

Anthony W. Schlimgen, Kade Head-Marsden, LeeAnn M. Sager, Prineha Narang, and David A. Mazziotti. 6/23/2021. “Quantum Simulation of Open Quantum Systems Using a Unitary Decomposition of Operators.” arXiv. Publisher's VersionAbstract

Electron transport in realistic physical and chemical systems often involves the non-trivial exchange of energy with a large environment, requiring the definition and treatment of open quantum systems. Because the time evolution of an open quantum system employs a non-unitary operator, the simulation of open quantum systems presents a challenge for universal quantum computers constructed from only unitary operators or gates. Here we present a general algorithm for implementing the action of any non-unitary operator on an arbitrary state on a quantum device. We show that any quantum operator can be exactly decomposed as a linear combination of at most four unitary operators. We demonstrate this method on a two-level system in both zero and finite temperature amplitude damping channels. The results are in agreement with classical calculations, showing promise in simulating non-unitary operations on intermediate-term and future quantum devices.

Jonathan B. Curtis, Andrey Grankin, Nicholas R. Poniatowski, Victor M. Galitski, Prineha Narang, and Eugene Demler. 6/15/2021. “Cavity magnon-polaritons in cuprate parent compounds.” arXiv. Publisher's VersionAbstract

Cavity control of quantum matter may offer new ways to study and manipulate many-body systems. A particularly appealing idea is to use cavities to enhance superconductivity, especially in unconventional or high-T

_{c}systems. Motivated by this, we propose a scheme for coupling Terahertz resonators to the antiferromagnetic fluctuations in a cuprate parent compound, which are believed to provide the glue for Cooper pairs in the superconducting phase. First, we derive the interaction between magnon excitations of the Neél-order and polar phonons associated with the planar oxygens. This mode also couples to the cavity electric field, and in the presence of spin-orbit interactions mediates a linear coupling between the cavity and magnons, forming hybridized magnon-polaritons. This hybridization vanishes linearly with photon momentum, implying the need for near-field optical methods, which we analyze within a simple model. We then derive a higher-order coupling between the cavity and magnons which is only present in bilayer systems, but does not rely on spin-orbit coupling. This interaction is found to be large, but only couples to the bimagnon operator. As a result we find a strong, but heavily damped, bimagnon-cavity interaction which produces highly asymmetric cavity line-shapes in the strong-coupling regime. To conclude, we outline several interesting extensions of our theory, including applications to carrier-doped cuprates and other strongly-correlated systems with Terahertz-scale magnetic excitations.Dominik M. Juraschek and Prineha Narang. 6/8/2021. “Highly confined phonon polaritons in monolayers of oxide perovskites.” Nano Letters. Publisher's VersionAbstract

Two-dimensional (2D) materials are able to strongly confine light hybridized with collective excitations of atoms, enabling electric-field enhancements and novel spectroscopic applications. Recently, freestanding monolayers of perovskite oxides have been synthesized, which possess highly infrared-active phonon modes and a complex interplay of competing interactions. Here, we show that this new class of 2D materials exhibits highly confined phonon polaritons by evaluating central figures of merit for phonon polaritons in the tetragonal phases of the 2D perovskites SrTiO

_{3}, KTaO_{3}, and LiNbO_{3}, using density functional theory calculations. Specifically, we compute the 2D phonon-polariton dispersions, the propagation-quality, confinement, and deceleration factors, and we show that they are comparable to those found in the prototypical 2D dielectric hexagonal boron nitride. Our results suggest that monolayers of perovskite oxides are promising candidates for polaritonic platforms that enable new possibilities in terms of tunability and spectral ranges.Georgios Varnavides, Yaxian Wang, Philip J.W. Moll, Polina Anikeeva, and Prineha Narang. 6/1/2021. “Finite-size effects of electron transport in PdCoO

_{2}.” arXiv. Publisher's VersionAbstractA wide range of unconventional transport phenomena have recently been observed in single-crystal delafossite metals. Here, we present a theoretical framework to elucidate electron transport using a combination of first-principles calculations and numerical modeling of the anisotropic Boltzmann transport equation. Using PdCoO

_{2}as a model system, we study different microscopic electron and phonon scattering mechanisms and establish the mean free path hierarchy of quasiparticles at different temperatures. We treat the anisotropic Fermi surface explicitly to numerically obtain experimentally-accessible transport observables, which bridge between the "diffusive", "ballistic", and "hydrodynamic" transport regime limits. We illustrate that distinction between the "quasi-ballistic", and "quasi-hydrodynamic" regimes is challenging and often needs to be quantitative in nature. From first-principles calculations, we populate the resulting transport regime plots, and demonstrate how the Fermi surface orientation adds complexity to the observed transport signatures in micro-scale devices. Our work provides key insights into microscopic interaction mechanisms on open hexagonal Fermi surfaces and establishes their connection to the macroscopic electron transport in finite-size channels.Derek S. Wang, Michael Haas, and Prineha Narang. 5/17/2021. “Quantum Interfaces to the Nanoscale.” ACS Nano. Publisher's VersionAbstract

Scalable quantum information systems would store, manipulate, and transmit quantum information locally and across a quantum network, but no single qubit technology is currently robust enough to perform all necessary tasks. Defect centers in solid-state materials have emerged as potential intermediaries between other physical manifestations of qubits, such as superconducting qubits and photonic qubits, to leverage their complementary advantages. It remains an open question, however, how to design and to control quantum interfaces to defect centers. Such interfaces would enable quantum information to be moved seamlessly between different physical systems. Understanding and constructing the required interfaces would, therefore, unlock the next big steps in quantum computing, sensing, and communications. In this Perspective, we highlight promising coupling mechanisms, including dipole-, phonon-, and magnon-mediated interactions, and discuss how contributions from nanotechnologists will be paramount in realizing quantum information processors in the near-term.

Georgia T. Papadakis, Christopher J. Ciccarino, Lingling Fan, Meir Orenstein, Prineha Narang, and Shanhui Fan. 5/3/2021. “Deep-Subwavelength Thermal Switch via Resonant Coupling in Monolayer Hexagonal Boron Nitride.” Physical Review Applied, 15, 5, Pp. 054002. Publisher's VersionAbstract

Unlike the electrical conductance that can be widely modulated within the same material even in deep-subwavelength devices, tuning the thermal conductance within a single material system or nanostructure is extremely challenging and requires a large-scale device. This prohibits the realization of robust

*on*/*off*states in switching the flow of thermal currents. Here, we present the theory of a thermal switch based on resonant coupling of three photonic resonators, in analogy to the field-effect electronic transistor composed of a source, a gate, and a drain. As a material platform, we capitalize on the extreme tunability and low-loss resonances observed in the dielectric function of monolayer hexagonal boron nitride (h-BN) under controlled strain. We derive the dielectric function of h-BN from first principles, including the phonon-polariton line widths computed by considering phonon-isotope and anharmonic phonon-phonon scattering. Subsequently, we propose a strain-controlled h-BN–based thermal switch that modulates the thermal conductance by more than an order of magnitude, corresponding to a contrast ratio in the thermal conductance of 98%, in a deep-subwavelength nanostructure.Christian Schäfer, Johannes Flick, Enrico Ronca, Prineha Narang, and Angel Rubio. 4/26/2021. “Shining Light on the Microscopic Resonant Mechanism Responsible for Cavity-Mediated Chemical Reactivity.” arXiv. Publisher's VersionAbstract

Strong light-matter interaction in cavity environments has emerged as a promising and general approach to control chemical reactions in a non-intrusive manner. The underlying mechanism that distinguishes between steering, accelerating, or decelerating a chemical reaction has, however, remained thus far largely unclear, hampering progress in this frontier area of research. In this work, we leverage a combination of first-principles techniques, foremost quantum-electrodynamical density functional theory, applied to the recent experimental realization by Thomas et al. [1] to unveil the microscopic mechanism behind the experimentally observed reduced reaction-rate under resonant vibrational strong light-matter coupling. We find that the cavity mode functions as a mediator between different vibrational eigenmodes, transferring vibrational excitation and anharmonicity, correlating vibrations, and ultimately strengthening the chemical bond of interest. Importantly, the resonant feature observed in experiment, theoretically elusive so far, naturally arises in our investigations. Our theoretical predictions in polaritonic chemistry shine new light on cavity induced mechanisms, providing a crucial control strategy in state-of-the-art photocatalysis and energy conversion, pointing the way towards generalized quantum optical control of chemical systems.

Xuezeng Tian, Xingxu Yan, Georgios Varnavides, Yakun Yuan, Dennis S. Kim, Christopher J. Ciccarino, Polina Anikeeva, Ming-Yang Li, Lain-Jong Li, Prineha Narang, Xiaoqing Pan, and Jianwei Miao. 4/18/2021. “Capturing 3D atomic defects and phonon localization at the 2D heterostructure interface.” arXiv. Publisher's VersionAbstract

The 3D local atomic structures and crystal defects at the interfaces of heterostructures control their electronic, magnetic, optical, catalytic and topological quantum properties, but have thus far eluded any direct experimental determination. Here we determine the 3D local atomic positions at the interface of a MoS2-WSe2 heterojunction with picometer precision and correlate 3D atomic defects with localized vibrational properties at the epitaxial interface. We observe point defects, bond distortion, atomic-scale ripples and measure the full 3D strain tensor at the heterointerface. By using the experimental 3D atomic coordinates as direct input to first principles calculations, we reveal new phonon modes localized at the interface, which are corroborated by spatially resolved electron energy-loss spectroscopy. We expect that this work will open the door to correlate structure-property relationships of a wide range of heterostructure interfaces at the single-atom level.

John P. Philbin, Joseph Kelly, Lintao Peng, Igor Coropceanu, Abhijit Hazarika, Dmitri V. Talapin, Eran Rabani, Xuedan Ma, and Prineha Narang. 4/13/2021. “Room temperature single-photon superfluorescence from a single epitaxial cuboid nano-heterostructure.” arXiv. Publisher's VersionAbstract

Single-photon superradiance can emerge when a collection of identical emitters are spatially separated by distances much less than the wavelength of the light they emit, and is characterized by the formation of a superradiant state that spontaneously emits light with a rate that scales linearly with the number of emitters. This collective phenomena has only been demonstrated in a few nanomaterial systems, all requiring temperatures below 10K. Here, we rationally design a single colloidal nanomaterial that hosts multiple (nearly) identical emitters that are impervious to the fluctuations which typically inhibit room temperature superradiance in other systems such as molecular aggregates. Specifically, by combining molecular dynamics, atomistic electronic structure calculations, and model Hamiltonian methods, we show that the faces of a heterostructure nanocuboid mimic individual quasi-2D nanoplatelets and can serve as the robust emitters required to realize superradiant phenomena at room temperature. Leveraging layer-by-layer colloidal growth techniques to synthesize a nanocuboid, we demonstrate single-photon superfluorescence via single-particle time-resolved photoluminescence measurements at room temperature. This robust observation of both superradiant and subradiant states in single nanocuboids opens the door to ultrafast single-photon emitters and provides an avenue to entangled multi-photon states via superradiant cascades.

Derek S. Wang, Tomáš Neuman, and Prineha Narang. 3/10/2021. “Spin emitters beyond the Point Dipole Approximation in Nanomagnonic Cavities.” The Journal of Physical Chemistry C. Publisher's VersionAbstract

Control over transition rates between spin states of emitters is crucial in a wide variety of fields ranging from quantum information science to the nanochemistry of free radicals. We present an approach to drive both electric and magnetic dipole-forbidden transitions of a spin emitter by placing it in a nanomagnonic cavity, requiring a description of both the spin emitter beyond the point dipole approximation and the vacuum magnetic fields of the nanomagnonic cavity with a large spatial gradient over the volume of the spin emitter. We specifically study the silicon vacancy (SiV

^{–}) defect in diamond, whose Zeeman-split ground states comprise a logical qubit for solid-state quantum information processing, coupled to a magnetic nanoparticle serving as a model nanomagnonic cavity capable of concentrating microwave magnetic fields into deeply subwavelength volumes. Through first-principles modeling of the SiV^{–}spin orbitals, we calculate the spin transition densities of magnetic dipole-allowed and -forbidden transitions and calculate their coupling rates to various multipolar modes of the nanomagnonic cavity. We envision using such a framework for manipulation of quantum spin states.Derek S. Wang, Tomáš Neuman, Johannes Flick, and Prineha Narang. 3/9/2021. “Light-matter interaction of a molecule in a dissipative cavity from first principles.” The Journal of Chemical Physics, 154, Pp. 104109. Publisher's VersionAbstract

Cavity-mediated light-matter coupling can dramatically alter opto-electronic and physico-chemical properties of a molecule. Ab initio theoretical predictions of these systems need to combine non-perturbative, many-body electronic structure theory-based methods with cavity quantum electrodynamics and theories of open quantum systems. Here we generalize quantum-electrodynamical density functional theory to account for dissipative dynamics and describe coupled cavity-molecule interactions in the weak-to-strong-coupling regimes. Specifically, to establish this generalized technique, we study excited-state dynamics and spectral responses of benzene and toluene under weak-to-strong light-matter coupling. By tuning the coupling we achieve cavity-mediated energy transfer between electronic excited states. This generalized ab initio quantum-electrodynamical density functional theory treatment can be naturally extended to describe cavity-mediated interactions in arbitrary electromagnetic environments, accessing correlated light-matter observables and thereby closing the gap between electronic structure theory and quantum optics.

Nicholas R. Poniatowski, Jonathan B. Curtis, Amir Yacoby, and Prineha Narang. 3/9/2021. “Spectroscopic signatures of time-reversal symmetry breaking superconductivity.” arXiv. Publisher's VersionAbstract

The collective mode spectrum of a symmetry-breaking state, such as a superconductor, provides crucial insight into the nature of the order parameter. In this context, we present a microscopic weak-coupling theory for the collective modes of a generic multi-component time-reversal symmetry breaking superconductor, and show that fluctuations in the relative amplitude and phase of the two order parameter components are well-defined underdamped collective modes, even in the presence of nodal quasiparticles. We then demonstrate that these "generalized clapping modes" can be detected using a number of experimental techniques including ac electronic compressibility measurements, ultrafast THz spectroscopy, and microwave power absorption. Finally, we discuss the implications of our work as a new form of "collective mode spectroscopy" that drastically expands the number of experimental probes capable of detecting time-reversal symmetry breaking in unconventional superconductors such as Sr

_{2}RuO_{4}, UTe_{2}, and moiré heterostructures.Dominik M. Juraschek, Derek S. Wang, and Prineha Narang. 3/4/2021. “Sum-frequency excitation of coherent magnons.” Physical Review B, 103, 9, Pp. 094407. Publisher's VersionAbstract

Coherent excitation of magnons is conventionally achieved through Raman scattering processes, in which the difference-frequency components of the driving field are resonant with the magnon energy. Here, we describe mechanisms by which the sum-frequency components of the driving field can be used to coherently excite magnons through two-particle absorption processes. We use the Landau-Lifshitz-Gilbert formalism to compare the spin-precession amplitudes that different types of impulsive stimulated and ionic Raman scattering processes and their sum-frequency counterparts induce in an antiferromagnetic model system. We show that sum-frequency mechanisms enabled by linearly polarized driving fields yield excitation efficiencies comparable or larger than established Raman techniques, while elliptical polarizations produce only weak and circularly polarizations no sum-frequency components at all. The mechanisms presented here complete the map for dynamical spin control by the means of Raman-type processes.

Kate Reidy, Georgios Varnavides, Joachim Dahl Thomsen, Abinash Kumar, Thang Pham, Arthur M. Blackburn, Polina Anikeeva, Prineha Narang, James M. LeBeau, and Frances M. Ross. 2/26/2021. “Direct imaging and electronic structure modulation of moiré superlattices at the 2D/3D interface.” Nature Communications, 12, Pp. 1290. Publisher's VersionAbstract

The atomic structure at the interface between a two-dimensional (2D) and a three-dimensional (3D) material influences properties such as contact resistance, photo-response, and high-frequency performance. Moiré engineering has yet to be explored for tailoring this 2D/3D interface, despite its success in enabling correlated physics at 2D/2D twisted van der Waals interfaces. Using epitaxially aligned MoS

_{2}/Au{111} as a model system, we apply a geometric convolution technique and four-dimensional scanning transmission electron microscopy (4D STEM) to show that the 3D nature of the Au structure generates two coexisting moiré periods (18 Angstroms and 32 Angstroms) at the 2D/3D interface that are otherwise hidden in conventional electron microscopy imaging. We show, via ab initio electronic structure calculations, that charge density is modulated with the longer of these moiré periods, illustrating the potential for (opto-)electronic modulation via moiré engineering at the 2D/3D interface.Kade Head-Marsden, Stefan Krastanov, David A. Mazziotti, and Prineha Narang. 2/25/2021. “Capturing non-Markovian dynamics on near-term quantum computers.” Physical Review Research, 3, 1, Pp. 013182. Publisher's VersionAbstract

With the rapid progress in quantum hardware, there has been an increased interest in new quantum algorithms to describe complex many-body systems searching for the still-elusive goal of “useful quantum advantage.” Surprisingly, quantum algorithms for the treatment of open quantum systems (OQSs) have remained underexplored, in part due to the inherent challenges of mapping non-unitary evolution into the framework of unitary gates. Evolving an open system unitarily necessitates dilation into a new effective system to incorporate critical environmental degrees of freedom. In this context, we present and validate a new quantum algorithm to treat non-Markovian dynamics in OQSs built on the ensemble of Lindblad's trajectories approach, invoking the Sz.-Nagy dilation theorem. Here we demonstrate our algorithm on the Jaynes-Cummings model in the strong-coupling and detuned regimes, relevant in quantum optics and driven quantum system studies. This algorithm, a key step towards generalized modeling of non-Markovian dynamics on a noisy-quantum device, captures a broad class of dynamics and opens up a new direction in OQS problems.

Conor Delaney, Kaushik P. Seshadreesan, Ian MacCormack, Alexey Galda, Saikat Guha, and Prineha Narang. 2/25/2021. “Demonstration of quantum advantage by a joint detection receiver for optical communications using quantum belief propagation on a trapped-ion device.” arXiv. Publisher's VersionAbstract

Demonstrations of quantum advantage have largely focused on computational speedups and on quantum simulation of many-body physics, limited by fidelity and capability of current devices. Discriminating laser-pulse-modulated classical-communication codewords at the minimum allowable probability of error using universal-quantum processing presents a promising parallel direction, one that is of both fundamental importance in quantum state discrimination, as well as of technological relevance in deep-space laser communications. Here we present an experimental realization of a quantum joint detection receiver for binary phase shift keying modulated codewords of a 3-bit linear tree code using a recently-proposed quantum algorithm: belief propagation with quantum messages. The receiver, translated to a quantum circuit, was experimentally implemented on a trapped-ion device -- the recently released Honeywell LT-1.0 system using

^{17}1Yb+ ions, which possesses all-to-all connectivity and mid-circuit measurement capabilities that are essential to this demonstration. We conclusively realize a previously postulated but hitherto not-demonstrated joint quantum detection scheme, and provide an experimental framework that surpasses the quantum limit on the minimum average decoding error probability associated with pulse-by-pulse detection in the low mean photon number limit. The full joint-detection scheme bridges across photonic and trapped-ion based quantum information science, mapping the photonic coherent states of the modulation alphabet onto inner product-preserving states of single-ion qubits. Looking ahead, our work opens new avenues in hybrid realizations of quantum-enhanced receivers with applications in astronomy and emerging space-based platformsDerek S. Wang, Christopher J. Ciccarino, Johannes Flick, and Prineha Narang. 2/18/2021. “Hybridized Defects in Solid-State Materials as Artificial Molecules.” ACS Nano. Publisher's VersionAbstract

Two-dimensional materials can be crafted with structural precision approaching the atomic scale, enabling quantum defects-by-design. These defects are frequently described as artificial atoms and are emerging optically-addressable spin qubits. However, interactions and coupling of such artificial atoms with each other, in the presence of the lattice, is remarkably underexplored. Here we present the formation of artificial molecules in solids, introducing a new degree of freedom in control of quantum optoelectronic materials. Specifically, in monolayer hexagonal boron nitride as our model system, we observe configuration- and distance-dependent dissociation curves and hybridization of defect orbitals within the bandgap into bonding and antibonding orbitals, with splitting energies ranging from ∼ 10 meV to nearly 1 eV. We calculate the energetics of

*cis*and*trans*out-of-plane defect pairs CH_{B}-CH_{B}against an in-plane defect pair C_{B}-C_{B}and find that in-plane defect pair interacts more strongly than out-of-plane pairs. We demonstrate an application of this chemical degree of freedom by varying the distance between C_{B}and V_{N}of C_{B}V_{N}and observe changes in the predicted peak absorption wavelength from the visible to the near-infrared spectral band. We envision leveraging this chemical degree of freedom of defect complexes to precisely control and tune defect properties towards engineering robust quantum memories and quantum emitters for quantum information science.John P. Philbin, Amikam Levy, Prineha Narang, and Wenjie Dou. 2/15/2021. “Spin-Dependent Transport Through a Colloidal Quantum Dot: The Role of Exchange Interactions.” arXiv. Publisher's VersionAbstract

The study of charge and spin transport through semiconductor quantum dots is experiencing a renaissance due to recent advances in nano-fabrication and the realization of quantum dots as candidates for quantum computing. In this work, we combine atomistic electronic structure calculations with quantum master equation methods to study the transport of electrons and holes through strongly confined quantum dots coupled to two leads with a voltage bias. We find that a competition between the energy spacing between the two lowest quasiparticle energy levels and the strength of the exchange interaction determines the spin states of the lowest two quasiparticle energy levels. Specifically, the low density of electron states results in a spin singlet being the lowest energy two-electron state whereas, in contrast, the high density of states and significant exchange interaction results in a spin triplet being the lowest energy two-hole state. The exchange interaction is also responsible for spin blockades in transport properties, which could persist up to temperatures as high as 77K for strongly confined colloidal quantum dots from our calculations. Lastly, we relate these findings to the preparation and manipulation of singlet and triplet spin qubit states in quantum dots using voltage biases.

Gavin B. Osterhoudt, Vincent M. Plisson, Yaxian Wang, Christina A. C. Garcia, Johannes Gooth, Claudia Felser, Prineha Narang, and Kenneth S. Burch. 1/27/2021. “Evidence for dominant phonon-electron scattering in Weyl semimetal WP

_{2}.” Physical Review X, 11, 1, Pp. 011017. Publisher's VersionAbstractTopological semimetals have revealed a wide array of novel transport phenomena, including electron hydrodynamics, quantum field theoretic anomalies, and extreme magnetoresistances and mobilities. However, the scattering mechanisms central to these behaviors remain largely unexplored. Here we reveal signatures of significant phonon-electron scattering in the type-II Weyl semimetal WP

_{2}via temperature-dependent Raman spectroscopy. Over a large temperature range, we find that the decay rates of the lowest energy A_{1 }modes are dominated by phonon-electron rather than phonon-phonon scattering. In conjunction with first-principles calculations, a combined analysis of the momentum, energy, and symmetry-allowed decay paths indicates this results from intraband scattering of the electrons. The excellent agreement with theory further suggests that such results could be true for the acoustic modes. We thus provide evidence for the importance of phonons in the transport properties of topological semimetals and identify specific properties that may contribute to such behavior in other materials.