2d materials and electromagnetic applicationsboag/conf1/pdfs/george_hanson_tel aviv...

2D Materials and Electromagnetic Applications

November 10th, 2016Tel Aviv University Antenna Symposium

George Hanson Dept. of Electrical Engineering and Computer Science

University of Wisconsin-Milwaukee

Introduction

Two-Dimensional Materials and Electromagnetic Applications

2D materials: summary of current materials and their properties(graphene, black phosphorus, hexagonal boron nitride, transitionmetal di-chalcogenides (TMDCs) such as molybdenum disulfide, etc.)

EM modeling of 2D materials

Local and nonlocal infinitesimal sheet models, 3D model,anisotropic models

Electronic and electromagnetic applications

Introduction

Prerequisite – concept of a bandgap

Conductors, semiconductors, and insulators can all be explained by these types of band diagrams.

wikipedia.org

ℏ𝜔𝜔ℏ𝜔𝜔 ℏ𝜔𝜔

Introduction

Nature Photonics 8, 899–907 (2014)

Two-dimensional Materials and the Four Minute Mile

For many years it was thought, based on thermodynamics, that it was impossible for a single atomic layer to exist independently.

In 2004 Andrei Geim and his PhD student Konstantin Novoselov, at the University of Manchester, obtained single-layer graphene using scotch tape exfoliation. This works for a wide range of van der Waals solids.

For many years it was thought that the human body was incapable of running a mile in under four minutes. In 1954 that barrier was broken by Roger Bannister at Oxford University. Two months later Australia's John Landy did it, and now it is commonplace.

articles.orlandosentinel.com

Science, 2004

Geim and Novoselov submitted a three-page paper to Nature, where it was rejected twice – one reviewer said that isolating a stable, two-dimensional material was “impossible,” and another said that it was “not a sufficient scientific advance.”

In October 2004, the paper, “Electric Field Effect in Atomically Thin Carbon Films,” was published in Science.

In 2010, Geim and Novoselov were awarded the Nobel Prize in Physics.

At the University of Manchester, the British government invested sixty million dollars to help create the National Graphene Institute.

In 2013 the EU awarded a €1bn ($1.35bn) grant to the Graphene Flagship Consortium based at Chalmers University (Sweden).

- Ten-year project involving 126 academic and industrial research groups. Goal is to commercialize graphene and spur economic growth.

2D Materials are Hot!

Now, there are a number of companiesproducing graphene commercially.

Blue: Monolayers stable under ambient conditions (room temperature in air) Green: Probably stable in airPink: unstable in air but may be stable in inert atmosphereGrey: 3D compounds that have been successfully exfoliated down to monolayers, but for which there is little further information.

A. K. Geim and I. V. Grigorieva, van der Waals heterostructures, Nature 499, 419, July, 2013

Current families of 2D materials

Universal Attribute of 2D Materials: Tunability, flexability(bendability), transparency

• 2D materials cover the usual classes of electronic materials: insulators (e.g., hex boron nitride), semiconductors (e.g., MoS2), and metals (e.g., graphene)

• A 2D material is “all surface,” and so

• the interface between the surface and a substrate, and the presence of adatomsand defects can dramatically alter the material’s properties.

• 2D materials can inherently be tuned using electrostatic or magnetostaticfields – little electronic screening takes place (there is no “inside”).

• Lattice strain plays an important role in modifying the lattice constants, leading to modifications of the energy bands. Strain engineering is an emerging field.

• Mechanically, their 2D nature is important as they are inherently flexible (weak out-of-plane bonds), strong (strong in-plane bonds), and, of course, extremely thin.

The initial excitement over graphene: Dirac Cones

In graphene (and a few other 2D materials), the E–k relation is linear (as occurs for photons) for low energies near the six corners of the two-dimensional hexagonal Brillouin zone.

The result is zero effective mass for electrons and holes, and an energy-independent velocity.

Electron behave as Dirac fermions (they obey the relativistic Dirac equation, and exhibit exotic effects such as Klein tunneling, half-integer quantum Hall effect, ultrahigh carrier mobility). Dirac cones come in pairs with opposite chirality.

A. K. Geim and I. V. Grigorieva, van der Waals heterostructures, Nature 499, 419, July, 2013.

Combining 2D Materials - van der Waals Heterostructures

General Electronic Applications

Digital Logic: electronic circuits are made mostly from metal–oxide–semiconductor field-effect transistors (MOSFETs).

There are three requirements for a good logic transistor:• high carrier mobility for fast operation, • high on/off ratio• low off-state conductance for low power consumption.

The 1–2 eV band gaps of Mo and W dichalcogenides can provide high on/off ratios and with low power dissipation.

Recent MoS2 top-gated FETs showed excellent on/off current ratio up to 108, room temperature mobility of >200 cm2 V-1 s-1.

Nat Nanotechnol 2011;6:147–50

General EM Applications

• Plasmonics and waveguiding/interconnects• Antennas• Optical devices, including modulators, detectors and emitters.

iopscience.iop.org

2D modulated waveguide

www.osapublishing.orgPlasmonic absorber

Synthesis Methods for 2D Materials

• Micromechanical exfoliation using scotch tape• Single and few atomic layer materials can be obtained by mechanical

exfoliation (first used by Grim for graphene). Material is peeled off using scotch tape.

• The monolayer yield is low; only suitable for laboratory scale, and samples are typically tens of microns in size.

• Resulting material has excellent structural integrity, high crystallinity, and superior electrical properties.

• Liquid exfoliation• Used to produce single and few layers 2D sheets at bulk scale. • Weak out-of-plane bonding in the layered materials and high surface

area results in the ability for molecules to be adsorbed between atomic layers. This intercalation weakens interlayer adhesion and lowers the energy barrier required for exfoliation.

A. Gupta, T. Sakthivel, and S. Seal, Recent development in 2D materials beyond graphene, Progress in Materials Science 73, 44-126, 2015.

• Sonication• Ultrasonic cleavage of weak out-of-plane bonding - bulk layered

material is dispersed in a solvent and sonicated for several hours.

• Chemical vapor deposition (CVD)• The material to be coated is placed inside a vacuum chamber. The

coating material is heated until the material vaporizes. The vapor settles on the material to be coated, forming a uniform coating. Adjusting the temperature and duration of the process makes it possible to control the thickness of the coating.

• CVD has enabled the synthesis of large area and uniform thickness 2D layers for large-scale fabrication.

A. Gupta, T. Sakthivel, and S. Seal, Recent development in 2D materials beyond graphene, Progress in Materials Science 73, 44-126, 2015. www.azonano.com

Synthesis methods for 2D materials

Synthesis methods for 2D materials

Kitchen blender method to make graphene:

Take a high-power (400-watt) kitchen blender, water, detergent(to act as surfactant), and 20–50 grams of graphite powder (pencil lead), run for 10-30 minutes.

The result was a large number of high-quality micrometre-sized flakes of graphene, suspended in the water.

K. R. Patton et al., “Scalable production of large quantities of defect-free few-layer graphene by shear exfoliation in liquids,” Nature Materials,13, 624, 2014.

Visualizing 2D materials

Optical micrograph of thin films of MoS2

AFM of MoS2

AFM of graphene

optical micrograph of graphene

Butler et al., Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene, ACS Nano, 2013, 7 (4), pp 2898–2926

An interference effect on dielectric-coated SiO2/Si substrates is commonly used to visualize and locate single and few layers.

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Infinite contiguous 2D sheets are most accurately modeled as a two-sided impedance surface having conductivity σ (S).

2D Materials – Electromagnetic Modeling

Conductivity may be anisotropic, non-local, non-linear, etc.

2D Materials – Electromagnetic Modeling

In general, all (or most) 2D materials can be characterized by the tensor, non-local Kubo conductivity.

This is a 2D conductivity (SI units Siemans (S), not S/m), accounting for both intraband and interband contributions.

Electromagnetic boundary conditions are

Usually nonlocal aspects add considerable complication to the model. However, for 2D nonlocal materials it is very easy! – simply replace σ(ω) with σ(ω,q).

2D Materials – Electromagnetic Modeling of Graphene

For graphene in the local case, relatively simple formulas result:

where the Drude weight is


Optics Express, 20, pp. 23201-23214, 2012.

Pauli blocking

22

101 103 104 104 104

f (GHz)-8-6-4-202468

101214

σ' (mS)

σ' ' (mS)

µc=0

101 102 103 1042 3 4 5 67 2 3 4 5 67 2 3 4 5 67

f (GHz)-8-6-4-202468

101214

σ' (mS)

σ' ' (mS)

µc=0

Graphene Optical Conductivity – intraband and interband components

𝜎𝜎 = 𝜎𝜎intraband + 𝜎𝜎interband

Intraband term is Drude-like

Drude conductivity

Onset of interband transitions

An alternative model is to consider the material to have some finite thickness, and to convert the 2D conductivity to a 3D conductivity. Effective thickness d is O(1 nm).

However, while convenient for some commercial simulators, one must be careful in using this model. It can lead to a fine mesh and long run times, plus inaccurate results.


Gonçalves and Peres , An Introduction to Graphene PlasmonicsWorld Scientific, 2016

The correct field behavior for a plasmon mode at the interface is to have an odd normal electric field.


Alù and Engheta, JOSA B, 23,p. 571, 2006.

Even mode

Odd mode

However, a negative epsilon slab waveguide supports both and odd and an even mode.


E. Forati, G.W. Hanson, A.B. Yakovlev,and Andrea Alù, A planar hyperlens based on a modulated graphene monolayer, Phys. Rev. B (Rapid Communications) 89, 081410(R) 2014.

2D Materials – Electromagnetic Modeling of GrapheneError in finite-thickness model

https://pantherfile.uwm.edu/george/public/Final%20Version_PhysRevB.89.081410.pdf

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Graphene's hexagonal lattice can be regarded as two interleaving triangular lattices.

grapheneindustries.com, titianmedia.ca/wordpress

Primary Attributes of Several 2D Materials:

Graphite consists of multiple graphene layers.

Carbon nanotubes are rolled up graphene sheets.

physicsforme.wordpress.com and photonics.com, nontrivialproblems.wordpress.com

Graphene: a single layer of carbon atoms arranged in a regular hexagonal pattern. Each atom participating in four bonds, one strong in-plane σ bond with each of its three neighbors and one weak out-of-plane π-bond.

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Graphene is the strongest material in the world

If a sheet of plastic cling wrap had the same strength as pristine graphene, to puncture it with a pencil would require applying a force on the order of 20,000 N.

Graphene is a zero-bandgap semimetal. At finite temperatures it is essentially a metal.

Graphene has the highest thermal conductivity of any material (2 x diamond).

Graphene has the one of the highest mobilities of any material (100 x Si).

Graphene can carry the highest current densities of any material (109 A/cm2 ; 1,000 x Cu).

One reason for these extraordinary properties is that graphene can be obtained that is nearly defect-free (at least in small areas).

http://titianmedia.ca/wordpress/?p=1480

Graphene

30

The E–k relation is linear (as occurs for photons) for low energies near the six corners of the two-dimensional hexagonal Brillouin zone.

The result is zero effective mass for electrons and holes, and an energy-independent velocity.

Graphene – electronic properties

• In this case the electron behave as Dirac fermions (They obey the relativistic Dirac equation, and exhibit exotic effects such as Klein tunneling.).

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General Potential Applications:

Transparent conducting electrodes, solar cells, batteries, ultracapacitors, nanoelectronics, THz plasmon oscillators, polarizers, filters, antennas, surface plasmon modulators, interconnects.

Graphene transistor:

• Many research teams (including IBM) have made graphene transistors.

• They can be extremely fast due to very high carrier mobility, but they lack a band gap.

• Typically, a bandgap of the order 1 eV at room temperature is needed for a FET.

• The best efforts to engineer a bandgap in graphene have produced modest band gaps (a few hundred meV). Thus, they dissipate lots of energy in the “off” state.

www.gajitz.com, www.sciencedirect.com,and vorbeck.com

Graphene

32

One “somewhat electronic” commercial example:

• Vorbeck produces Vor-ink™, a graphene-based conductive ink for electronics.

There are some sports equipment (tennis racquets, bike frames, etc.) that purport to use graphene.

It seems that there are no electronic/optical commercial devices yet.

khelmart.wordpress.com, www.gajitz.com, www.sciencedirect.comand vorbeck.com

Graphene

In the optical range the low-temperature conductivity is dominated by the interband term, and 𝜎𝜎 = 𝜋𝜋𝑒𝑒2/2ℎ. The transmittance is |T|²≃1-πα≃97.7%, where 𝛼𝛼 = 𝑒𝑒2𝜂𝜂0/2ℎ is the fine structure constant, which is in excellent agreement with measurements.

Science 6 June 2008 320: 1308.

The intraband Drude conductivity has also been experimentally verified for graphene.

Pauli blocking

Graphene

Special Cluster on Graphene and Two-Dimensional Materials for Antenna Applications. Antennas and Wireless Propagation Letters, Co-guest Editors: Hao Xin and George W. Hanson

Vor-Ink on cardboard.

Graphene

Graphene ink G-102E (BGT Materials, U.K.) on paper.

Graphene

Using graphene strips to form a hyperbolic metasurface

Graphene

2D Mater. 2 ( 2015) 049501

Crystal lattice

Transitional Metal Dichacolgenides

These are essentially semiconductors with parabolic bands.

• They generally have the form MX2, where • M is one of the transition metals: partially filled d sub-shell; Molybdenum

(Mo), Wolfram (W), etc.) and • X a chalcogen atom: Sulfur (S), Selenium (Se), Tellurium (Te)

• One layer of M atoms is sandwiched between two layers of X atoms, forming a very thin structure (for MoS2, a ‘monolayer’ has thickness 0.65 nm, not much more then the ‘thickness’ of graphene).

• Mo and W dichalcogenides are semiconductors with band gaps around 1–2 eV.• The most common varieties are MoS2 and WS2.• TMD layers have a thickness of 0.6–0.7 nm, with strong covalent bonds in-plane

and weak van der Walls interactions out-of-plane.• Monolayer MoS2 has broken inversion symmetry, which leads to interesting

valley physics.• Applications as transistors and optical emitters

and detectors.

A. Gupta et. al, Recent development in 2D materials beyond graphene, Progress in Materials Science 73, 44-126, 2015.

The band gaps of MoS2 and WS2 are indirect for the bulk materials but direct for monolayers.


Primary applications seem to be electronic (transistors), but also can be used for optical applications in the range of energies of its bandgap.

EM Application – Saturable Absorber


Black Phosphorus (and phosphorene)

Phys. Rev. B 89, 201408(R) (2014)

• Black phosphorus (BP) is essentially a narrow-gap semiconductor.• BP forms a puckered surface due to sp3 hybridization. • It is one of the thermodynamically more stable phases of phosphorus, at ambient

temperature and pressure (white/yellow phosphorus is highly flammable and self-igniting upon contact with air - it is stored under water).

• BP is fairly reactive (hours to days), and must be passivated.

BP is highly anisotropic


Phys. Rev. B 89, 201408(R) (2014)

• BP has recently been exfoliated into its thin multilayers (5-30 nm) and single layers (phosphorene), showing good electrical transport properties.

• Optical absorption spectra of BP vary sensitively with thickness, doping, and light polarization.

• Potential for mid- to near-infrared spectrum applications: optoelectronics, imaging, and detection.

• BP band gap is direct for both monolayers and finite-thickness films, but the value of the band gap is controllable, and depends on the number of layers:

• A large gap (~2 eV) for monolayers decreasing gradually to a narrow band gap (about 0.3 eV) for bulkblack phosphorus.

J. Phys. Chem. Lett., 2015, 6 (21), pp 4280–4291

Possible applications: • Electronics (transistors).• Photovoltaic energy harvesting (optimized for

semiconductors with 1.2 eV−1.6 eV band gap)• Fiber optic telecommunications (wavelengths in

the range of 1.2 μm−1.5 μm, corresponding to photon energies of 0.8 eV−1 eV)

• Thermal imaging (typically requires semiconductors with gaps spanning from 0.1 to 1.0 eV).


The band gap spans a wide range of the electromagnetic spectrum, and bridges the gap between graphene (zero band gap) and TMDs (wide band gap).

J. Phys. Chem. Lett., 2015, 6 (21), pp 4280–4291

BP transistors

BP bridges the gap between graphene (very high mobility and poor current on/off ratio) and transition metal dichalcogenides (low mobility and excellent on/off ratio).

Black Phosphorus Optical Conductivity

10nm thick BP obtained at doping level 10x1013/cm2 (a,b) and 5x1012/cm2 (c,d) normalized to 𝜎𝜎0 = 𝑒𝑒2/4ℏ. Regions 1 and 3 show anisotropic inductive and capacitive responses, respectively, and region 2 shows the hyperbolic regime.

Conductivity tensor

𝜎𝜎 =𝜎𝜎𝑥𝑥𝑥𝑥 00 𝜎𝜎𝑦𝑦𝑦𝑦

hyperbolic regime

Some EM applications – hyperbolic surface

𝑘𝑘𝑥𝑥2 + 𝑘𝑘𝑦𝑦2

𝜀𝜀∥+𝑘𝑘𝑧𝑧2

𝜀𝜀⊥= 𝑘𝑘02𝜇𝜇𝑟𝑟

Equifrequency surface for hyperbolic medium, 𝜀𝜀∥ ⋅ 𝜀𝜀⊥ < 0

Can also be implemented using layered metal-dielectric


Isotropic case – black circleHyperbolic case (𝜎𝜎𝑥𝑥𝑥𝑥′′ ⋅ 𝜎𝜎𝑦𝑦𝑦𝑦′′ < 0):

blue and green hyperbola

Slopes of hyperbola asymptotes

𝑞𝑞𝑦𝑦 = ±𝑞𝑞𝑥𝑥 𝜎𝜎𝑦𝑦𝑦𝑦′′ /𝜎𝜎𝑥𝑥𝑥𝑥′′

Different slopes of hyperbolas correspond to different levels of doping.

Equifrequency contour for 2D materials

Hyperbolic Black Phosphorus

Energy flow in anisotropic material is defined by direction of group velocity

𝐯𝐯𝑔𝑔 = 𝛻𝛻𝐪𝐪𝜔𝜔(𝐪𝐪)i.e. orthogonal to equi-frequency surface

Normals to hyperbolas point to the same direction, causing propagation of energy along narrow rays.

Different slopes correspond to different values of doping, allowing for tunable control of the propagation direction.

Directional flow of energy in hyperbolic 2D materials

PRL 116, 066804 (2016)

Ray optics with hyperbolic surface plasmons

Hexagonal Boron Nitride (hBN)

• hBN is essentially an insulator.

• Boron Nitride (BN) arises from the reaction of boric oxide and potassium cyanide.

• The microcrystalline powder form of BN has many uses as a lubricant or coating (as is graphene) when chemical inertness at high temperature is required. It is a white slippery solid.

• BN can assume many crystalline phases; hexagonal (h-BN) including boron nitride and cubic (c-BN) films are most common.

• The spacing between successive layers is 0.334 nm, similar to graphene (0.333 nm). The bond length between two successive B and the N atoms is 1.44 A.

A. Gupta, T. Sakthivel, and S. Seal, Recent development in 2D materials beyond graphene, Progress in Materials Science 73, 44-126, 2015.

Hexagonal Boron Nitride (hBN)

Electronic Applications: 2D hBN has been considered a promising material 2D electronics, including substrates, and gate dielectrics for 2D transistors.Electromagnetic Applications: hBN supports phonon polaritons with extremely high confinement and low loss (much smaller than graphene plasmonspolaritons)• hBN has natural hyperbolicity.• Both graphene plasmons and hBN phonons reside

in the mid-IR; promising as hetrostructures. • hBN is now being used as a substrate of choice for

graphene due to the preservation of high (graphene) carrier mobility, as opposed to conventional SiO2 substrates.

• Phonon modes of hBN can couple to graphene plasmons providing the possibility of interesting effects, e.g., such as phonon-induced transparency. Nano Lett. 2015, 15, 3172−3180

Silicene and Germanene

• Bulk silicon (Si) cannot form a layered phase like graphite.

• Experiments (first done in 20101) show that surface-assisted epitaxial growth can produce 2D monolayers of silicon, termed silicene, showing a honeycomb structure similar to those of graphene.

• Free-standing silicene is expected to have a zero band gap (like graphene), but a tiny gap can be opened in epitaxial silicene, due to the symmetry-breaking induced by the interaction with the substrate.

• Germanene is the germanium analogue of silicone, first produced in 20142.

• As in silicene, it has zero bandgap.

• EM applications?

1Applied Physics Letters 97 (22): 223109, 20102New Journal of Physics 16 (9): 095002, 2014

Summary

2D materials are of interest for many reasons

• No screening - they are very tunable with applied static bias (or optical pumping)

• Some have Dirac cones

• They tend to have very good mechanical and electrical properties

• They come as metals, insulators, and semiconductors

• They don’t take up much space

2d materials and electromagnetic applicationsboag/conf1/pdfs/george_hanson_tel aviv...

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