2D Materials 2D Materials Polaritonics Polaritonics 2D ...

2D Materials 2D Materials PolaritonicsPolaritonics-- QuickQuick tutorial tutorial --

Tony LowUniversity of Minnesota, Minneapolis, USA

Email: tlow@umn.eduWeb: http://people.ece.umn.edu/groups/tlow/

IMA UMN, 6-10th Feb 2017

About us

Nanoelectronics Nanophotonics

Multiphysics and multiscale modeling of 2D materials electronics and photonics for computing and communication devices.

Mission:

• 2D materials polaritonics• Photodetectors• Reflectarray• Modulators• Sensors

• 2D materials and transport physics• Tunneling devices• Spintronics• Valleytronics• Strain and piezoelectronics

Graphene

plasmon

Polaritons – marrying the best of both worlds

Transition metal

dichalcogenides

exciton

Quick overview

Basics on graphene plasmonsA pedagogical tutorial on graphene plasmonics starting from Maxwell eq.

Graphene plasmonics Graphene plasmonics A review on graphene plasmonics experiments and its applications

Beyond graphene plasmonicsA forward looking perspective on what’s new with other 2D materials

Polaritons in 2D materials

Graphene Boron nitride Transition metal dichalcogenides

T.Low, J.Caldwell, F. Koppens, L.M.Moreno, P. Avouris, T. Heinz et al, Nature Materials (2016)

Plasmon as collective electronic excitations

External perturbation screened within a Thomas Fermi length

( )( ) 1 ( ) 1

σ ωε ω χ ωωε

= + = + ( ) 0plε ω ω= =

Collective electronic oscillation i.e. plasmons

2D materials carrier concentration tunable up to 0.01 electrons per atom THz and mid-IR plasmon 6

Technologies across electromagnetic spectrum

Terahertz to Mid-infrared Contains atmospheric transmission window Super high-speed wireless communication Imaging for military, security & medical Detections of molecules for bio. and chem.

Possible applications for graphene plasmonics

IBM, Nature Nano (2012)

EPFL, Science (2015)

IBM, Nature Com (2013)

IBM, Nature Nano (2012)

U Penn, Science (2012)

Far field communications, e.g. modulator, reflectarray for far-field MIR

Applications of polaritons in 2D materials

T.Low, J.Caldwell, F. Koppens, L.M.Moreno, P. Avouris, T. Heinz et al, Nature Materials (2016) 9

Quick overview

( , )( , )

B r tE r t

∂∇× = −∂

Faraday’s Law

( , )( , ) ( , )

D r tH r t J r t

∂∇× = +∂

Ampere’s Law

( , ) 0, ( , ) ( , )B r t D r t r tρ∇ ⋅ = ∇ ⋅ = Gauss’s Law

Maxwell equations in SI units

Constitutive relations

Maxwell equations in a nutshell

Constitutive relations

( , ) 0r t

J r tt

ρ∂ + ∇ ⋅ =∂

( , ) ' ( ', ') ( ', ')

D r t dt dr r r t t E r t

B r t dt dr r r t t H r t

J r t dt dr r r t t E r t

−∞

= − − ⋅

∫ ∫

Fields are related by permittivity,

permeability, conductivity tensors

Continuity equation

Ohm’s law11

Assumptions

• Mediums are not spatially dispersive, i.e. local response

• Linear response, monochromatic waves

• No free charges or currents

( , ) ( ) ( , )E r i H rω ωµ ω ω∇× = ⋅

( , ) ( , )exp( )E r t E r i tω ω→ −

( , ) ( ) ( , )H r i E rω ωε ω ω∇× = − ⋅

( ) ( , ) 0E rε ω ω∇ ⋅ ⋅ =

Maxwell equations Boundary conditions

( ) 0n j i

e E E× − =

( ) ( , )n j i

e H H J r ω× − =

Current at interface to be included i.e.

where 2D materials is also described!

( ) ( , ) 0E rε ω ω∇ ⋅ ⋅ =

( ) ( , ) 0H rµ ω ω∇ ⋅ ⋅ =

( , ) ( ) ( , )D r E rω ε ω ω= ⋅ ( , ) ( ) ( , )B r H rω µ ω ω= ⋅

Constitutive

where 2D materials is also described!

( , ) ( ) ( , )E r i H rω ωµ ω ω∇× = ( , ) ( ) ( , )H r i E rω ωε ω ω∇× = − ( , ) 0E r ω∇ ⋅ =

Maxwell equations

In the isotropic case,

( , ) ( ) ( , ) ( ) ( ) ( , )

( ( , )) ( , ) ( ) ( ) ( , )

E r i H r E r

E r E r E r

ω ωµ ω ω ω µ ω ε ω ωω ω ω µ ω ε ω ω

∇×∇× = ∇× =∇ ∇ ⋅ − ∆ =

( , ) 0E r ω∇ ⋅ = ( , ) 0H r ω∇ ⋅ =

( , ) ( ) ( , )D r E rω ε ω ω= ( , ) ( ) ( , )B r H rω µ ω ω=

Constitutive

( ( , )) ( , ) ( ) ( ) ( , )

( , ) ( ) ( ) ( , )

E r E r E r

E r E rω ω µ ω ε ω ω∆ = −

Solution

0( , ) ( , )exp( )E r E k ik rω ω= ⋅

22 2 2

( , ) ( ) ( ) ( , )

( ) ( )

k E r E r

ω ω µ ω ε ω ωωω µ ω ε ω

− = −

− = − ⇒ =

2exp( ) exp( )ik r k ik r∆ ⋅ = − ⋅Using

We are interested in finding EM modes localized at the interface, 2D material

x zk e ieβ γ= ±

This localized EM mode is reflected in the following ansatz

0 exp( )exp( ) for 0( , )

exp( )exp( ) for 0

A i x z zA r t

A i x z z

β γβ γ

− >= <

Confined EM modes, TE plasmons

0 exp( )exp( ) for 0A i x z zβ γ <

We start with the electric field for the TE plasmons

1 1 1 1( ) ( , ) exp( )exp( ), 0y y

E r e E x z e E z i x zγ β= = <

2 2 2 1( ) ( , ) exp( )exp( ), 0y y

E r e E x z e E z i x zγ β= = − >

0where j j

γ β ω µ ε= −

The magnetic field takes the form,

( )1 11 1 1

( , ) ( , )1 1( , ) ( , )

z x z x

E x z E x zH x z e e e i e E x z

i x z iβ γ

ωµ ωµ∂ ∂ = − = − ∂ ∂

( )2 2 2

1( , ) ( , )

z xH x z e i e E x z

iβ γ

ωµ= +

From boundary conditions

1 2E E=

2 12 1 1

E E i Eγ γ σωµ µ

We obtain the solution for electric field

11 2 0

( ) 0E

iγ γ σωµµ

+ − =

Which has non zero solution if,

Confined EM modes, TE plasmons

1 2 0 0iγ γ σωµ+ − =This is also the pole of the Fresnel coefficients for TE waves!

We can obtain plasmon dispersion in free-standing case,

2 2 2 2 22 2 2 20 0

0 0 0 0

0 0 0 0 0 0

2 14 4

where , . Thus the TE plasmon is

σ ω µ σ ηγ σωµ β β

ω ε µ η µ ε

σ ηβ

= → − = − → = −

= −15

x zk e ieβ γ= ±

This localized EM mode is reflected in the following ansatz

exp( )exp( ) for 0( , )

exp( )exp( ) for 0

A i x z zA r t

A i x z z

β γβ γ

− >= <

We start with the magnetic field for the TM plasmons

1 1 1 1

2 2 2 2

( ) ( , ) exp( )exp( ), 0

H r e H x z e H z i x z

γ βγ β

= = − >

0where j j

γ β ω µ ε= −

The electric field takes the form,

Confined EM modes, TM plasmons

1( , ) ( , )

E x z e i e H x zi

β γωε

= − −

= − +

From boundary conditions

1 2 11 2 2 1 1

2 1 1 1 21 1 2 2 1

we obtain 1 0 0

H H H H Hi

γ γ γσε ε ωε

ε γ γ γ γσ ε γ ε γ σε γ ωε ω

+ = − = −

+ − = → + − =

We can obtain plasmon dispersion in free-standing case,2 2 2

2 20 00 0 0 0 0

0 0 0 0 0 0

42 0 2

where , . Thus the TM plasmon is

σγ ε ωε γ γ ε ω σ βω σ

ω ε µ η µ ε

βσ η

− = → = → − = −

This is also the pole of the Fresnel coefficients for TM waves!

Confined EM modes, TM plasmons

The equation of motion of free electrons in metal

electron

momentum

relaxationtime

, we also have

Hence,

p mvp dpeE

J envdt

dJ J neE

=− − =

Assuming time dependence exp( ) , exp( ), we obtain, E E i t J J i tω ω= − = −

Drude conductivity

Assuming time dependence exp( ) , exp( ), we obtain,

( 1 ) ( )

To map the relation to graphene, we use the relation,

E E i t J J i t

ne ne ii J E

E km n

ω τ σω τ

= − = −

− + = → =+

where (also known as Drude weight)( )

Fe EiD

σω τ π

= ≡+ ℏ

TE plasmons, 14

4TM plasmons, 1

σ ηβ

βσ η

2 where

Fe EiD

σω τ π

= ≡+ ℏ

η µ ε σ

−= = Ω = ×∼ℏ

∼ ≪

Lets consider some typical numbers,

376.6 and | | 6.1 104

Hence, | | 0.02 1. This implies that,

Graphene plasmons

η σβ

ω ωπ π ε ωβση η η

∼ ≪

ℏ ℏ∼ ∼ ∼

0 0 00 2 2

Hence, | | 0.02 1. This implies that,

TE plasmons,

2 2 22TM plasmons,

D e E e E

βω βπ ε ε

= =ℏ

Quick overview

E Light

Maxwell

Exciting plasmons in graphene

0.2eV+ -+

e qωπ

µε ε

M.Jablan et al, Phys. Rev. B (2009)F. Koppens et al, Nano Lett. (2011)

L.Ju et al, Nature Nano (2011)H.Yan et al, Nature Nano (2012)

J.Chen et al, Nature (2012)Z.Fei et al, Nature (2012)

T.Low and P.Avouris, ACS Nano (2014)H.Yan, T.Low et al, Nature Phot.(2013)

Understand Graphene plasmonic resonator and what we can do with it

RPA Loss function1( , ) Im RPAL q ω ε − =

e qµωπ ε ε

Mid-infrared plasmons with graphene nanostructures

1 2~ , ( tan [ 4 / (4 )])RR

π π π−− Φ Φ = − +

momentum

A.Y.Nikitin, T.Low, L.M.Moreno, PRB Rapid (2014)

H.Yan, T.Low et al, Nature Photonics (2013)T.Low and P.Avouris, ACS Nano (2014)

( , ) 1 per

Tω = −

Measuring extinction:

20Optical Phonon

Peak 3Peak 2

Width (nm) 60 70 85 95 100 115 125 140 150 170 190 240

2nd order

Peak 1

Graphene on SiO2

First peak Second peak Third peak 2nd order mode ω ~ q1/2

Mid-infrared plasmons with graphene nanostructures

1000cm-1 ~ 10um ~ 30THz

1000 1500 2000 2500 3000

Wave number (cm-1)0 2 4 6 8

1000ω

Wave vector q (x105 cm-1)

H.Yan, T.Low, F.Guinea et al, Nature Phot. (2013)T.Low and P.Avouris, ACS Nano (2014)

Plasmon dispersion can be engineered with substrates

( ) ( ) ( )

βε β ω β ω β ωβε π ω τ

ε β ω

= − ∏ = ∏ =+

0Coulomb Polarizabilitypotential

Dielectric function of graphene (free electron contributions only)

, 1 , where and ,2 ( )

Plasmon occurs when , 0, ignori

ββε π ω

ββε ω

→ = −

ng damping, we have,

Graphene plasmons

βε ωβ βω ωε ε

ωε β ω

→ = −

→ = → =

= −+

Hence, we can also express dielectric function as,

, 1( )

αε ω ωω τ ω

ω αε β ωω τ ω τ ω

ε β ω

=+ −

= − −+ + −

Dielectric function of polar phonons can be described by

where is phonon frequency( )

Total dielectric function becomes

, 1( ) ( )

Hybrid modes when , 0, assu

ω α− − =2

ming again zero damping,

Plasmons and phonons hybridization

ω αω ω ω

ω ω ω ω ω ω αω

ω ω ω ω ω ω ω αω

ω ω ω ω α ω ω

ω ω α ω ωω ω αω

− − =−

− − − − =

− − + − =

− + − + =

+ − −+ −= ±

2 2 2 2 2 2 2

4 2 2 2 2 2 2 2

4 2 2 2 2 2

2 2 2 2 22 20 002

( ) ( ) 0

pl plpl

E3x106 ms-

13x106 ms-1

LightDirac e100 x

100 xSound Vel.

op BE k T≫

Graphene, Dirac electrons

Highest mobility µ=1x106 cm2Vs-1

D.C.Elias et al, Nature Physics (2011)

Quantum Hall effect at Room Temp.

K.S.Novoselov et al, Science (2007)

( )Re σ ω

ωFrequency

VisibleNear-IRMid-IR

Terahertz

απ=2% light absorption

Graphene absorption spectrum

Intraband

Z.Q.Li et al, Nature Physics (2008)

R.R.Nair et al, Science (2008)

Disorder-

mediated

Interband

1eV ~ 8000cm-1 ~ 1.25um ~ 240THz 27

We seek the scattering coefficients (transmission, reflection), due to a 2D material at z=0

for plane monochromatic waves i.e.

Fresnel coefficients

0( , ) exp( )A r A ik r i tω ω= ⋅ −

for plane monochromatic waves i.e.

Without loss of generality, assume xz being plane of incidence2 2 2 where | | and

x zk e e k k kβ γ ω εµ β γ= + = = + =

1tan ( )

βθ γ−=

Angle of incidence

Any EM waves can be expressed as linear superposition of TE and TM waves.

Incident EM wave gives rise to reflected wave in medium 1 and transmitted wave in

medium 2

1 1 1 1( ) ( , ) [ exp( ) exp( )]exp( )y y i r

E r e E x z e E i z E i z i zγ γ β= = + −

2 2 2( ) ( , ) exp( )exp( )y y t

E r e E x z e E i z i zγ β= =

The magnetic fields can be obtained from Maxwell equation i.e. Faraday’s law

1 1 1 1 1 1

1( ) [ ( ) ( exp( ) exp( ))exp( )]

z x i rH r e E r e E i z E i z i xβ γ γ γ γ β

ωµ= − − −

Fresnel coefficients, TE waves

2 2 2 2

1( ) [ ( ) exp( )exp( )]

z x tH r e E r e E i z i xβ γ γ β

ωµ= −

Using boundary conditions

2 1 1 2( ) 0 ( ,0) ( ,0)z

e E E E x E x× − = → =

2 1 1 2 1 2( ) ( ,0) ( ,0)z y x x

e H H e E x H H E xσ σ× − = → − =

i r tE E E+ =

( )t i r t

E E E Eγ γ σωµ µ

− + − =29

Then transmission and reflection coefficients are

γγ γ σωµ

= =+ +

γ γ σωµγ γ σωµ

− −= =+ +

We measure reflection and transmission probably with respect to energy carried by the

EM wave, using the Poynting vector,

2 21 11 1 1

1 1( 0) Re[ ( ( ) ( ))] Re[( ) ( ) ] | | | |

2 2 2z i r i r i r i r

S z e E r H r E E E E E E S Sγ γ

ω µ ωµ

∗∗ ∗ ∗

= = ⋅ × = − + − − = − = −

Re[ ]1 1 γ γ∗∗ ∗

= = ⋅ × = − = =

Fresnel coefficients, TE waves

22 22 2 2

Re[ ]1 1( 0) Re[ ( ( ) ( ))] Re[ ] | |

2 2 2z t t t t

S z e E r H r E E E Sγ γ

ω µ ωµ

∗∗ ∗

= = ⋅ × = − = =

Then, the probabilities are

| || |

S ER r

S E= = = 22

Re[ ]| |t

Total internal reflection,

1 2 2 2 2 and is pure imaginary, 0k k Tε ε β γ β> > → = − =

α βσ ωω τ

−= −

− + + −

ℏℏ

, ', , ' ' '

Optical conductity can be computed from the Kubo formula,

', ' , , ', '( ( ')) ( ( ))( , )

( ') ( ) ( ) ( ') ( )

where ' , as is due to ph

k k j j j j j j

k j v k j k j v k jf E k f E kq ie

E k E k i E k E k

k k q q∼

oton momentum.

In this work, we are interested in the local limit, i.e. 0q

Graphene low energy Hamiltonian at point,

Optical conductivity of graphene

= ⋅ = +

= ± Ψ = ± +

ℏ ℏ

ℏ, ,

with the following eigenvalues and eigenfunctions,

11 and

we also have the veloci

c v F c v

q iqH v v

E v qq iq q

σ σ= =

= = − = −, , ,

ty operators

and and

x F x y F y

x cc F x vv F x cv F

v v v v

ikk kv v v v v v

k k k31

α βσ ωω τ

π ω τ

−= −

− + + −

+ −= −+

ℏℏ

, ', , ' ' '

Consider intraband contributions in conduction band

', ' , , ', '( ( ')) ( ( ))(0, )

( ') ( ) ( ) ( ') ( )

( ( 0)) ( ( ))4

k k j j j j j j

k j v k j k j v k jf E k f E kie

E k E k i E k E k

f E k f E kied k

π ω τ

θ θπ ω τ π ω τ

∞ ∞

∂= − + ∂+ −

∂ ∂ = − = − + ∂ + ∂

∫ ∫

∫ ∫ ∫

2 2 22 2

2 2 22

2 20 0 0

( ( ))( )

( )( 0) ( )

( ) ( )cos

( ) ( )

f E kie v kv k d k

i E kk E k

f Eie v ie f Ed dkk dEE

i E i E

σ ωπ ω τ

−∞ ∂ = − + ∂ ∫ℏ

Straightforward to show that contributions from valence band

( )(0, )

ie f EdEE

σ ωπ ω τ π ω τ

π ω τ

∞ −∞ ∞

−∞

∂ ∂ ∂ = − + − = − + ∂ ∂ + ∂

∫ ∫ ∫ℏ ℏ

2 20 0

Thus the total intraband conductivity is

( ) ( ) ( )( )

( ) ( )

2ln 2cosh

ie f E f E ie f EdEE dEE dE E

i E E i E

ik Te E

σ ωπ ω τ

≈ →

Thus the total intraband conductivity is

2( ) ln 2cosh

For low temperture, , we can simplify further. In this case,

1cosh exp ln 2cosh

ik Te E

k T k T

σ ωπ ω τ

Hence, we obtain, ( )( )

k T k T

π ω τ+ℏ ( )i

ω τσ ωπ ω τ π ω τ

∞ + − −= + + + −

∫ℏ ℏ

2 2 2 20

Including both intraband and interband conductivity, we get,

( )2 ( ) ( )( ) ln 2cosh

( ) 2 ( ) 4

For low temperture, , we can simplify to,

ie iik Te E f E f EdE

i k T i E

ie ω τπ ω τ π ω τ

− ++ + + +

ℏ ℏ ℏ

2 ( )ln

( ) 4 2 ( )

E iE ie

Universal optical conductivity

0 0 0 0

The absorption is defined as

and we can show that

4( ) Re

γ σωµγ σωµ γ σωµ

γ ωµω σγ σωµ

= − −

−= =+ +

Nair et al, Science (2008)

ℏ ℏ

ℏ ℏ ℏ

Note for high frequency,

2 ( )( ) ln

4 2 ( )

2 ( )Re[ ( )] Im ln

4 2 ( ) 4

E iie e

ω τσ ωπ ω τ

− +→ ∞ ≈ + +

− +→ ∞ ≈ = + +

[ ][ ] ℏ

For normal incidence and high frequencies,

Re 0.022

where 377 and Re4

γ σωµ

Photocurrent mechanisms

Bipolar junction

PVI en µξ∗≈

( )I S S Tσ δ∗≈ −

Visible light

Unipolar junction

PVI en µξ∗≈

1 2( )PTEI S S Tσ δ∗≈ −

N.Gabor et al, Science (2012)35

Photoconductivity experiment

M.Freitag, T.Low et al, Nature Phot.(2013) 36

Bolometric vs photovoltaic

0GV ≈PCI DCI

Photovoltaic

0GV ≫

PCI DCI

Bolometric

M.Freitag, T.Low et al, Nature Phot.(2013) 37

Non-radiative decay < ps

Mid infrared plasmons Thermal photo-response Room temperature operation

Mid-infrared photodetector

Key Ingredients

M.Freitag, T.Low et al, Nature Comm. (2013) M.Freitag, T.Low et al, Nature Photonic (2013)

Drives a bolometric current

-40 -20 0 20 400.0

W=140nm

Gate Voltage VG (V)

Mid-infrared photodetector

0 2 4 60

Wave vector q (x105 cm-1)

Intraband Landau Damping

M.Freitag, T.Low et al, Nature Comm. (2013)

Gate tunability, thanks to hybrid plasmon-phonon polariton

Driven mechanical oscillator Credit: MIT TechTV

Resonator can acquire a phase from its driving force which is determined by its detuning from resonance 40

Vg=1V Vg=3V Vg=4V

Electrically controlled terahertz and mid-infrared beam reflectors

Mid-infrared light bending

Graphene

High-κ dielectric

Metal Reflector

C.Eduardo, T.Low, et al, Nanotechnology (2015)

A. Nemilentsau, T.Low, arXiv:1610.05236 (2016)

sin( ) sin( )2

λ φθ θπ ε

Generalized Snell’s law

ω µφ ωτ ε

− ≈ −

Graphene plasmonics for THZ and MIR applications

T.Low and P.Avouris, ACS Nano (2014) 42

Quick overview

A new class of 2D crystals

Eg=0 eV Eg=6.0 eV

TMOs (Transition Metal Oxides): MoO3, LiCoO2

Graphene Boron Nitride

TMDs (Transition Metal Dichalcogenides): MoS2,WS2,NbSe2 (MX2)

III-VI/V-VI Compounds (Ga,In)2Se3, Bi2(Se,Te)3

Strong in-plane bonds Weak van der Waals interlayer coupling Surfaces ideal self-passivation, intrinsically

good electrical properties Pathway for large scale growth Full range of material properties Black Phosphorus 44

Polaritons in 2D materials

Graphene Boron nitride Transition metal dichalcogenides

Mid-infrared plasmons with near field optical micro scopy

J.Chen et al, Nature (2012)Z.Fei et al, Nature (2012)

Complex wavevector q of plasmon can be measured as function of frequency ω

Re[ ]qπ

exp( Im[ ] )q x−

Spacing between fringes

Exponential decay of fringes

Figure of merits

A.Woessner et al, Nature Mat (2015)

Figure of merits

Im[ ] Re[ ]q qγ =1 2γ π− Number of cycles plasmon propagates

before amplitudes decay by 1/e

0Re[ ]q kβ =

Damping

Confinement Light confinement by the polariton mode

Current state-of-the-art, γ-1 >25 and β~150

Mid-infrared plasmons with near field optical micro scopy

bilayer

monolayer

Goldgraphene

F. Koppens et al, Nature Materials 2015R. Hillenbrand et al, Science, 2015 47

Comparison of plasmon figure of merits

2D materials has better FOM than 1nm Au in the infrared

Plasmonics beyond graphene

Phonon induced transparencySlow light

T.Low et al, Phys. Rev. Lett. (2014)H.Yan, T.Low et al, Nano Lett. (2014)

Anisotropic plasmonHyperbolic plasmon

T.Low et al, Phys. Rev. Lett. (2015)A.Nemilentsau, T.Low et al, Phys. Rev. Lett. (2016)

Non-reciprocal plasmonGiant Faraday rotation

A.Kumar et al, PRB Rapid. (2015)

From monolayer to bilayer graphene

σ/σ 0]

phononinterband

T.Low, F.Guinea et al, Phys. Rev. Lett. (2014)

0.0 0.2 0.4 0.6 0.80.0

Frequency ω (eV)

Pronounced plasmonic enhancements of IR phonon absorption

Narrow optical transparency window at zero detuning

Plasmon coupled to interband resonance 50

Observing phonon-induced transparency

Higher dopingW = 100 nm

Chemical doping

T.Low, F.Guinea et al, Phys. Rev. Lett. (2014)

Substrate

Dielectric

SiO2/Si

Experiments

1200 1500 1800 21000

Frequency (cm-1)

1200 1500 1800 21000

Frequency (cm-1)

Higher doping

W = 100 nm Chemical doping

H.Yan, T.Low et al, Nano Lett. (2014)

plasmon phonon

From narrow absorption to narrow transparency 51

Coupled two harmonic oscillators

1κ 2κ1 1,m γ 2 2,m γ

cos( )F tω

1 1 1 1 1 1 1 2

2 2 2 2 2 2 2 1

( ) cos( )

m x x x x x F t

m x x x x x

γ κ κ ωγ κ κ

+ + + − =+ + + − =

ɺɺ ɺ

Equation of motion:

1 2γ γ≫

“plasmon” “phonon”

1.90 1.95 2.00 2.05 2.100.0

Freq. ω

1 2ω ω>Narrow

absorption

1.51 2ω ω=Narrow

cos( )

ω θω θ

= += +

Solutions:

Power absorption by “plasmon”:

11 1 12

( ) cos( ) ( )

( ) sin( )

P t F t x t

P t Fa

ωω θ

== −

1.90 1.95 2.00 2.05 2.100.0

Freq. ω

1 2ω ω=Narrow transparency

Destructive interference, mass becomes stationary

From induced-transparency to slow light

( ) ( ) ( )g

dω ω ω ω

( )( )g

( ) Re ( )n ω ε ω =

Highly dispersive n will yields Induced-transparency

Highly dispersive n will yields modified group velocity

A.H.Safavi-Naeini et al, Nature (2011)T.J.Kippenberg et al, Science (2008)

Slowing light for integrated photonic memory

Anisotropic 2D materials

Class of anisotropic materials

Highly anisotropic in-plane effective masses

Intraband optical process

Anisotropic intraband Drude conductivity

Interband optical process

Grp 5, e.g. black phosphorus1T TMD, e.g. ReS2, ReSe2

Transition metal trichalcogenides

L.Li et al, Nature Nano. (2014)S.Tongay et al, Nature Comms. (2014)

Interband conductivity also anisotropic

J.Qiao et al, Nature Comms. (2014)

Absorption edge at different energies for different polarization due to symmetry of the bands

Anisotropic 2D materials, optical conductivity

( ) lng

ie n is

ω ωσ θ ω ωω η π ω ω

−= + − + + +

Graphene

Intraband Interband

σσσ

( ) ln ,jjj j j

ie n is j x y

ω ωσ θ ω ω

ω η π ω ω −

= + − + = + +

Anisotropic semiconductor

Imaginary part follows from Kramers-Kronig

Im( ) Im( ) 0xx yyσ σ >

Im( ) Im( ) 0xx yyσ σ <Hyperbolic

Anisotropic

Anisotropic versus Hyperbolic plasmons

Anisotropic case Im( ) Im( ) 0xx yyσ σ > Hyperbolic case Im( ) Im( ) 0xx yyσ σ <

Hyperbolic plasmons, ray optics

Hyperbolic plasmon rays can be electrically control

A.Nemilentsau, T.Low, G.Hanson, Phys. Rev. Lett. (2016)57

Massive Dirac systems

Graphene/hBNR.V.Gorbachev et al, Science (2014)

Valley hall current Optical circular dichroism

TMDK.F.Mak et al, Nature Nano (2012)R.V.Gorbachev et al, Science (2014) K.F.Mak et al, Nature Nano (2012)H.Zeng et al, Nature Nano (2012)X.Xu et al, Nature Phys. (2014)

MoS2K.F.Mak et al, Science (2014)

( )1( )nE k e

v E kk

∂= − × Ω

ℏ ℏ

These phenomena are static and dynamic manifestation of Berry physics

Berry curvature: can be viewed as an effective magnetic field due

to orbital part Bloch states

( ) ( )n nK KΩ = Ω −

( ) ( )n nK KΩ = −Ω −K and K’ valleys are related by time reversal symmetry

Spatial inversion symmetry requires thatBreaking spatial inversion

produces finite Berry curvature i.e. sublattice asymmetry

Massive Dirac system, optical conductivity

σ σσ

= − '

σ σσ

σ σ−

22( ) ( )xy K K

ek k dkσ ρ= Ω∫ℏ

Optical pumping to create non-equilibrium valley imbalance

A.Kumar et al, PRB Rapid. (2015) 59

Non-reciprocal edge modes

0 20 0 0

( )g g xyi i k

σ σ σε κκ ωε ε ε

+ ⋅ − + =

2 20k qκ ε= −

Bulk plasmons, continuous film

[3 2 2 sgn( )] | | 0g xyg xyq i q q

σ σσ σ

ωε ε +

− + + =

Edge plasmons, semi-infinite film

Linear dipole

Massive Dirac material

A.Kumar et al, PRB Rapid. (2015) 60

Isotropic plasmon(graphene)

bilayer

monolayer

Gold graphene

Experimental realization?

Graphene plasmons can be launched by nano Au optical

antenna, and mapped with SNOM

P.Alonso-Gonzalez et al, Science. (2014)

Chiral plasmon(gapped Dirac materials)

Hyperbolic plasmon(anisotropic materials)

Phonon-polaritons in boron nitrides

Optical mode

Acoustic mode

S.Dai et al, Science (2014)

hBN as natural hyperbolic material

2 2, ,

, , 2 2,

LO m TO mm m m

TO m mi

ω ωε ε ε

ω ω ω∞ ∞

−= +

− − Γ

hBN permittivity

Out-of-plane phonon modes

1 1, ,780cm , 830cmTO LOω ω− −= =

In-plane phonon modes

1 1, ,1370cm , 1610cmTO LOω ω− −= =

Elliptic Hyperbolic type I Hyperbolic type II 63

Phonon-polaritons in hBN

Dispersion within the Reststrahlen band

A.Kumar, T.Low, et al, Nano Lett. (2015)S. Dai et al, Science (2014)

A.Woessner, Nature Mat. (2014)

1 0( ) 2 tanhBN

εψω πε ψ

= − +

ψ ε ε⊥= ±

Plasmon-Phonon-polaritons in graphene-hBN

A.Kumar, T.Low, et al, Nano Lett. (2015)S. Dai et al, Science (2014)

A.Woessner, Nature Mat. (2014)

( )0 0 01 1 0( ) tan tanhBN

i q k Zq n

ε σ εψω πε ψ ε ψ

− −

⊥ ⊥

+ = − + +

( ) ( )( )( ) ( )( )

( )( )

0 0 0 0

1( ) ln

i i q k Z

i i q k Ziq

ψ ε ε σ εψ ε ε σ εψω

ψ ε εψ ε ε

− + + + = − × +

D. Basov et al, Science, 2014 J. Caldwell et al, Nature Comm. 2014 Hillenbrand et al, Nature Phot. 2015

Hyperbolic phonon polaritons in hBN

Hyperbolic polaritons beyond hBN

T.Low et al, Nature Materials (2016) 67

Exciton polaritons in 2D materials

Designers’ polaritons with 2D heterostructure

Acknowledgement

IBMHugen Yan, Marcus Freitag, Fengnian XiaWenjuan Zhu, Damon Farmer, Phaedon Avouris

SpainFrancisco Guinea, Luis Martin Moreno,Alexey Nikitin, Rafael Roldan, Frank Koppens

MIT – Nick FangU Wisconsin Milwaukee – George HansonNRL – Josh CaldwellNRL – Josh CaldwellStanford – Tony HeinzBrazil – Andrey Chaves

UMNRoberto Grassi, Eng Hock Lee, Yongjin Jiang, Kaveh Khaliji, SudiptaBiswas, Javad Azadani, Anshuman Kumar, Andrei Nemilentsau

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