2D Materials Polaritonics -- Quick tutorial --

Tony Low University of Minnesota, Minneapolis, USA Email: [email protected] Web: http://people.ece.umn.edu/groups/tlow/ IMA UMN, 6-10 th Feb 2017 About us

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

Nanoelectronics Nanophotonics

• 2D materials and transport physics • 2D materials polaritonics • Tunneling devices • Photodetectors • Spintronics • Reflectarray • Valleytronics • Modulators • Strain and piezoelectronics • Sensors 2 Polaritons – marrying the best of both worlds

plasmon - - + + - - - - + + - -

Graphene

exciton + - - + - +

Transition metal dichalcogenides

3 ß Quick overview

ß Basics on graphene plasmons A pedagogical tutorial on graphene plasmonics starting from Maxwell eq.

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

ß Beyond graphene plasmonics A forward looking perspective on what’s new with other 2D materials

4 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 , 5 Nature Materials (2016) Plasmon as collective electronic excitations

σ ω( ) ε χ ω ω()1=+ ()1 =+ ε ω ω = = ωε (pl ) 0 i 0 External perturbation screened Collective electronic oscillation within a Thomas Fermi length 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 ß Contains atmospheric transmission window ß Super high-speed wireless communication ß Imaging for military, security & medical ß Detections of molecules for bio. and chem.

7 Possible applications for graphene plasmonics

IBM, Nature Nano (2012)

EPFL, Science (2015) IBM, Nature Com (2013)

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

U Penn, Science (2012)

8 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 ß Quick overview

ß Basics on graphene plasmons A pedagogical tutorial on graphene plasmonics starting from Maxwell eq.

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

ß Beyond graphene plasmonics A forward looking perspective on what’s new with other 2D materials

10 Maxwell equations in a nutshell Maxwell equations in SI units ∂B( r , t ) ∇×E( r , t ) =− Faraday’s Law ∂t ∂D( r , t ) ∇×Hrt(,) = + Jrt (,) Ampere’s Law ∂t ∇⋅Brt(,) =∇⋅ 0, Drt (,) = ρ (,) rt Gauss’s Law

Constitutive relations ∂ρ(r , t ) +∇⋅J(,) r t = 0 Continuity equation ∂t t Drt( , )=∫ dtdr ∫ 'ε ( r −−⋅ rt ', t ') Ert ( ', ') −∞ Fields are related by permittivity, t permeability, conductivity tensors Brt( , )=∫ dtdr ∫ 'µ ( r −−⋅ rt ', t ') Hrt ( ', ') −∞ t Jrt( , )=∫ dtdr ∫ 'σ ( r −−⋅ rt ', t ') Ert ( ', ') Ohm’s law −∞ 11 Maxwell equations in a nutshell Assumptions • Mediums are not spatially dispersive, i.e. local response • Linear response, monochromatic waves Ert(,)→ Er (, ωω )exp( − it ) • No free charges or currents

Maxwell equations Boundary conditions ∇×ω ωµ =ω ω ⋅ × − = Er(,) i () Hr (,) en( E j E i ) 0 × − = ω ∇×Hr(,) ωεω ω ω =− i ()(,) ⋅ Er en( H j H i ) Jr (,) Current at interface to be included i.e. ∇⋅ε ω ω () ⋅E (,) r = 0 where 2D materials is also described! ∇⋅µ ω ω () ⋅H (,) r = 0

Constitutive Dr(,)ω ε ω ω = () ⋅ Er (,) Br(,)ω µ ω ω = () ⋅ Hr (,)

12 Maxwell equations in a nutshell

In the isotropic case,

Maxwell equations ∇×Er(,)ω ωµ = ω i ()(,) Hr ∇×Hr(,) ωεω ω =− i ()(,) Er ∇×∇×Eri(,)ω ωµ ω ω = ω µ ω ε ω () ∇× Hr (,) = 2 ()()(,) Er ∇∇⋅ ω ωω µ ω ε −∆ ω = 2 ∇ ⋅E(, r ω ) = 0 (Er (,)) Er (,) ()()(,) Er ∆Er(,)ω ω µ ω = ε − ω 2 ()()(,) Er ∇⋅H(, r ω ) = 0

Constitutive Solution ω ε ω = ωω = ⋅ Dr(,) ()(,) Er Er(, ) Ek0 (, )exp( ikr ) Br(,)ω µ ω = ()(,) Hr −kEr2(,)ω ω µ ω = ε − ω 2 ()()(,) Er ω2 −k2 = − ω µ 2 ω ε( ω )( ) ⇒ k 2 = v2 Using ∆exp(ikr ⋅=− ) k2 exp( ikr ⋅ ) 13 Confined EM modes, TE plasmons

We are interested in finding EM modes localized at the interface, 2D material z ε 2 z |E | x ε 1 This localized EM mode is reflected in the following ansatz Aexp( γ ixβ )exp(− z ) for z > 0 k= e γβ ± ie A( r , t ) =  0 x z γβ <  A0 exp( ix )exp( z ) for z 0 We start with the electric field for the TE plasmons = = βγ < Er1() eExzeEy 1 (,) y 11 exp( z )exp( ix ), z 0 β ωγ µ ε=2 − 2 where j0 j ==− βγ > EreExzeE2( )y 2 ( , ) y 21 exp( z )exp( ix ), z 0 The magnetic field takes the form, 1∂Exz(,) ∂ Exz (,)  1 Hxz(,) = e1 − e 1 =−() eieExz γβ ( , ) 1 ωµ ωµ z∂ x ∂  zx 1 1 i1  x zi  1 1 Hxz(,)=() ei γβ + e Exz (,) 2ωµ z x 2 2 i 2 14 Confined EM modes, TE plasmons

From boundary conditions = E1 E 2 γγ 2E+ 1 EiE = σω µµ 2 1 1 0 0 We obtain the solution for electric field E ( σωµ γ + −i )1 = 0 1 2 0 µ 0 Which has non zero solution if, σωµ γ + − = 1 2i 0 0 This is also the pole of the Fresnel coefficients for TE waves!

We can obtain plasmon dispersion in free-standing case,

σ ησ ω222 µ  22  ω β σωµγ β = →−=−220 →= 22 − 0 20i 0 k 0 k 0  1  4 4  =ω ε µ µ η ε = where k0 000 , 00 . Thus the TE plasmon is σ η2 2 β = − 0 k0 1 4 15 Confined EM modes, TM plasmons This localized EM mode is reflected in the following ansatz Aexp( γ ixβ )exp(− z ) for z > 0 k= e γβ ± ie A( r , t ) =  0 x z γβ <  A0 exp( ix )exp( z ) for z 0 We start with the magnetic field for the TM plasmons Hr()= eHxzeH (,) = exp( βγ z )exp( ix ), z < 0 1y 1 y 11 β ωγ µ ε=2 − 2 where j0 j ==− βγ > Hr2() eHxzeHy 2 (,) y 22 exp( z )exp( ix ), z 0 The electric field takes the form, 1 Exz(,)= −() ei γβ − e Hxz (,) 1 ωε z x 1 1 i 1 1 Exz(,)= −() ei γβ + e Hxz (,) 2 ωε z x 2 2 i 2 From boundary conditions γ γ γ 1HH+ 2 =0, HH −=− σ 1 H ωε ε ε 12 21 1 1 2 i 1 γ γ  γ ε γ  we obtain 1+−2 ε εγ σγ 1σ 1 H =→+− 0 1 2 = 0 εω  ωε ε γ  1 1221 1 2i 1  i 16 Confined EM modes, TM plasmons

We can obtain plasmon dispersion in free-standing case, σγ 2 4ε ω2 2 ε ωγ β σ 2ε γ −0 =→= 0 2i →−=− 2 k 2 0 00iω 00 0 σ =ω ε µ µ η ε = where k0 000 , 00 . Thus the TM plasmon is 4 β =k 1 − 0 σ η2 2 0

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

17 Drude conductivity

The equation of motion of free electrons in metal

electron momentum p dp p= mv − −eE = , we also have τ dt J= − env relaxation time dJ J ne 2 Hence, + = E dtτ m = ω −ω =− Assuming time dependence EE0 exp( it ) , JJ 0 exp( it ), we obtain, ne2 ne 2 i σ(−+iω τ 1 ) J = E →= 0m 0 m(ω τ + i ) To map the relation to graphene, we use the relation, E k 2 m=F and n = F 2 π vF then , iD e2 E σ = where D ≡ F (also known as Drude weight) π (ω τ + i ) ℏ2 18 Graphene plasmons

σ η2 2 0 β = − 2 TE plasmons, k0 1 iD e E 4 σ = where D ≡ F π (ω τ + i ) ℏ2 4 TM plasmons, β =k 1 − 0 σ η2 2 0

Lets consider some typical numbers, e2 σ µη ε = =Ω376.6 and | |∼ =× 6.1 10 −5 S 0 0 0 4ℏ η σ Hence, 0 | |∼ 0.02 ≪ 1. This implies that, β = TE plasmons, k0 2 2k ωπω π ε ω 2 k ℏ2 2 ℏ 2 2 TM plasmons, β = ik ∼0 ∼ 0 ∼ 0 0 ση η η 2 2 0 0D 0 eEF eE F e2 E Dβ → βω =F = pl επ ε 2 2ℏ 0 2 0

19 ß Quick overview

ß Basics on graphene plasmons A pedagogical tutorial on graphene plasmonics starting from Maxwell eq.

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

ß Beyond graphene plasmonics A forward looking perspective on what’s new with other 2D materials

20 Exciting plasmons in graphene

Maxwell E Light 2µ ω = e q pl π 2ε ε 2 ℏ 0 r -IR E + - 0.2eV + - + - Terahertz to Mid Mid to to Terahertz Terahertz + -

q 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) Understand Graphene plasmonic Z.Fei et al , Nature (2012) resonator and what we can do with it

T.Low and P.Avouris, ACS Nano (2014) 21 H.Yan, T.Low et al , Nature Phot.(2013) H.Yan, T.Low Mid-infrared plasmons with graphene nanostructures T.Low and P.Avouris, and T.Low ACSNano(2014) A.Y.Nikitin, T.Low,L.M.Moreno, Measuringextinction: WZ et al

1 ) , (

PRB Rapid(2014) , Nature PhotonicsNature(2013) , ω − = T T Par per q

(tn[4/( )]) (4 / 4 [ tan ,( ~

π ω n pl

W energy Φ− = RPA Lossfunction qL

Im ) , ( R 2 ε ε π

q e ℏ ε ω 2

+ − = Φ 2 µ = 0 R r     RPA − 1 − 2 1

momentum π π 22 Mid-infrared plasmons with graphene nanostructures -1 20 Peak 1 1000cm ~ 10um ~ 30THz Width (nm) Peak 2 2500 Optical Phonon 60 Graphene on SiO Peak 3 70 2 85 First peak nd 95 Second peak 16 2 order 100 2000 Third peak nd 115 2 order mode 125 1/2 ) ω ~ q -1 ω 140 op 12 (%) 150 1500 // 170 /T

190 ω per 240 sp2

1-T 8 1000 ω sp1

Wave numbernumberWaveWave (cm (cm 4 500

0

0 1000 1500 2000 2500 3000 0 2 4 6 8 -1 Wave vector q (x10 5 cm -1 ) Wave number (cm ) Plasmon dispersion can be H.Yan, T.Low, F.Guinea et al , Nature Phot. (2013) engineered with substrates T.Low and P.Avouris, ACS Nano (2014) 23 Graphene plasmons

Dielectric function of graphene (free el ectron contributions only) e2 E β 2 β ω β ω ε β ω (),=−∏ 1v() , where v = and ∏=() , F c c π ω τ βε 2+ 2 20 ℏ (i ) Coulomb Polarizability potential Plasmon occurs when ε β ω() ,= 0, ignori ng damping, we have, e2 E β 2 → 0 = 1 − F βε π ω 2 2 2 0 ℏ e2 D β 2 → 0 = 1 − βε ω 2 2 2 0 e β Dβ D → ω ω 2 = → = pl ε ε pl 20 2 0 Hence, we can also express dielectric function as, ω2 ε β ω(),= 1 − pl (ω τ + i ) 2

24 Plasmons and phonons hybridization

Dielectric function of polar phonons can be described by α ω ε ω ()= where is phonon frequency ω ω τ +2 − 2 0 (i ) 0 Total dielectric function becomes ω2 α ε β ω(),= 1 −pl − ω ω τ τ +2 + 2 − 2 (i )( i ) 0 Hybrid modes when ε β ω() ,= 0, assu ming again zero damping, ω2 α 1−pl − = 0 ω ω ω2 2− 2 0 ω ω ω222 ω ω ω αω−− 222 −− 2 = (0 )(pl 0 ) 0 ω ω4 ω ω− ω ω ω αω 22 − 22 + 22 −= 2 0pl pl 0 0 ω ω ω4222 ω α− ω ω +−+ 22 = (0pl ) pl 0 0 ω ω α2+ 2 − (ω ω ω α 2 ω+ 2 − ) 2 − 4 22 ω2 =0 pl ± 0pl pl 0 2 2

25 Graphene, Dirac electrons

Light 100 x E e Dirac e k y 3x103x10 66 msms --1 1 y K Eop≫ k B T kx 100 x K’ Sound Vel. x ky

kx

Quantum Hall effect at Room Temp. Highest mobility µ=1x10 6 cm 2Vs -1 K.S.Novoselov et al , Science (2007) D.C.Elias et al , Nature Physics (2011) Graphene absorption spectrum

Re σ ω( ) 

e2 απ =2% light absorption 4ℏ Frequency 1 2 3 ω 2µ Terahertz Mid-IR Near-IR Visible

Disorder- Intraband mediated

µ

Interband

1 2 3

Z.Q.Li et al , Nature Physics (2008) 1eV ~ 8000cm -1 ~ 1.25um ~ 240THz R.R.Nair et al , Science (2008) 27 Fresnel coefficients

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

Et ε 2 z x ε 1 Ei Er

We seek the scattering coefficients (transmission, reflection), due to a 2D material at z=0 for plane monochromatic waves i.e. ωω = ⋅ − Angle of incidence Ar(,) A0 exp( ikrit ) θ = −1 β Without loss of generality, assume xz being plane of incidence tan (γ ) ω=+ εµ β γ γ β ==2 += 2 2 keex z where kk | | and k Any EM waves can be expressed as linear superposition of TE and TM waves. = TE: E ey E = TM: H ey H

28 Fresnel coefficients, TE waves

Incident EM wave gives rise to reflected wave in medium 1 and transmitted wave in medium 2 = = β γ γ +− EreExzeE11()y (,) yi [ exp( izE 1 ) r exp( iz 1 )]exp( iz ) = = βγ Er2() eExzy 2 (,) eE y t exp( iz 2 )exp( iz ) The magnetic fields can be obtained from Maxwell equation i.e. Faraday’s law 1 Hr()= β γ γ γ [ eEreEγ β () − ( exp( iz ) −− E exp( iz ))exp( ix )] 1ωµ z 1111 xi r 1 0 1 Hr()= β γ [ eEreE γ β () − exp( iz )exp( ix )] 2ωµ z 222 x t 0 Using boundary conditions ×−=→ = eEEz (21 ) 0 Ex 1(,0) Ex 2 (,0) ×−= σ σ →−= eHHeExz(21 ) y 1 (,0) HH 212 xx Ex (,0) Then, + = Ei E r E t γγ −2E + 1 ( EE −= ) σω E µµ t ir t 0 0 29 Fresnel coefficients, TE waves

Then transmission and reflection coefficients are E 2γ E σωµγ − − t =t = 1 r =r = 1 2 0 σωµγ + + σωµγ + + Ei 1 2 0 Ei 1 2 0 We measure reflection and transmission probably with respect to energy carried by the EM wave, using the Poynting vector, ∗ 1 ∗ 1 γγ ∗ ∗  Sz(== 0) Re[ eErHr ⋅× (() ())] =− Re[( EE +− )1 ( EE −= )] 1  |||| EESS2 − 2  =− 1z 1 1 ωµ µ ω ir ir   irir  2 20  2 0 ∗ 1 ∗ 1 γγ ∗  Re[ ] Sz(==⋅×=− 0) Re[ eErHr (() ())] Re[ EE2 ] = 2 || ES2 = 2z 2 2 ωµ µ ω tt  tt 2 20  2 0 Then, the probabilities are S| E | 2 S Re[γ ] R=r = r = | r | 2 T=t = 2 | t | 2 2 γ Si| E i | Si 1 Total internal reflection, β γ β ε > >→=−2 2 = 12 and k 2 22 k is pure imaginary, T 0

30 31 ∼ ' q y ik i.e. 0

α β

the Kubo formula, Kubo the ω τ ', ' , , ', ' functions, ℏ kjv kj kjv kj 1 K x y q iq q ( ) oton momentum. momentum. oton ± +     0 − 2 − − ++− x y 1 q iq (') () ( ) (') () ' jj jj ' j j ty operators ty EkEk i EkEk ( ( ')) ( ( )) 0 + fEk fEk x y q iq      ∑ , ', , ' kk jj ℏ 2 and and and and and σ x x = + k k k and and k k = − kk q q q ℏ

F F σ σ = =− =− ℏ ℏ =± Ψ= q ie = = = ⋅ = ( , ) Optical conductivity of graphene of conductivity Optical , , , , , cv F cv xccF xvv F xcv F xFx yFy where 'as , ph dueisto Optical conductity can be computed from computedcanconductityfrom be Optical

σ ω local the inlimit, interested are we work, this In we also have the veloci have alsothe we with the following the with eigenvalues and eigen Graphene low energy Hamiltonian at energy lowGraphene point, H v v E v q vv vv vv vv vv 32 2 c c ( ( )) ∞ −∞ f E k ∂ ( )     ∂ 2     ( ) 2 ℏ

π ω τ i E k F x ∞ 0 + ∂ 2 2 2 iev k ( ) ' 2

π ω τ 2 = − ie fE 2 ( )

α β ℏ , 2 2 x cc

ω τ vk dk uction band uction ', ' , , ', ' ns from valencefromband ns ℏ kjv kj kjv kj c c ( ) f E c ∂ ( ) () () () ∂      ∂ ∂ ∂ − − ++−     + − + −     k E k B ∞ ( 0) ( ) c (') () ( ) (') () dEE c c ' E jj jj ' j j       dEE dEE dE E ( ( 0)) ( ( )) −∞ EkEk i EkEk 2 0 ∞ −∞ fEk fEk ∫ 0 0 ( ( ')) ( ( )) ∫ ∫ ∫ fEk fEk     cos 2

θ θ i E d k d dkk dEE ∫ ∫       + ∂ ∑ , ', , ' π 2 i E Ei E kk jj 2 0 0 ln 2cosh + ∂ ∂+ ∂ ie fE 2 ℏ ∫ ∫ ∫ ( ) 2 i 2 ie fE fE ie fE () 2 ℏ ie + 2 i k T 2

iπ ω τ EiE ie F ℏ + 4 + ∂+∂ = − = − π ωB τ F 2 2 2 ( ) 2 ik Te E = −+−= − π ωie v τ 2 () () 2 4 ( ) 2 ℏ Optical conductivity of graphene of conductivity Optical πωτπ ω πωτ τ (0, ) (0, ) ( ) = −= − = − Consider intrabandConsidercond incontributions σ ω Thus the total Thustheintraband conductivity is = Straightforward to show that contributio showthat to Straightforward σ ω σ ω 33 + − − − ( ) () 2 2 2 ℏ dE ≈ ∞ 0 ∫ rther. In this case, this In rther. F F B B kT kT E E 2 2 +    

ω τ ℏ ( ) onductivity, we get, onductivity,we 2 ie i     ω τ i F − + + B B F 2 F E i ( ) ie E 2 2 ( )           ℏ     π ω τ ln = 2 ≫ ≫ F B F B           E k T E k T

σ ω + 2 2 i E i i kT iE i k T F 1 + + + + + E ie ≈ → 2 B F B F ( )4 2( ) () 2 ()4 ( ) 2 ikTe E fEfE ik Te E ie 2 2 2 2 2 ℏ ℏ ℏ ℏ ℏ F F B B kT kT π ω τ E E πωτπωτπ π ωτ ωτ 2 2 2 = + = ≈            ( ) ln 2cosh ( ) ln 2cosh ( ) Optical conductivity of graphene of conductivity Optical Thus the total intrabandtheThus conductivity is cosh exp ln 2cosh Hence, we obtain,Hence,we ( ) For low temperture, lowtemperture, For wecansimplify , fu σ ω Includingbothintraband and interband c σ ω lowtemperture, For σ ω wecansimplify , to, Universal optical conductivity Nair et al , Science (2008) The absorption is defined as A=1 − T2 − R 2 where σωµ 2γ − T=0 and R = 0 ω σωµγ σωµ γ + + 200 2 00 and we can show that 4γ ωµ σ A()ω = 0 0 Re [] σωµγ + 2 2 0 0 For normal incidence and high frequencie s, = ση ≈ A 0 Re[] 0.022 2 where σ η ≈ 377 and Re [] = e 0 4ℏ Note for high frequency, ie 2 2E− (ω τ + i ) ℏ  σ ω(→ ∞ ) ≈ ln F ω τ π + +  4ℏ 2(EF i ) ℏ  ie2 2E− (ω τ + i ) ℏ   e 2 Re[(σ ω →∞≈ )] ImlnF  = ω τ π + +  4ℏ 2(EF i )4 ℏ   ℏ 34 Photocurrent mechanisms

Visible light

Bipolar junction

≈ ∗µξ IPV en δ≈σ ∗ − IPTE ( SST1 2 )

Unipolar junction

≈ ∗µξ IPV en

et al δ≈σ ∗ − N.Gabor , Science (2012) IPTE ( SST1 2 ) 35 Photoconductivity experiment

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

Photovoltaic

≈ VG 0 IPC IDC

Bolometric

VG ≫ 0 IPC IDC

M.Freitag, T.Low et al , Nature Phot.(2013) 37 Mid-infrared photodetector

Key Ingredients Light ß Mid infrared plasmons ß Thermal photo-response ß Room temperature operation

Non-radiative decay < ps

P-pol S-pol Drives a bolometric current

M.Freitag, T.Low et al , Nature Comm. (2013) M.Freitag, T.Low et al , Nature Photonic (2013) 38 Mid-infrared photodetector

2000 2.0 100 100 140 140 160 160 120

1.5 W=140nm W = 200nm 200nm = W

ω 1500 op

1.0 ) -1

0.5 ωsp2 Loss (a.u.) Function 1000 ωexp

0.0 CO2 -40 -20 0 20 40 Gate Voltage V (V) ωsp1 G Wave numbernumber(cm (cm WaveWave 500 E

Intraband Landau Damping x15 0 0 2 4 6 Wave vector q (x10 5 cm -1 ) Gate tunability, thanks to E hybrid plasmon-phonon polariton

M.Freitag, T.Low et al , Nature Comm. (2013) 39 Driven mechanical oscillator Credit: MIT TechTV

Resonator can acquire a phase from its driving force which is determined by its detuning from resonance 40 Mid-infrared light bending Vg=1V Vg=3V Vg=4V

θ Light i Electrically controlled terahertz and θ(V) mid-infrared beam reflectors

Graphene Generalized Snell’s law λ dφ sin( θθ )− sin( ) = 0 r i π ε V 2 1 dx

High-κ dielectric 2 − µω 3e   ω φ ≈tan 1 2 −   W SiO W ε τ 2 1 2 N Metal Reflector 8W ℏ   C.Eduardo, T.Low, et al, Nanotechnology (2015) A. Nemilentsau, T.Low, arXiv:1610.05236 (2016) 41 Graphene plasmonics for THZ and MIR applications

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

ß Basics on graphene plasmons A pedagogical tutorial on graphene plasmonics starting from Maxwell eq.

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

ß Beyond graphene plasmonics A forward looking perspective on what’s new with other 2D materials

43 A new class of 2D crystals

Eg=0 eV Eg=6.0 eV Graphene Boron Nitride TMOs (Transition Metal Oxides):

MoO 3, LiCoO 2

TMDs (Transition Metal Dichalcogenides): III-VI/V-VI Compounds MoS 2,WS 2,NbSe 2 (MX 2) (Ga,In)2Se 3, Bi 2(Se,Te)3 ß Strong in-plane bonds ß Weak van der Waals interlayer coupling ß Surfaces ideal self-passivation, intrinsically good electrical properties ß Pathway for large scale growth ß Full range of material properties Black Phosphorus 44 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 , 45 Nature Materials (2016) Mid-infrared plasmons with near field optical microscopy

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

Spacing between fringes π Re[q ] Exponential decay of fringes exp(− Im[q ] x )

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

Figure of merits Damping γ = Im[q ] Re[ q ]

πγ −1 2 Number of cycles plasmon propagates before amplitudes decay by 1/e

β = Confinement Re[q ] k 0 Light confinement by the polariton mode

Current state-of-the-art, γ-1 >25 and β~150 A.Woessner et al , Nature Mat (2015) 46 Mid-infrared plasmons with near field optical microscopy

monolayer

bilayer Gold graphene

Gold

F. Koppens et al, Nature Materials 2015 R. Hillenbrand et al, Science, 2015 47 Comparison of plasmon figure of merits

2D materials has better FOM than 1nm Au in the infrared T.Low, J.Caldwell, F. Koppens, L.M.Moreno, P. Avouris, T. Heinz et al , Nature Materials (2016) 48 Plasmonics beyond graphene

Phonon induced transparency Non-reciprocal plasmon Slow light Giant Faraday rotation

T.Low et al , Phys. Rev. Lett. (2014) A.Kumar et al , PRB Rapid. (2015) H.Yan, T.Low et al , Nano Lett. (2014) Anisotropic plasmon Hyperbolic plasmon T.Low et al , Phys. Rev. Lett. (2015) 49 A.Nemilentsau, T.Low et al , Phys. Rev. Lett . (2016) From monolayer to bilayer graphene T.Low, F.Guinea et al , Phys. Rev. Lett. (2014)

2.5 E E ] 0 2.0 σ/σ µ k

1.5 phonon interband

1.0

Conductivity Re[ Re[ Conductivity Conductivity 0.5 γ =0 0.0 0.0 0.2 0.4 0.6 0.8 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 Experiments

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

15 W = 100 nm Higher doping Substrate SiO 2/Si Vg

10

W = 100 nm Chemical doping

5 15 Higher doping Loss Function (a.u.) (a.u.) FunctionFunction Loss Loss

10

0 (%) S

1200 1500 1800 2100 Frequency (cm -1 )

1-T/T 5

0 plasmon phonon 1200 1500 1800 2100 -1 From narrow absorption to Frequency (cm ) narrow transparency H.Yan, T.Low et al , Nano Lett. (2014) 51

52

> = 1 2 1 2 ω ω ω ω ω ω Freq. Freq. Freq. Freq. Narrow Narrow Narrow Narrow absorption becomes stationary becomes transparency 1.90 1.95 2.00 2.05 2.10 1.90 1.95 2.00 2.05 2.10

1.5 1.0 0.5 0.0 1.5 1.0 0.5 0.0

1 1 Power P Power Power P Power Destructive mass interference, Destructive 2 ≫ κ 1 2 γ γ

γ , 2 2 m ( ) 0 ( ) cos( )

κ ɺ ω θ

ω ω ω θ 2 γ ω θ 1 , 1 1 cos( ) m cos( ) = −

γ κcos( κ ) γκκ ω F t = “plasmon” “phonon” + + + −= + + + −= ( ) sin( ) = + 1 = + ɺɺ ɺ 1 1 1 ( ) cos( ) ( ) ɺɺ ɺ

κ 22 22 22 2 1 1 1 2 2P 2 t Fa 11 11 11 1 2 1 1 1 Pt F txt x ax a t t Power Power absorption by “plasmon”: Solutions: mxmx x x x x xxF xx t Equation of Equationmotion: Coupled two harmonic oscillators harmonic two Coupled From induced-transparency to slow light

ε ωω =   n() Re () 

dn n() ωω ω = n () + () g dω c v (ω ) = g ω ng ( )

Highly dispersive n will yields modified group velocity Induced-transparency

Slowing light for integrated photonic memory A.H.Safavi-Naeini et al , Nature (2011) T.J.Kippenberg et al , Science (2008)

53 Anisotropic 2D materials

Class of anisotropic materials Intraband optical process Highly anisotropic in-plane effective masses Anisotropic intraband Drude conductivity

Interband optical process

Grp 5, e.g. black phosphorus J.Qiao et al , Nature Comms. (2014) 1T TMD, e.g. ReS 2, ReSe 2 Transition metal trichalcogenides Interband conductivity also anisotropic Absorption edge at different energies for different L.Li et al , Nature Nano. (2014) polarization due to symmetry of the bands S.Tongay et al , Nature Comms. (2014)

54 Anisotropic 2D materials, optical conductivity Imaginary part follows from Kramers-Kronig Graphene σ 2 g 0  ien i ω ω−  σ =   θ ω ω σ = +s ( −+ ) ln 0  σ  πω ωg ω η +i m 0 + 0 g  0 

Intraband Interband Anisotropic semiconductor σ 0  ien2  i ω ω−  σ = xx  θ ω ω σ = +−+s()lnj  jxy = , 0 σ πω ωjjω η + j j + yy  i m j j 

Anisotropic σσ > Im(xx )Im( yy ) 0

Hyperbolic σσ < Im(xx )Im( yy ) 0

55 Anisotropic versus Hyperbolic plasmons

σσ > σσ < Anisotropic case Im(xx )Im( yy ) 0 Hyperbolic case Im(xx )Im( yy ) 0

56 Hyperbolic plasmons, ray optics

Hyperbolic plasmon rays can be electrically control

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

Valley hall current Optical circular dichroism

Graphene/hBN TMD R.V.Gorbachev et al , Science (2014) K.F.Mak et al , Nature Nano (2012) H.Zeng et al , Nature Nano (2012) MoS2 X.Xu et al , Nature Phys. (2014) K.F.Mak et al , Science (2014)

These phenomena are static and dynamic manifestation of Berry physics 1 ∂E( k ) e v=n − E ×Ω ( k ) Berry curvature: can be viewed ℏ∂k ℏ as an effective magnetic field due to orbital part Bloch states K and K’ valleys are related by time reversal symmetry Ω()K =−Ω () − K n n Breaking spatial inversion Spatial inversion symmetry requires that produces finite Berry curvature Ω =Ω− i.e. sublattice asymmetry n()K n ( K ) 58 Massive Dirac system, optical conductivity

σ σσ − g xy    σ =   σ = g xy  K − σ  K '  σσ  xy g  xy g 

2e2 Optical pumping to create non- ρσ =()k Ω () kdk equilibrium valley imbalance xyℏ ∫ K K

A.Kumar et al , PRB Rapid. (2015) 59 Non-reciprocal edge modes

Bulk plasmons, continuous film Edge plasmons, semi-infinite film

σ σ σ 2 2 2 2ε g  g  xy σ σ +  +i  ⋅−2κ i k  + = 0 −q2 g xy  ++[3 σ iσ 22 sgn()]| qq | = 0 εεε ε ωε κ c0 ( c ) 2 ωε ε g xy 0  0  0 0 

εκ =2 − 2 k0 q

Linear dipole

Massive Dirac material

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

Graphene plasmons can be monolayer launched by nano Au optical antenna, and mapped with SNOM bilayer Gold graphene P.Alonso-Gonzalez et al , Science. (2014) Gold

Isotropic plasmon (graphene)

Hyperbolic plasmon (anisotropic materials)

Chiral plasmon (gapped Dirac materials)

61 Phonon-polaritons in boron nitrides

Acoustic mode

Optical mode

S.Dai et al , Science (2014)

62 hBN as natural hyperbolic material

hBN permittivity ωω 2− 2 ε ε ε = + LOm, TOm , m∞, m ∞ , m ωω 2− 2 − Γ TOm, i m

Out-of-plane phonon modes ω ω =−1 = − 1 TO,780cm , LO , 830cm

In-plane phonon modes ω ω =−1 = − 1 TO,1370cm , LO , 1610cm

Elliptic Hyperbolic type I Hyperbolic type II 63 Phonon-polaritons in hBN

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

Dispersion within the Reststrahlen band

ψ ε   π q(ω )= − 2 tan −1 0 + n ε ψ   thBN ⊥  

ε εψ = ± ⊥

64 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)

ψ  σ− ε + i( qk) Z  − ε   q(ω )=− tan1 0 0 0 + tan 1 0 + πn   ε ψε ψ    thBN  ⊥   ⊥   σ− εψ ε ε ( ) + ( )( ) 1 i⊥ 0 iqkZ 000   σ+ εψ ε ε +  ψ 1 i()⊥ iqkZ()() ω = i 0 000  q( ) ln   2t 1− iψ ε ε()  hBN ×⊥ 0   + ψ ε ε()   65 1 i ⊥ 0   Hyperbolic phonon polaritons in hBN

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

66 Hyperbolic polaritons beyond hBN

T.Low et al, Nature Materials (2016) 67 Exciton polaritons in 2D materials

T.Low et al, Nature Materials (2016) 68 Designers’ polaritons with 2D heterostructure

T.Low et al, Nature Materials (2016) 69 Acknowledgement

IBM Hugen Yan, Marcus Freitag, Fengnian Xia Wenjuan Zhu, Damon Farmer, Phaedon Avouris Spain Francisco Guinea, Luis Martin Moreno, Alexey Nikitin, Rafael Roldan, Frank Koppens

MIT – Nick Fang U Wisconsin Milwaukee – George Hanson NRL – Josh Caldwell Stanford – Tony Heinz Brazil – Andrey Chaves

UMN Roberto Grassi, Eng Hock Lee, Yongjin Jiang, Kaveh Khaliji, Sudipta Biswas, Javad Azadani, Anshuman Kumar, Andrei Nemilentsau