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2D Materials 2D Materials Polaritonics Polaritonics 2D Materials 2D 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 Quick overview Basics on graphene plasmons A pedagogical tutorial on graphene plasmonics starting from Maxwell eq. Graphene plasmonics A review on graphene plasmonics experiments and its applications 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 ) εω()1=+ χω ()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 Contains atmospheric transmission window Super high-speed wireless communication Imaging for military, security & medical 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 Quick overview Basics on graphene plasmons A pedagogical tutorial on graphene plasmonics starting from Maxwell eq. Graphene plasmonics A review on graphene plasmonics experiments and its applications 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 d d d d ∂B( r , t ) ∇×E( r , t ) =− Faraday’s Law ∂t d d d d∂D( r , t ) d ∇×Hrt(,) = + Jrt (,) Ampere’s Law ∂t dd d d d ∇⋅Brt(,) =∇⋅ 0, Drt(,) = ρ (,) rt Gauss’s Law Constitutive relations d ∂ρ(r , t ) d +∇⋅J(,) r t = 0 Continuity equation ∂t ddt dddf d d Drt( , )=∫ dtdr ∫ 'ε ( r −−⋅ rt ', t ') Ert ( ', ') −∞ Fields are related by permittivity, ddt dddf d d permeability, conductivity tensors Brt( , )=∫ dtdr ∫ 'µ ( r −−⋅ rt ', t ') Hrt ( ', ') −∞ ddt dddf d d 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 dd d d • Linear response, monochromatic waves Ert(,)→ Er (,ω )exp( − it ω ) • No free charges or currents Maxwell equations Boundary conditions d d d d ∇×dω = ωµf ω ⋅ d ω d × − = Er(,) i () Hr (,) en( E j E i ) 0 d d ddf d d d × − = ω ∇×Hr(,)ω =− i ωε ()(,) ω ⋅ Er ω en( H j H i ) Jr (,) f d d Current at interface to be included i.e. ∇⋅ε() ω ⋅E (,) r ω = 0 where 2D materials is also described! f d d ∇⋅µ() ω ⋅H (,) r ω = 0 Constitutive ddf d d Dr(,)ω= εω () ⋅ Er (,) ω ddf d d Br(,)ω= µω () ⋅ Hr (,) ω 12 Maxwell equations in a nutshell In the isotropic case, Maxwell equations dd d d ∇×Er(,)ω = i ωµ ()(,) ω Hr ω dd d d dd d d d d ∇×Hr(,)ω =− i ωε ()(,) ω Er ω ∇×∇×Eri(,)ω = ωµω () ∇× Hr (,) ω = ωµωεω2 ()()(,) Er ω d d d d d ∇∇⋅dω −∆ d ωωµωεω = 2 d ω ∇ ⋅E(, r ω ) = 0 (Er (,)) Er (,) ()()(,) Er dd d d d d ∆Er(,)ω = − ωµωεω2 ()()(,) Er ω ∇⋅H(, r ω ) = 0 Constitutive Solution dd d d d d d d ω= εω ω dω= ω ⋅ d Dr(,) ()(,) Er Er(, ) Ek0 (, )exp( ikr ) dd d d dd d d Br(,)ω= µω ()(,) Hr ω −kEr2(,)ω = − ωµωεω 2 ()()(,) Er ω ω2 −k2 = − ω 2 µ( ω )( ε ω ) ⇒ k 2 = v2 dd d d 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 d d d d d d Aexp( ixβ )exp(− γ z ) for z > 0 k= eβ ± ie γ A( r , t ) = d0 x z β γ < A0 exp( ix )exp( z ) for z 0 We start with the electric field for the TE plasmons d d= d = d γ β < Er1() eExzeEy 1 (,) y 11 exp( z )exp( ix ), z 0 γ= β2 − ωµε 2 d where j0 j d==− d d γ β > EreExzeE2( )y 2 ( , ) y 21 exp( z )exp( ix ), z 0 The magnetic field takes the form, d 1d∂Exz(,) d ∂ Exz (,) 1 dd Hxz(,) = e1 − e 1 =−() eieExzβ γ ( , ) 1 ωµz∂ x ∂ ωµ zx 1 1 i1 x zi 1 d 1 d d Hxz(,)=() eiβ + e γ Exz (,) 2ωµ z x 2 2 i 2 14 Confined EM modes, TE plasmons From boundary conditions = E1 E 2 γ γ 2E+ 1 EiE = σω µ2 µ 1 1 0 0 We obtain the solution for electric field E (γ+ γ −i σωµ )1 = 0 1 2 0 µ 0 Which has non zero solution if, γ+ γ − σωµ = 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, σωµ222 ση 22 γ= σωµ →−=− β220 →= β 22 − 0 20i 0 k 0 k 0 1 4 4 =ωεµ η = µε where k0 000, 00 . Thus the TE plasmon is σ2 η 2 β = − 0 k0 1 4 15 Confined EM modes, TM plasmons This localized EM mode is reflected in the following ansatz d d d d d d Aexp( ixβ )exp(− γ z ) for z > 0 k= eβ ± ie γ A( r , t ) = d0 x z β γ < A0 exp( ix )exp( z ) for z 0 We start with the magnetic field for the TM plasmons d d d d Hr()= eHxzeH (,) = exp(γ z )exp( ix β ), z < 0 1y 1 y 11 γ= β2 − ωµε 2 d where j0 j d==− d d γ β > Hr2() eHxzeHy 2 (,) y 22 exp( z )exp( ix ), z 0 The electric field takes the form, d 1 d d Exz(,)= −() eiβ − e γ Hxz (,) 1 ωε z x 1 1 i 1 d 1 d d Exz(,)= −() eiβ + e γ Hxz (,) 2 ωε z x 2 2 i 2 From boundary conditions γ γ γ 1HH+ 2 =0, HH −=− σ 1 H ε12 ε 21 ωε 1 1 2 i 1 εγγ γγ we obtain 1+−2 1σ 1 H =→+− 0 εγεγσ 1 2 = 0 εγωε 1 1221 ω 1 2i 1 i 16 Confined EM modes, TM plasmons We can obtain plasmon dispersion in free-standing case, σγ 2 4ε2 ω 2 2εγ−0 =→= 0 γ2i εωσ →−=− β 2 k 2 0 00iω 00 0 σ =ωεµ η = µε where k0 000, 00 . Thus the TM plasmon is 4 β =k 1 − 0 σ2 η 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 md d d d pd dp p= mv − −eE = , we also have d d τ dt J= − env relaxation time d d dJ J ne 2 d Hence, + = E dtτ m dd dd = −ω =− ω Assuming time dependence EE0exp( it ) , JJ 0 exp( it ), we obtain, dne2 d ne 2 i (−+iω 1 τ ) J = E →= σ 0m 0 m(ω+ i τ ) 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) (ω+ i τ ) π 2 18 Graphene plasmons σ2 η 2 0 β = − 2 TE plasmons, k0 1 iD e E 4 σ = where D ≡ F (ω+ i τ ) π 2 4 TM plasmons, β =k 1 − 0 σ2 η 2 0 Lets consider some typical numbers, e2 ηµε= =Ω376.6 and | σ | =× 6.1 10 −5 S 0 0 0 4 η σ Hence, 0| | 0.02 1. This implies that, β = TE plasmons, k0 2 2kω 2 k ωπ2 2 π 2 ε ω 2 TM plasmons, β = ik 0 0 0 0 ση η η 2 2 0 0D 0 eEF eE F e2 E Dβ → ω= β F = pl π2 ε ε 2 0 2 0 19 Quick overview Basics on graphene plasmons A pedagogical tutorial on graphene plasmonics starting from Maxwell eq. Graphene plasmonics A review on graphene plasmonics experiments and its applications 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 0 r -IR d 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) Mid-infrared plasmons with graphene nanostructures RPA Loss function ω= ε −1 L( q , ) Im RPA energy 2µ ω = e q pl π2 ε ε 2 0 r Measuring extinction: T Z( W ,ω )= 1 − per TPar H.Yan, T.Low et al , Nature Photonics (2013) momentum T.Low and P.Avouris, ACS Nano (2014) π n − Φ − A.Y.Nikitin, T.Low, L.M.Moreno, q ~R ,(Φ= tan[4/(41 −π + π 2 )]) PRB Rapid (2014) R W 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.
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