Supporting Information Wedge Dyakonov Waves and Dyakonov Plasmons in Topological Insulator Bi 2 Se 3 Probed by Electron Beams

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1 Suppoting Infomation Wedge Dyakonov Waves and Dyakonov Plasmons in Topological Insulato Bi 2 Se 3 Pobed by Electon Beams Nahid Talebi, Cigdem Osoy-Keskinboa, Hadj M. Benia, Klaus Ken, Chistoph T. Koch, Pete A. van Aken Max Planck Institute fo Solid State Reseach, D Stuttgat, Gemany Institut de Physique de la Matièe Condensée, Ecole Polytechnique Fédéale de Lausanne, 1015 Lausanne, Switeland Humboldt Univesity of Belin, Depatment of Physics, Belin, Gemany Contents: 1- Evanescent modes at the inteface of a uniaxial anisotopic and an isotopic medium 2- EFTEM images fo seveal Bi 2 Se 3 paticles 3- Complete Band diagam of a Bi 2 Se 3 ib waveguide

2 1- Evanescent modes at the inteface of a uniaxial anisotopic and an isotopic medium In ode to obtain the chaacteistic equations fo the slab waveguide shown in Figue S1, a vecto-potential appoach is utilied. This appoach helps us to moe efficiently distinguish and deive the possible modal goups, athe than the usual appoach to constuct the solutions at the field level. 1, 2 This is because that in contast to the field appoach one can deive a Helmholt equation fo vecto potentials even fo an anisotopic medium, though such an equation is not valid fo field components in geneal. We fist conside the Maxwell equations as:, ib, E (1a) H, id, (1b) B, 0 (1c) D, 0 (1d) whee the hamonic epesentation as expi t is consideed, with i 2 1. We conside also the constitutive elations including the ME effect as: D, ˆ : E, (2a), B, H 1 (2b) Since the magnetic flux density B, is a pue solenoidal vecto quantity it can be descibed as the cul of anothe vecto, which is called the magnetic vecto potential, as: B A (3) in which we have dopped the agument, fo simplicity. Inseting eq. (3) in eq. (1a) gives: E ia (4) in which is the electic scala potential. Using (1) to (4), we can deive the following equation fo the magnetic vecto potential: 1 2 A A ˆ : A i ˆ : 0 Equation (3) only defines the otational pat of the magnetic vecto potential. We ae fee to employ a gauge theoy to fix its solenoidal pat. In ode to do so, we can define a genealied Loent gauge in the fom of: 1 ˆ A (6) i Using (6), (5) is futhe simplified to a Helmholt equation in the fom of: 2 2 A k 0 ˆ : A 0 (7) (5)

3 in which k0 is the wave vecto of the light in fee space. Solutions to (7) ae called wave potentials. The field components obtained using eqs. (3), (4) and (6). We conside hee the solutions to the optical modes excited at the inteface of a uniaxial anisotopic mateial and an isotopic dielectic located at 0and 0, espectively. The optic axis is paallel to the inteface, and without any loss of geneality we conside the wave to popagate along the x-axis and evanescent along the -axis. Moeove, the only non-eo elements of the pemittivity tenso ae xx yy and. The optical modes in such a system ae decomposed into two individual sets; namely TM x and TM, whee they ae constucted by the choice of the magnetic vecto potential as A A x,0,0 and A 0,0, A, espectively. It is easily veified that a choice as A 0, A y,0 solutions to wave potentials can be constucted as A, 2, will not satisfy the bounday conditions. Fo each case, the d A exp exp i and A, A exp exp i fo 0, and 0, espectively, whee i is the popagation constant ( is the phase constant and is the attenuation constant), 2 2, 2 k 0 2 and d 2 d k 0. Moeove, x,. We fist conside solutions to the TM x modes. Satisfying the tangential bounday conditions fo the E x and H y field components, the popagation constant of the optical modes at the inteface is obtained as: TM x d 0 d k (8) Inteestingly, it is only the component of the pemittivity which affects the popagation constant of the TMx mode. Moeove, eq. (8) is quite simila to the popagation constant of plasmon polaitons at the inteface of two isotopic mateials with the elative pemittivity of and d, espectively. In ode to have evanescent modes, the eal pats of be positive, which lead us to the citeion d 2,x and d should. This means that in ode to have evanescent TM x modes, the mateial should be metallic at least in the diection of the optic axis. The popagation constant fo the TM mode can be deived as: TM d k0 d d (9) In ode to have bound TM modes, the eal pats of the damping factos positive, which leads to moe difficult citeion, as shown in table S1. 2, and d should be

4 The last citeion (C4) fo TM modes has been so fa consideed in the liteatue 1 as the citeion to excite the bound Dyakonov mode at the inteface of uniaxial anisotopic/isotopic mateials, as d, when both and can be positive. Moeove fo d and d Table S1. Bound modes at the intefaces of isotopic/isotopic and anisotopic/isotopic mateials. Scale ba is 2 d k 0.

5 both TM x and TM modes can be excited, even if 0, fo which the mateial is called hypebolic. Inteestingly, wheeas the only non-eo field components fo both TM x and TM modes ae E x, E, and H, the field pofiles ae diffeent as shown in table S1, and the oveall y polaiation state would be diffeent fo these cases. Notably fo an isotopic/isotopic inteface, TM and TM x modes ae degeneate. All the citeia above ae only valid fo lossless mateials. Howeve, fo eal mateials, the dielectic loss is not negligible as the imaginay pat of the pemittivity can be even lage than the eal pat, especially at the enegies nea to the inte band tansitions, as happens also fo Bi 2 Se 3 mateial. Fo the sake of completeness fo ou discussions egading the inteface modes, we have computed hee the popagation constants of the optical modes at the inteface of Bi 2 Se 3 /ai and Bi 2 Se 3 /a-c, whee a-c stands fo amophous cabon. As noted in the main text, Bi 2 Se 3 has thee distinguished enegy anges: It is dielectic at E < 1.06 ev, hypebolic type II ( 0 and 0 ) at 1.06 ev < E < 1.73 ev, and hypebolic type I ( 0 and 0 ) at E >1.73 ev. These fequency anges ae denoted by D, H II, and H I Figue S1: Popagation constant of a) TM x mode at Bi 2 Se 3 /ai, b) TM mode at Bi 2 Se 3 /ai fo c) TM x modes at Bi 2 Se 3 /a-c, and d) a) TM modes at Bi 2 Se 3 /a-c intefaces. Spatial field distibution fo E x field component at a given time and selected enegies ae depicted at the insets. espectively. both TM x and TM modes can be excited in the whole fequency ange at the Bi 2 Se 3 /ai inteface, while thee is a clea gap at the excitation enegies of the fowadly popagating (

6 TM d k0 d d, see eq. (9)) TM modes, as attenuation constant becomes negative in some enegy anges. At these enegies, a backwadly popagating mode (( TM d k0 d d, see eq. (9))) with a negative phase velocity is still possible, which can only be excited at discontinuities and tapes. 3 Moeove, despite the case of isotopic plasmons in a Dude metal like silve o aluminum, the hypebolic plasmon dispesion in Bi 2 Se 3 ae in geneal moe attached to the light line, due to the huge dielectic loss. 2- EFTEM images fo seveal Bi 2 Se 3 paticles Figue S2: shows the EFTEM seies fo a Bi 2 Se 3 paticle with the thickness of 55 nm positioned on a cabon substate, which suppots wedge modes. Complete Band diagam of a Bi 2 Se 3 ib waveguide In ode to emphasie the extemely huge numbe of PDOS in the Bi 2 Se 3 ib waveguide, the full photonic dispesion diagam as well as the computed PDOS is demonstated in Figue S3. In ode to calculate the PDOS fom the numeical data, the method descibed in ef 4 is used.

7 Figue S3: (a) Dispesion diagam a,d (b) photonic local density of states (PDOS) sustained by a Bi 2 Se 3 ib waveguide on a amophous cabon substate. L = 400 nm, H = 50nm, and h = 55 nm. Refeences 1. Polo, J. A.; Lakhtakia, A., Suface Electomagnetic Waves: A Review. Lase Photonics Rev. 2011, 5, Walke, D. B.; Glytsis, E. N.; Gaylod, T. K., Suface Mode at Isotopic-Uniaxial and Isotopic-Biaxial Intefaces. J. Opt. Soc. Am. A 1998, 15, Jang, M. S.; Atwate, H., Plasmonic Rainbow Tapping Stuctues fo Light Localiation and Spectum Splitting. Phys. Rev. Lett. 2011, 107, Busch, K.; John, S., Photonic Band Gap Fomation in Cetain Self-Oganiing Systems. Phys. Rev. E 1998, 58,