Plexciton Dirac points and topological modes

Abstract

Plexcitons are polaritonic modes that result from the strong coupling between excitons and plasmons. Here, we consider plexcitons emerging from the interaction of excitons in an organic molecular layer with surface plasmons in a metallic film. We predict the emergence of Dirac cones in the two-dimensional band-structure of plexcitons due to the inherent alignment of the excitonic transitions in the organic layer. An external magnetic field opens a gap between the Dirac cones if the plexciton system is interfaced with a magneto-optical layer. The resulting energy gap becomes populated with topologically protected one-way modes, which travel at the interface of this plexcitonic system. Our theoretical proposal suggests that plexcitons are a convenient and simple platform for the exploration of exotic phases of matter and for the control of energy flow at the nanoscale.

ARTICLE

Received 23 Oct 2015 | Accepted 28 Apr 2016 | Published 9 Jun 2016

Joel Yuen-Zhou1, Semion K. Saikin2,3, Tony Zhu4,5, Mehmet C. Onbasli6, Caroline A. Ross6, Vladimir Bulovic5,7 & Marc A. Baldo5,7

Plexcitons are polaritonic modes that result from the strong coupling between excitons and plasmons. Here, we consider plexcitons emerging from the interaction of excitons in an organic molecular layer with surface plasmons in a metallic lm. We predict the emergence of Dirac cones in the two-dimensional band-structure of plexcitons due to the inherent alignment of the excitonic transitions in the organic layer. An external magnetic eld opens a gap between the Dirac cones if the plexciton system is interfaced with a magneto-optical layer. The resulting energy gap becomes populated with topologically protected one-way modes, which travel at the interface of this plexcitonic system. Our theoretical proposal suggests that plexcitons are a convenient and simple platform for the exploration of exotic phases of matter and for the control of energy ow at the nanoscale.

DOI: 10.1038/ncomms11783 OPEN

Plexciton Dirac points and topological modes

1 Department of Chemistry and Biochemistry, University of CaliforniaSan Diego, La Jolla, California 92093, USA. 2 Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA. 3 Department of Physics, Kazan Federal University, Kazan 420008, Russian Federation.

4 Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 5 Center for Excitonics, Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 6 Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 7 Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. Correspondence and requests for materials should be addressed to J.Y.-Z. (email: mailto:[email protected]

Web End [email protected] ).

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When ultravioletvisible light is absorbed by an organic molecular aggregate, it promotes molecules from their ground states to their excited electronic states. The

resulting excitations, known as excitons, can migrate between molecules via a mixture of coherent and incoherent processes1. Understanding and controlling how this migration of energy occurs is a fundamental problem of chemistry and physics of condensed phases. Furthermore, it is also a technological problem which is relevant to the development of efcient organic solar cells and light-emitting devices as well as all-optical circuitry2. Many strategies to control the motion of excitons exist, a particularly interesting one being where they couple to surface plasmons (SPs)3. In such strategy, the spatial coherence of plasmons assists the transport of an exciton across length-scales that are orders of magnitude larger than regular exciton diffusion lengths. When the coupling is strong, meaning that the energy exchange between the exciton and plasmon is faster than the respective decay times48, plexcitons (a class of polaritons) emerge9,10 and energy can migrate ballistically over the coherence length of the plasmon (between 10 and 40 mm)11. Besides their importance for energy transport, organic plexcitons promise to be an exciting room-temperature laboratory for the study of light-matter and many-body interactions at the nanoscale8.

On the other hand, topologically nontrivial states of matter have been a topic of great interest in condensed matter physics owing to the discoveries of the quantum Hall effect12, and more recently, of topological insulators13,14. The systems supporting these states are characterized by topological invariants15, integer numbers that remain unchanged by weak perturbations. Physically, a nontrivial topological invariant signals the presence of one-way edge modes that are immune against moderate amount of disorder. Even though these phenomena were rst conceptualized for fermions in solids, they have been successfully generalized to bosonic systems including photons in waveguides1618, ring resonator arrays19, ultracold atoms in optical lattices20 and classical electric circuits21,22. Furthermore, we have recently proposed an excitonic system consisting of a two-dimensional porphyrin lm, which becomes topologically nontrivial in the presence of a magnetic eld23. A challenging feature of that proposal is the requirement of large magnetic elds (X10 T) and cryogenic temperatures to preserve exciton coherence.

In this work, we consider a conceptually different platform, which, by recreating Dirac cones and topologically protected edge modes in plexcitons, avoids the use of large magnetic elds and, under appropriate circumstances, may work at room temperature. In the last year, Dirac and topological polaritons have been proposed in other contexts, such as optomechanical arrays and inorganic materials in optical cavities. All of these works share a common goal to ours, which is the design of exotic modes in strongly coupled light-matter systems. However, there are substantial qualitative and quantitative differences arising from the choices of material (organic exciton versus inorganic exciton2426 or mechanical mode27,28) and electromagnetic (SP versus microcavity2426 or photonic crystal27,28) excitations. Hence, the physics involved in our plexciton system contrasts with the other proposals in terms of the energy and length-scales involved in the excitations, the magnitude of the couplings, the generation of nontrivial topology and the experimental conditions for its realization. Organic excitons differ from their inorganic counterparts in that they have large binding energies and are associated with large transition dipole moments. In general, SP electromagnetic elds are strongly conned compared with those in microcavities because of the hybridization of light with charge oscillations in the metal29. The combination of all these properties in the organic plexciton context gives rise to

strong light-matter interactions even at room temperature and in an open cavity8. We believe that, by introducing topological band theory concepts into the realm of plexciton systems, the present article yields a fresh perspective to the degree of control of energy transport which is achievable in the nano- and mesoscales.

ResultsPlexciton Dirac points. The setup of interest is depicted in Fig. 1. It consists of three layers: a plasmonic metal modelled with a

Drude permittivity EmoE1 o

2P

o2 , with constants E1 3:7,

oP

8.8 eV, which are representative parameters for Ag), an a 80 nm thick dielectric spacer (Ed1) and an organic layer

(Eorg1). The spacer is placed to avoid quenching of organic

excitons by single-particle excitations in the metal upon close contact30. Later on, we will consider the case when this dielectric spacer is also endowed with magneto-optical (MO) properties. For purposes of band-structure engineering, we take the organic layer to be an oblique superlattice of organic nanopillars (Fig. 1a) with unit cell angles b, g, d and an angle y with respect to the x axis of the setup (in the absence of this superlattice, a standard molecular crystal may be used with some trade-offs, as explained below). The nanopillars are rectangular parallelepipeds of densely packed organic chromophores (assuming a van der Waals distance between chromophores of 0.3 nm, rnp 38

chromophores per nm3) with volume Vnp WxWyWz (Fig. 1b),

obtained from growing a J-aggregate lm31,32. J-aggregation of chromophores results in a collective transition dipole ln for the n-th nanopillar. Dipolar interactions between the collective dipoles couple the various nanopillars. Assuming perfect periodicity of the superlattice and only nearest and next-nearest neighbour (NN and NNN) dipolar interactions, we can write the Hamiltonian for excitations in the plexciton setup as H

PkHk

z

a b

Organicsuperlattice Nanopillar

y

x

[afii9797]h

Wy

Wz

[afii9829] [afii9826]

[afii9828] [afii9835]

[afii9797][afii9840]

Dielectric spacer Metalfilm

Wx

Figure 1 | Plexciton setup. (a) It consists of a plasmonic metal lm, a dielectric spacer and an organic layer. The latter is taken to be a monoclinic superlattice of organic nanopillars, which makes an angle y with respect to the x axis and is further characterized by angles b, g, d as well as distances between nanopillars Dh and Dv. When the density of emitters in the organic nanopillars is big enough, the coupling between the excitons in the organic layer and the surface plasmons (SPs) in the metal becomes larger than their linewidths, giving rise to polaritonic eigenmodes that are superpositions of excitons and plasmons, or more succintly, plexcitons. In this article, we shall also consider the case where the dielectric spacer is a magneto-optical (MO) material. The superlattice design is not essential for our purposes, except for guaranteeing a global gap for topological modes spanning all wavevectors. Hence, one may substitute it with a standard molecular crystal with some trade-offs. (b) Zoom-in view of a nanopillar. It is a parallelepiped of dimensions Wi along each axis, and consists of closely packed organic emitters (represented by balls and sticks) each featuring a transition dipole (blue arrows).

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(see the Methods for details),

Hk oexc;ksy

k sk

ay

k ak

|{z}

h i

|{z}

aksy

k

h:c:

|{z}

1

Hexc;k

o k

HSP;k

J k

Hexc SP;k

where,

oexc;k

oeff

2Jxcos kxDx 2Jycos kyDy

;

2

J k

: 3

Here, oexc,k and o(k) are dispersions of the exciton and plasmon, respectively. Ji and Di are effective NN hoppings and spacings between nanopillars along the i-th axis, respectively,

oeff is the

effective nanopillar site energy, and we have taken ln l for all

n. Equation (3) denotes the k-dependent coupling between a plasmonic mode ayk and a collective exciton state syk

throughout the organic layer. It is proportional to the square root of the total effective number of nanopillars

NxNy

NxNy

S

o k

2E0Lk

s

e aorg k z kl E k

, where Nnp is

the number of chromophores per nanopillar (see the Methods, Modelling of superlattice). Overall, J k

scales as

NxNyNnp

pcoherently coupled to each SP mode3335 (Ni is the number of nanopillars along the ith axis) and to the collective nanopillar transition dipole moment l, which scales as

Nnp

p

p

,

the total number of chromophores in the superlattice, as expected from a plexciton problem7. S and Lk denote the SP mode surface area and mode length, respectively, and e0 is the permittivity of vacuum.

Figure 2a shows superimposed dispersion curves for Hexc,k and

HSP,k independently (setting Hexc SP 0 in the calculations, see

the Methods for parameters). We assume that l is parallel to the xy plane, and design our superlattice to obtain Jx, Jy40, yielding a dome-like dispersion for Hexc,k; that is, it features a maximum at

k 0, behaving as a two-dimensional (2D) H-aggregate1. As for

HSP,k, its (time-reversal symmetric) dispersion has the shape of a rotationally symmetric fountain, and is nothing more than the 2D rendering of the standard one-dimensional textbook result36, featuring a linear dependence of the energy at short wavevectors and a plateau at large ones, indicating excitations that are qualitatively closer to light or to charge oscillations in the metal, respectively (Supplementary Notes 1 and 2, as well as Supplementary Figs 14). Figure 2b shows the two-plexciton-branch band-structure arising from the diagonalization of equation (1). We notice that anticrossing gaps are opened in the vicinity of where the dispersion curves for Hexc,k and HSP,k

used to cross in Fig. 2a. This is a signature of exciton-SP coupling Hexc,SP,k. Given a xed wavevector direction, whenever these

anticrossings occur, the lower-plexciton (LP) branch starts off as being mostly SP at short k values, but parametrically morphs into mostly exciton at large ones; the opposite happens with the upper-plexciton (UP) branch. However, the most striking feature of Fig. 2b is the appearance of two Dirac cones (see dashed circles) at critical wavevectors k* where anticrossings do not happen. Their onset coincides with the directions at which k is orthogonal to l. Their physical origin is explained in Fig. 2e, which, in its top panel, shows that the in-plane electric eld for the k-th SP mode, E>(k) E(k) E(k) ^z^z, is purely parallel to k,

E>(k) Ek(k) (blue vector eld). If all the dipoles in the organic

layer are aligned (in-plane) along l, their projection onto the SP electric eld, which gives rise to the exciton-SP coupling (see equation (3) as well as Supplementary Note 3 and Supplementary Figs 58), will wax and wane as a function of the azimuthal angle j between the xed dipole and the varying SP wavevector according to l E(k)pcosj. Clearly, this projection will vanish if

k happens to be orthogonal to l, that is, at the special angles j p2 ; 3p2, so that any degeneracy between the exciton and the SP

modes will remain unlifted along these directions. From this physical picture, we can extract the two essential ingredients for the emergence of the plexciton Dirac cones. First, the dipoles need to be aligned to create an anisotropic J k

as a function of

a c e

4 4

4

ExcitonSP LP

ky (107 m1) ky (107 m1)

ky (107 m1) ky (107 m1)

UP

Exciton

Energy (eV)

Energy (eV) Energy (eV)

3

2 2

0 0

1

5 5

0 5 5

0 0

0 0 0

5 5

5 5

5 5

5 5

0 0

5 5

5 5

f

Magneto-SP

kx (107 m1) kx (107 m1)

103

kx (107 m1) kx (107 m1)

b d 2

0

2

0

UP LP

7 m1 )

Berry curvature

3

2

2

k y(10

1

4

6

0

2 2

Ek

E[afii9835]

0

kx (107 m1)

Figure 2 | Bulk plexciton properties. Dispersion relations: (a) for SP and exciton (organic layer) modes independently, (b) when they couple in the absence of the MO effect, yielding lower (LP) and upper (UP) plexciton branches, which feature two Dirac cones (dashed black circle), and (c) when they couple in the presence of the MO effect (g 0.3), lifting the Dirac cones. (d) (unnormalized) Berry curvature associated with the LP in c. Physical

mechanism for the appearance of plexciton Dirac points: (e) we specialize to nanopillar transitions, which are parallel to the xy plane and (f) plot of magnitude of the electric eld of magneto-SP modes as a function of wavevector. In the absence of the MO effect, only the wavevector-parallel components Ek(blue) are present. Thus, the nanopillars experience no coupling with modes whose wavevectors are perpendicular to the transition dipole. Along these directions, degeneracies between the SP and the exciton modes are not lifted, yielding two plexciton Dirac points. Nonzero tangential components Eh (red)

emerge upon inclusion of the MO effect, lifting these degeneracies.

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j. Second, this alignment needs to be horizontal, as a vertical component of the dipole will couple to the vertical component of the electric eld E k

^z^z, and this coupling, unlike its horizontal

counterpart, does not vanish for any j. It is important to note that neither of these properties require the use of the superlattice, which will be exploited for a different purpose (the design of a global gap for topologically protected edge modes, as explained in the next paragraphs). Therefore, a standard organic molecular crystal with aligned transition dipoles lying on the horizontal xy plane will sufce. These plexciton Dirac cones, which, to our knowledge, have not been reported in the past, should be easily detectable by collecting the reected light spectra upon excitation of the plexcitonic system in a grating, Otto or Kretschmann congurations36, by systematically scanning across |k| and j values. For a general (|k|, j), the spectrum should consist of two dips as a function of dispersed energy, each associated with the corresponding eigenenergies of the LP and UP. However, the two dips merge at the Dirac cones. In a standard plexciton dispersion measurement, one only scans across |k|. As we are interested in a two-dimensional dispersion, the scan must also be performed across j. To trace the formation of the Dirac points,

Supplementary Note 3 and Supplementary Figs 911 show two-dimensional cross-sections of the dispersion curves in Fig. 2.

Plexciton topological modes. Having elucidated the mechanism for the formation of plexciton Dirac cones, we proceed to entertain a more ambitious goal. We aim to engineer topologically protected plexcitons by opening the Dirac cones using a time-reversal symmetry breaking (TRSB) perturbation15. To accomplish this, we now assume that the dielectric spacer has MO properties; that is, upon application of a perpendicular magnetic eld, its permittivity becomes anisotropic, Ed ! E

$MO,

$MO

Ed ig 0

ig Ed 0

0 0 Ed

E

2

4

3

5: 4

Materials associated with this dielectric tensor exhibit Faraday effect and are of great interest in the fabrication of optical isolators37. For the near-infrared and visible spectra, yttrium iron garnets (Y3Fe5O12, YIG) substituted with Bi (BiYIG), Ce (CeYIG)

or other rare earths provide high Faraday rotation with low optical absorption38,39. In this article, we are interested in the new SP modes, denoted as magneto-SP modes, arising at the interface of the plasmonic metal and the MO layer. The solution for the latter is highly non-trivial, and we refer the reader to our solution in the Supplementary Notes 1 and 2, which build a perturbation theory on the small MO parameter g, generalizing a simpler calculation of Chiu and Quinn40. Figure 2c shows the same calculation as Fig. 2b except for the inclusion of the MO effect. Interestingly, we notice that the Dirac cones have been lifted, yielding a global gap because of the H-aggregate superlattice dispersion. A physical understanding of the latter phenomenon can be obtained by appealing to Fig. 2e again. Within our perturbation theory, the magneto-SP modes differ from their original SP counterparts in that there are additional tangential components (Ey, red) to the electric eld. The clockwise vortex vector eld is a signature of TRSB; it becomes counterclockwise upon change of direction of the magnetic eld. This tangential electric eld is solely responsible for opening the Dirac cones at the critical angles j p2 ; 3p2, where the original eld (blue) ceased

to couple to the excitons. Hence, we have concocted a situation where anticrossings occur for all azimuthal angles j. To characterize the topology of the resulting band-structure, we numerically compute the Berry curvature for each plexciton branch41; we show that of the LP in Fig. 2d. Its integral with

respect to the Brillouin zone is the so-called Chern number C, an integer which, if nonzero, signals a topologically nontrivial phase. Figure 2d clearly shows that this integral is non-vanishing, and in fact, adds up to C 1 (by the sum rule of Chern numbers, the

upper branch necessarily has C 1). Intuitively, it is also clear

that most of the nontrivial topology, and hence, Berry curvature, is concentrated in the vicinity of what used to be the Dirac cones. In passing, we note that considerable attention has recently been given to magneto-SPs, where the magnetic eld is applied parallel (instead of perpendicular) to the metal lm itself, yielding dispersion relations, which are nonreciprocal40,42. Curiously, this arrangement does not give us the vortex vector eld we are looking for, although it might be intriguing to explore the connection between these magneto-SPs and the ones exploited in our present work, arising from a perpendicular magnetic eld.

So far, all the described calculations have been carried out in the bulk. By virtue of the bulk-boundary correspondence15, we expect topologically protected one-way edge modes associated with this setup. To compute them, it is convenient to keep periodic boundary conditions for the magneto-SP modes, yet consider two domains of excitons on top (Fig. 3a), one with (in-plane) dipoles pointing along l (red dipoles) and the other one with vertical dipoles along ^z (blue dipoles). Pictorially, this setup resembles a donut with two icings, where the donut is the metal with toroidal geometry, and the two icings are the domains of excitons separated by two interfaces located at y Ly2 and

y 0, where Li is the total width of the simulated sample along

i (in our calculations, we take Lx 40 mm, Ly 6 mm). The Chern

numbers associated with the bulk LP branch of each domain are C 1 and C 0, respectively. Hence, the plexcitons for the

blue domain are topologically trivial. This can be understood by recalling that no plexciton Dirac points occur when dipoles are vertically aligned, regardless of the MO effect. As this system is perfectly periodic along the x-direction, kx is still a good quantum number, and Fig. 3b shows the corresponding plexciton dispersion. This band-structure is essentially a projection of the gapped 2D bulk band-structures of both domains of plexcitons onto one axis kx with additional states spanning the gap between the LP and UP branches. Inspection of the nature of these mid-gap states reveals that they have substantial exciton and magneto-SP character, and that they are precisely the edge states we are searching for: one band has positive (negative) dispersion

4

a b

y

x

Energy (eV)

2

0 2 2

0

kx (107 m1)

Figure 3 | Topologically protected edge modes. (a) Simulation of edge modes where magneto-SPs are computed in the torus geometry. Two domains of organic layers are placed on top of it (just like two icings on a donut). In-plane (red) and out-of-plane (blue) transition dipoles yield topologically nontrivial and trivial plexcitons, respectively. Topologically protected one-way plexcitons appear at the interfaces (white domains). Each interface features a different plexciton direction of motion. (b) One-dimensional dispersion relation o(kx) for the setup in a. Bulk LPs (blue) and

UPs (red) separated by edge modes (green) featuring positive and negative dispersions, respectively, and localized along each interface.

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a

2

0

t = 0 fs t = 300 fs t = 600 fs

t = 264 fs

y (m)

y (m)

y (m)

2

2

0

2

b

2

0

2

20

20

10

10

0

10

10

20

10

0

0 0

10

10 10

20

10

0

10

x (m)

x (m) x (m)

c

2

0

2

t = 0 fs

0

t = 132 fs

d

y (m)

20 20

10 10

x (m) x (m) x (m)

Figure 4 | Snapshots of dynamics of topologically protected edge modes. (ad) The dynamics of an initially localized plexciton along the y 0 and

y L2 3 mm interfaces of our simulation is shown. (ad) The exciton and magneto-SP components are shown, respectively. These modes, which are

robust to disorder, have substantial excitonic and magneto-SP components and travel in opposite directions along different interfaces.

and is localized along y 0 y Ly2. Thus, by preparing a

plexciton wavepacket localized along one of the interfaces, and making sure it is composed of energy states within the global topological gap, one ensures that transport occurs robustly without much probability of backscattering. The reason being that elastic backscattering requires coupling between counterpropagating modes, which are separated by a distance

Ly

2 , which is large compared with the width of the corresponding wavefunctions along ^y.

Figure 3b was generated with the parameters Ed1, g 0.3,

yielding a minimum gap between plexciton branches (at the wavevectors k* of the original Dirac points) of 2jJ k

j0:23 eV.

The crossing of the SP and exciton dispersion curves happens at2.86 eV. Given typical linewidths associated with the various dissipative mechanisms at room temperature (gexc,relB5 MeV,

gexc,dephB40 MeV, gSP,relB10 MeV, where rel and deph stand for

relaxation and dephasing), our parameters lie within the strong-coupling regime, namely, 2jJ k j4 10 meV (ref. 7). Ideally,

we would like to have all edge states lying within the global topological gap. For the present simulation, only a fraction of the latter (close to the anticrossings) lies within the global topological gap; the rest are degenerate with bulk plexcitons. This scenario arises as a consequence of the large exciton dispersion anisotropy (JxaJy) compared with the strength of jJ kj for all k. The latter

is a result of optimization: thicker MO spacers yield larger values of jJ k j but, due to the evanescent nature of SP modes, also

reduce the overall couplings jJ kj for kak* (Supplementary

Note 3). Typical MO garnets have similar g but higher Ed values (for an external magnetic eld of 0.01 T, Ed 6:25 and g 0.1

(ref. 38)), yielding a strong index mismatch in our setup. As explained in Supplementary Note 3, our future aim is to boost

jJ k j by considering various MO layer thicknesses a as well as

novel MO garnet compositions, which maximize the g=Ed ratio, the latter of which is not fundametally limited. Strategies may include the consideration of MO garnet sphere arrays43, plasmonic/magnetic metal nanostructures44, Ce substituted YIGs45 or Eu nanocrystals46.

Figure 4a,b shows snapshots of the dynamics associated with the edge states lying within the global gap. Figures 4a,b and c,d show an edge plexciton wavepacket that starts localized at x Lx4 and x Lx4, respectively, and track the one-way

(to the left or to the right) nature of its micrometre-scale motion within the subpicosecond timescale. The dispersion of the edge plexcitons is such that a subset falls within the light cone o4ckx=

Ed

p so far-eld excitation and detection

of this fraction is possible via direct interaction with the organic layer. The rest of the plexcitons can be probed using the already mentioned SP measurement techniques, by launching plexcitons exciting the metallic layer itself. Furthermore, the ballistic and one-way nature of these modes can in principle be demonstrated using uorescence microscopy47. As explained, the robustness of one-way transport for these edge modes, even in the presence of disorder, relies on the global topological gap, which is a consequence of our 2D H-aggregate-type superlattice design (see the Methods for details). In terms of experimental feasibility, we note that some compromise on topological protection might be acceptable if using a plain molecular crystal is much easier than constructing

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1 Pk syk e ik r (here, Ni is the effective number of nanopillars along the i-th direction) and rewriting equation (5) in reciprocal space, we obtain

H

modes syn

the superlattice as long as scattering between kx states is not strong.

DiscussionTo summarize, we have described the design of exotic plexcitons via a judicious choice of material and electromagnetic excitation modes. We showed that Dirac cones and topologically protected edge states emerge from relatively simple hybrid organic/ inorganic nanostructures. Even though we have not precisely identied an explicit MO material which fully satises our requirements, we believe its design is within reach, and is the subject of our present investigations. It is also worth noting that the physical origin of the described edges states is different from that of edge plasmons in disk geometries48,49, although the connections are worth exploring. An interesting extension of this work is the consideration of plexcitonic Dirac point opening without TRSB, which might give rise to rich phases in analogy to other Dirac systems50. In general, the possibility of directed migration of excitation energy at the nano- and mesoscale offers exciting prospects in light-harvesting and all-optical circuit architectures. Furthermore, given the recent experimental discovery of nonlinear many-body effects such as BoseEinstein condensation of organic cavity-polaritons5153 and plexcitons54 at room temperature, the introduction of the novel features described in this letter enriches the scope of these materials as a test-bed for novel many-body quantum phenomena.

Methods

Hamiltonian of the plexciton setup. A quantum mechanical description of the plexciton setup is given by a Hamiltonian,

H Hexc HSP Hexc SP; 5 where each of the terms denotes the energetic contributions from the excitons in the organic layer, the SPs and the coupling between them. More specically (: 1),

Hexc

Pn;s onsyn sn

Pn 6 nJnn syn sn h:c:

;

6

N N

p

Pk Hk, where Hk is given by equation (1).

Modelling of superlattice. For our simulations, both of the band-structure and the edge states, we chose the length parameters Dh 100, Dv 88, Wx 10,

Wy 75, Wz 70 nm, and take b 13.1 (see Supplementary Note 3 for expla

nation of choice of parameters). Denoting the transition dipole l m^

l, we estimate

l

p

12 D1:7 104 D (D Debye), where Nnp rnpVnp is the number of

chromophores in the nanopillar, rnp 37 chromophores per nm3; ^

Nnp

l is the in-plane unit vector making an angle of a 224) with respect to ^x, that is,

^

l cosa^x sina^y. Choosing the simulation values for Hexc in equation (2) to be

Dx Dy 50 nm, we obtained the effective parameters

oeff

0.57 eV, Jx 1.04

meV and Jy 0.31 meV. As explained in Supplementary Note 3, for strong exci

ton-plasmon coupling, the relevant density of chromophores is the weighed average between the density in the nanopillars and the null one in the void space. The density and vertical thickness of this superlattice are in line with the typical parameters that yield strong coupling in plexciton systems7.

Experimental considerations. In terms of the fabrication of the plexciton setup, we warn that the creation of BiYIG layers typically need high temperature and oxygen, which is incompatible with deposition on Ag or organic materials. Hence, the MO layer could rst be deposited on garnet substrates such as GGG (Gd3Ga5O12) (111), and subsequently oated off by dissolving or polishing the substrate. One should then transfer the lm on a Ag-coated substrate and the organic layer may be deposited and patterned on garnet.

Effects of disorder and global gap. It is important to note that, owing to the topological nature of these states, perfect lattices are not required, so the robustness of the one-way edge states holds as long as orientational and site energy disorder induce perturbations which are smaller than the topological anticrossings. We tested these ideas by simulating lattices with disorder in the site energies

oeff

!

oeff

as well as in the orientations of the dipoles ^

l ! cos Df cos a Da

^x sin a Da

^y

sinDf^z, where Dj are chosen to be

Gaussian random variables centred at 0 and having disorder widths sj for each j

D

oeff

oeff , a, f. By systematically varying these widths independently and keeping track of the presence of the one-way edge states, we noticed that the latter survive under large amounts of disorder, whose thresholds are approximately located at s

o

0:25 eV, saB57 and sfB28. Furthermore, we suspect that these values are lower bounds, as the disorder in these simulations is exacerbated by the torus conguration of our simulation, where periodicity along the x axis is conserved. In other words, owing to the latter, for a given superlattice coordinate along the y axis, we xed the same values of disorder Dj across all x (that is, the disorder was perfectly correlated along x).

One may check that if Jxr0 or Jyr0, the global topological gap is not guaranteed anymore, and edge modes may become degenerate with bulk modes. Under certain circumstances, these two types of modes could hybridize because of impurity potentials, yielding channels connecting one edge to the other, opening backscattering channels. Importantly, however, the resulting band-structure would formally remain topologically nontrivial (in terms of the Chern numbers Ca0 for the plexciton bands) even in the absence of such global gap, so even if perfect oneway transport is not observed in these cases, signatures of the latter may remain. The latter observation applies if it is experimentally more convenient to use of a standard molecular crystal (with negligible exciton dispersion) rather than the proposed superlattice. These issues will be explored in future work. In the mean time, it sufces to note that, as a proof of concept, a global gap that hosts topologically protected edge states can be obtained by using an H-aggregate-type organic superlattice like the one suggested in this article.

HSP

Xko k aykak; 7

HexcSP

Xk;nJknaksyn eik r h:c: 8

Here, syn sn

and ayk ak

label the creation (annihilation) operators for the

collective exciton at the n-th nanopillar and the k-th SP mode, respectively, where n and k are (two-dimensional) in-plane vectors denoting a position and a wavevector, respectively. J-aggregation of chromophores results in a collective transition dipole ln at an excitation energy on, whereas the dispersion energy of the k-th SP mode is denoted o(k). Dipolar interactions Jnn couple the various nanopillars. The coupling between the exciton and the SP depends on the average in-plane location rn of the n-th nanopillar, and is also dipolar in nature,

J

kn

o k

2E0SLk

s e a

k

z k

eik r mn E k

: 9

Here, S is the SP mode surface area, Lk denotes a vertical (z-direction) mode-length of the SP, which guarantees that the total energy of a SP prepared at the kth mode is quantized at the energy o(k), E(k) is an appropriately scaled electric eld of the corresponding mode, and e a k

z k

yields a mean-eld average of the interaction of the evanescent SP eld (with decay constant aorg in the organic layer) over the chromophores at different vertical positions of the nanopillar; it optimizes the interaction such that one may assume the nanopillar is a point-dipole located at the mean height z k

. The latter average renders the originally 3D system into an

effectively 2D one. Detailed derivations of equations (6)(9) are available in the Supplementary Notes 13.

Assuming perfect periodicity of the superlattice (on

o, ln l) and only NN

and NNN dipolar interactions, we can re-express Hexc (equation (6)) in terms of k modes. As explained in Supplementary Note 3, it is possible to approximate Hexc

up to O(|k|2) as arising from an effective simplied rectangular (rather than monoclinic) lattice aligned along the x, y axes, and with NN interactions only. This is a reasonable thing to do, as the topological effects we are interested arise at relatively long wavelengths. Within this approximation, we may construct Fourier

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Acknowledgements

J.Y.-Z thanks Justin C.W. Song for a discussion on edge magnetoplasmons. J.Y.-Z acknowledges partial support from startup funds at UC San Diego. J.Y-Z., T.Z., V.B. and M.B. were supported by an Energy Frontier Research Center funded by the US Department of Energy, Ofce of Science, Ofce of Basic Energy Sciences under Award Number DESC0001088. S.K.S. was supported by the Defense Threat Reduction Agency grant HDTRA1-10-1-0046. S.K.S. is also grateful to the Russian Government Program of Competitive Growth of Kazan Federal University. Finally, C.A.R. and M.C.O. acknowledge support of the Solid-State Solar-Thermal Energy Conversion Center (S3TEC), award DE-SC0001299, and FAME, a STARnet Center of SRC supportedby DARPA and MARCO.

Author contributions

J.Y.Z. and S.K.S. conceived the original idea, developed the theoretical formalism in the main text and Supplementary Information and carried out the numerical work. J.Y.Z., S.K.S., T.Z., M.C.O., C.A.R., V.B., and M.A.B. developed and discussed ideas, contributed their technical expertise at every stage of the project, and wrote the nal version of the manuscript and Supplementary Information.

Additional information

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How to cite this article: Yuen-Zhou, J. et al. Plexciton Dirac points and topological modes. Nat. Commun. 7:11783 doi: 10.1038/ncomms11783 (2016).

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