Surface plasmons induce topological transition in graphene/α-MoO3 heterostructures

Polaritons in hyperbolic van der Waals materials—where principal axes have permittivities of opposite signs—are light-matter modes with unique properties and promising applications. Isofrequency contours of hyperbolic polaritons may undergo topological transitions from open hyperbolas to closed ellipse-like curves, prompting an abrupt change in physical properties. Electronically-tunable topological transitions are especially desirable for future integrated technologies but have yet to be demonstrated. In this work, we present a doping-induced topological transition effected by plasmon-phonon hybridization in graphene/α-MoO3 heterostructures. Scanning near-field optical microscopy was used to image hybrid polaritons in graphene/α-MoO3. We demonstrate the topological transition and characterize hybrid modes, which can be tuned from surface waves to bulk waveguide modes, traversing an exceptional point arising from the anisotropic plasmon-phonon coupling. Graphene/α-MoO3 heterostructures offer the possibility to explore dynamical topological transitions and directional coupling that could inspire new nanophotonic and quantum devices.

wavevector. Given fixed frequency , we need to determine the in-plane momenta of the polaritons. The quasistatic Maxwell's equations inside α-MoO3, = 0, lead to the bulk dispersion relation The boundary conditions for on the top and bottom surfaces of α-MoO3 give two equations satisfied by 1,2 , which can be interpreted as two complex reflection "phase shifts": Solving the boundary condition equation on the top and bottom interfaces, one obtains the wellknown expression for the complex phase shift 1,2 : where sub is the permittivity of the substrate under α-MoO3, is the two-dimensional sheet conductivity of the top interface (if there is any 2D conducting layer there), and , ∥ are the complex dielectric permittivities of the slab. We have defined ∥ = cos 2 + sin 2 as the effective in-plane dielectric permittivity of α-MoO3 where is the angle between q and the x direction. Note that are not necessarily real numbers and can be imaginary for evanescent waves. Existence of solutions to Equation S2 leads to the polariton eigenmode condition in the near-field limit: where ∈ ℕ 0 is the mode order. Graphene affects the mode via its effect on the phase shift 1 , which leads to hybridization between graphene plasmons and α-MoO3 phonon polaritons. Equations S1, S3, and S4 determine the isofrequency contours (IFCs) in Supplementary Figure 1b for < 0, > 0, > 0, which happens in the experimental frequency range. Combining Equations S1, S3, and S4 gives: The first-order hybrid mode ( = 0) crosses over from a hyperbolic plasmon-phonon polariton (HP 3 ) for < to a surface plasmon-phonon polariton (SP 3 ) for > where the critical angle is defined such that Re ∥ ( ) = 0. The critical wavevector = ( ) can be computed analytically when → 0 (assuming lossless), 1 → , and 2 → − . With these assumptions, Equation S5 simplifies to where ≡ 2 , and the dielectric constant of air has been replaced by unity. Solving Equation

S6
gives In the limit of charge neutrality ( → ∞), the value of diverges. This is the asymptote of the unbounded hyperbolic IFC when there is no doping. As soon as ≠ 0, becomes finite. We know also ( = 90°) corresponds to an unhybridized graphene plasmon: likewise will only diverge when = 0. We assume then that between ≤ ≤ 90° all diverge together, and we track the value of to check the boundedness of the IFC. Bounded and unbounded IFCs are not homeomorphic and thus we say that a topological transition occurs upon incremental doping away from = 0. This topological transition admits discontinuous changes in the polaritonic density of states, analogously to the Lifshitz transition for Fermi surfaces 3 .

c. Two-dimensional limit
In certain limits, the whole device can be viewed as a two-dimensional plane with net sheet conductivity ̂=̂g +̂M oO 3 with contributions from both graphene and α-MoO3, where ̂M oO 3 = 4 ( − 1) is related to the in-plane components of the dielectric tensor of α-MoO3. Hybrid polaritons can be viewed as the plasmonic modes of the 2D plane, whose dispersions are determined by Therefore, the dispersion is in-plane hyperbolic if Im and Im are opposite in sign. By increasing the doping level of graphene, one can obviously render them the same sign, thus changing the polaritons to non-hyperbolic. However, the 2D limit makes the approximation that the in-plane electric field is independent on z inside the α-MoO3 slab, which is generally not true. Note that there is no HP 3 mode in the 2D limit. Hybrid polaritons in the strong coupling regime are always SP 3 and the hyperbolic α-MoO3 phonon is effectively a surface phonon at .

d. Complex root finding algorithm
Equation S5 must be solved numerically when ≠ 0 since the polariton appears in the phase shift term. We search for roots using the LMFIT 4 wrapper for scipy.optimize methods in Python with the Nelder-Mead simplex algorithm. Since logarithms are multivalued for complex arguments, one must be careful when choosing how to represent the multivalued function. Branch cuts can cause problems for local root-finding algorithms like Nelder-Mead and indeed do cause our algorithm to yield unusual results for the imaginary part of at intermediate . To improve the stability of the root finding algorithm, we do not search for roots of Equation S5 directly, but rather of the pointwise product of its first several Riemann sheets 5 .

Supplementary Note 2: Coupled plasmons and in-plane hyperbolic phonons a. Derivation of Equation 1
Supposing the Fermi energy | | ≫ , the graphene optical conductivity computed with the Kubo formula using the bare bubble approximation in Gaussian units (m/s) is given by 6 : where is the scattering rate of graphene. Only intraband contributions are considered and nonlocal effects are neglected since ≪ . The dielectric tensor components of α-MoO3 are modeled by the TO-LO equation 7 with a single oscillator along each crystallographic direction: where and are the frequencies of the transverse and longitudinal optical phonon modes, respectively. In the middle reststrahlen band, only the x-direction or [100] phonon is strongly dispersive, so we set the y-direction permittivity to a constant value for simplicity. By combining Equations S8-S10, we can then write: with ( , ) ≡ 2 cos 2 serving as a "coupling strength" parameter. Setting Equation S12 to zero, the complex corresponding to the self-sustained polariton modes were computed analytically in symbolic mathematical software.

b. Hybrid mode scattering rate
From Equation S8, we can also obtain an approximate semi-analytical equation for the direction-dependent scattering rate in the two-dimensional limit. The scattering rate of the hybrid mode Γ can be obtained self-consistently by satisfying the following relation: where Υ(E F ) ≡ 2 2 ℏ 2 0 ( 2 − 2 )/( 2 | |) > 0. When = 90°, the righthand side goes to zero, forcing Γ = on the lefthand side. In contrast, at = 0°, Γ is not forced to equal the phonon scattering rate. Since the righthand side is always positive, it must be the case that ≤ Γ(θ) ≤ given that > . Numerical solutions to Equation S13 are plotted in Figure 4d in the main text.
To obtain the hybrid mode scattering rate Γ = ′′ for a finite thickness slab, we can compute ′′(ω) and ′( ± ℎ) numerically using Equation S5 and then use a central difference formula with ℎ = 10 −9 cm -1 to estimate the group velocity: We remark that solutions obtained through Equations S13 and S14 are qualitatively consistent, but Equation S14 can be unstable at intermediate angles.

Supplementary Figure 2: Raman spectroscopy on graphene/α-MoO3 heterostructures with varying
WSe2 layer number. a, G and 2D peaks in Raman spectra of graphene/α-MoO3 heterostructures with 0 (blue), 1 (red), and 2 (purple) layers of tungsten diselenide (WSe2) stacked on top and oxidized. b, the oxidized WSe2 hole dopes graphene by a charge-transfer process, as evidenced by the shifted G and 2D Raman peaks. Since oxidation is self-limited, only the topmost layer is oxidized. The pristine bottom WSe2 layer in the bilayer structure acts as a spacer that reduces the doping level. Also, the bilayer G peak has a secondary undoped peak due to the many bubbles and tears in the bilayer sample.

a. Extracting doping level from near-field data
Experimental Fermi energies reported in the main text for WOx/graphene/α-MoO3 (monolayer) and WOx/1L-WSe2/graphene/α-MoO3 (bilayer) samples were determined by fitting polariton dispersions to energy-momentum ( , ) data with the of graphene as a free parameter. In other words, we solve the following nonlinear least squares problem assuming homoscedasticity: ̂= argmin ∑ ( ( , ) − ( )) 2 (S15) We report to the nearest 0.05 eV, since likelihood-based 95% confidence intervals for this type of estimation were previously determined to be about ±0.03 eV 8 . The best-fit dispersion for the 0.45 eV sample is shown in Figure 3e in the main text. Similar dispersions were obtained for the 0.60 eV samples.

b. Raman spectroscopy
To corroborate the doping levels extracted from fitting polariton dispersions, we also performed Raman spectroscopy on these samples. Supplementary Figure 2a shows Raman spectra on monolayer, bilayer, and undoped samples. The undoped graphene G and 2D peaks appear at 1582 cm -1 and 2689 cm -1 , respectively. The bilayer G peak shifts to 1591.5 cm -1 and the monolayer G peak shifts further to 1596 cm -1 while 2D peaks shift to 2695 cm -1 and 2691 cm -1 for bilayer and monolayer samples, respectively. The gray lines in Supplementary Figure 2b represent the G and 2D peak positions corresponding to approximate doping levels from a standard analysis 9 using empirical parameters from Supplementary Reference 10. The undoped sample was used as a zerocarrier reference point. Qualitatively, our analysis agrees with trends determined from dispersion fitting.
Additionally, the bilayer sample has a secondary undoped G peak likely corresponding to the many bubbles and tears seen in topography (Supplementary Figure 5a). Finally, we remark that the spectral weight ratio of G to 2D peaks changes visibly upon doping. Overall, Raman spectroscopy is consistent with the appearance of plasmon polaritons in near-field images from substantial doping of the graphene layer.  ). a, near-field amplitude image at laser frequency =980 cm -1 of an undoped graphene/α-MoO3 heterostructure with a gold antenna deposited by electron beam lithography. b, black and blue profiles correspond to black and blue dashed lines in a on and off graphene, respectively. The graphene edge is indicated by the white dashed line. There is no significant difference in polariton momentum (look at red dashed lines), suggesting that there is little to no work-function-mediated doping of graphene on crystalline α-MoO3. c, the out-of-plane hyperbolic upper reststrahlen band [001] polariton mode is also consistent with little to no doping. d, plasmons are not visible at =1050 cm -1 , where there are no phonon hybridization effects. Also, the graphene-covered region is not brighter than bare α-MoO3.

c. Samples without WOx
In this section, we present an extended dataset for an undoped graphene/α-MoO3 heterostructure without WOx. We show that this data is consistent with low to no doping, that is, a Fermi energy in graphene approximately at charge neutrality. This is in contrast to Raman data on graphene on amorphous, oxygen-deficient MoO3-x heterostructures showing evidence of holedoping graphene to ~0.28 eV from charge transfer between constituent layers 11 .
In Supplementary Figure 4a

d. Effect of WSe2 layer on optical properties
A 1.4 nm monolayer of pristine WSe2 was included in Figure 3e calculations in the main text. The mid-infrared dielectric tensor of bulk WSe2 from Supplementary Reference 12 was used in calculations. We assume the topmost, oxidized WOx layer has a dielectric permittivity of unity. A thin layer of WSe2 on a polaritonic medium has a significant effect on the confinement of polariton modes due its high dielectric permittivity 13 , but it cannot induce a topological transition of α-MoO3 phonon polaritons without doped graphene. Supplementary Figure 7 shows the calculated dispersions with and without WSe2 both with and without graphene. The WSe2 layer can increase confinement of α-MoO3 modes, but will not allow them to surpass and initiate a topological transition. Lastly, note that Figure 2f in the main text shows that [100] modes are actually less confined on WOx, implying that hybridization with graphene plasmons is the dominant effect on α-MoO3 phonon polaritons.