Tunable unidirectional surface plasmon polaritons at the interface between gyrotropic and isotropic conductors

Tunable unidirectional surface plasmon polaritons at the interface between gyrotropic and isotropic conductors Yi Liang,1,2 Samaneh Pakniyat,3 Yinxiao Xiang,1 Jun Chen,4 Fan Shi,1,5 George W. Hanson,3 AND Cheng Cen1,* Department of Physics and Astronomy,West Virginia University, Morgantown,West Virginia 26506, USA Guangxi Key Lab for Relativistic Astrophysics, Center on Nanoenergy Research, School of Physical Science and Technology, Guangxi University, Nanning, Guangxi 530004, China Department of Electrical Engineering, University ofWisconsin-Milwaukee,Milwaukee,Wisconsin 53211, USA Department of Electrical andComputer Engineering and Peterson Institute of NanoScience and Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, USA Key Laboratory of Computer Vision and System ofMinistry of Education, Tianjin University of Technology, Tianjin 300384, China *Corresponding author: cheng.cen@mail.wvu.edu


INTRODUCTION
Through the coupling with surface plasmon polaritons (SPPs), it is possible to confine light at the nanoscale [1][2][3][4]. While the excitation of SPPs with tailored directionalities can be achieved using various metacoupler designs [5], the propagation of SPPs in solid state systems is usually reciprocal [6], which means that a forwardtraveling SPP wave can be easily backscattered by disorders. Since such backscattering is detrimental for SPP-based information processing applications, the development of nonreciprocal plasmonic platforms [7][8][9][10][11][12][13][14][15][16][17][18][19][20] that enforce unidirectional SPP propagations is of great importance. For a typical metal-dielectric plasmonic interface, when the metallic medium is gyrotropic, several earlier theoretical works have predicted the existence of frequency windows in which SPPs travel unidirectionally [21][22][23][24][25][26][27]. However, more recent investigations revealed a thermodynamic paradox [28,29] associated with such prediction. When considering the nonlocal effects present in realistic materials, there are no longer any strictly unidirectional frequency ranges, even in lossless systems. Theoretically, the unidirectionality of many classes of SPPs, including conventional surface magnetoplasmons, is rooted in a nonphysical flat dispersion asymptote at large wavenumbers.
This kind of SPP would become bidirectional in the presence of nonlocal effects.
In order to find alternative interfaces that can support truly unidirectional SPPs, less common plasmonic material combinations need to be considered. Interestingly, it is also possible to excite SPPs at interfaces formed between two metals, given one of them is gyrotropic. At such metal-metal interfaces [23,25,30,31], not only are strictly unidirectional frequency windows present without any thermodynamic paradox [29], the unidirectionality of SPPs in these frequency windows is further protected by the topology of the bulk optical band structure [32][33][34][35][36]. The unidirectionality of such topological SPPs arises from a bias-induced asymmetry, and thus it is preserved when nonlocal effects are present. To experimentally demonstrate this effect, in this work we excite and characterize terahertz (THz) surface magnetoplasmons at a grating-coupled gyrotropic-metal/isotropic-metal interface at low temperatures. The theoretically predicted nonreciprocal SPP dispersions are clearly observed. This system also allows active control over both the SPP directionality and frequency with weak magnetic fields. To bridge our work with future applications, we also discuss how (τ = 0+) − k 2 c 2 |. Parameters used are ω p /2π = 6 THz, ω c /2π = 1.4 THz, ω m /2π = 10 THz, and k max c /2π = 100 THz. (I) Topologically protected unidirectional SPPs (red) emerging from the bulk gaps. (K) and (L) Simulated electric field intensities for the sample structure illustrated in (J). The size of the simulation area is 2 mm × 0.7 mm. The grating period used for SPP launching is d = 2π/k 0 = 84 µm. the nonreciprocal SPP propagation is affected by practical material characteristics.

THEORETICAL BACKGROUND
We first introduce the basic concepts of the topologically protected interfacial SPPs studied here by considering a lossless heterostructure formed from a gyrotropic conductor that is biased by B = Bŷ and a regular Drude metal ( Fig. 1(A)). The permittivity of the isotropic metal is a scaler (ε m < 0), and the permittivity tensor of the gyrotropic conductor can be expressed as In a continuous medium, due to the behavior of the Hamiltonian at infinite momentum in the 2D planar momentum space, the usual assumption of a local material model can lead to noninteger Chern numbers [32,33]. Here, in order for the isotropic metal and the gyrotropic conductor to share a common bandgap and have Chern numbers that are meaning for the bulkedge correspondence, we use an interpolated material model to calculate the dispersion, which uses a parameter τ ∈ [0, 1] to represent a continuous transition between the isotropic (τ → 0+) and gyrotropic (τ → 1−) limits. Thus, the permittivity elements are defined as where ε ∞ is the infinite frequency permittivity, ω p and ω m are the plasma frequencies of the two materials, and ω c is the cyclotron frequency. In Eq. (2), a spatial frequency cutoff k max is introduced following the method in Ref [32,33]. This value is chosen to be large enough (>100ω p /c ) such that its effect is only significant at the limit of k → ∞. At small or moderate momentums, the electromagnetic response is very similar to a local model with k max → ∞. It should be noted that a hydrodynamic model would be more accurate than the spatial cutoff model at large wavenumber, but for the low-wavenumber SPPs considered here, the spatial cutoff model is expected to provide accurate results.
Inside the bulk materials, plane waves propagating perpendicular to the field include both TE and TM modes. It can be shown that the TE modes are all topologically trivial and thus will not be discussed further. The TM bulk modes, on the other hand, follow the dispersion relation where ε eff = (ε t 2 − ε g 2 )/ε t is the effective permittivity for the field-orthogonal polarizations. Equation (3)  In a continuous medium, by mapping the unbounded k-space onto the Riemann sphere [33], it is possible to assign an integer Chern number to each bulk band that is separated from the others by a complete bandgap [37][38][39]. At the limit of τ → 0 (isotropic metal), the Chern numbers of the three bands are all zero, while for τ → 1 (gyrotropic conductor), the nontrivial Chern numbers are found to be [40] C U = 1, C M = −2, and C L = 1. As τ varies from 0 to 1, the band structure changes are shown in Figs. 1(B)-1(H). Two topological phase transitions occur during this process, each associated with changes in the Chern numbers. The first transition, marked by the opening of BG2 at k = 0, takes place near τ → 0. The second transition occurs between τ = 0.6 and τ = 0.8, where C M and C L abruptly change from (1,0) to (−2, 1). Similar to what is discussed in Ref [32,33], this transition is driven by the closing and reopening of BG1 at very large k.
The distinct topologies at the two sides of the interface mandates the formation of in-gap surface states. The sum of the Chern numbers for each band below the gap (gap Chern number) indicates the number of topologically protected surface modes crossing the gap. The gap Chern numbers of BG1 and BG2 are 1 and −1. Thus, two in-gap unidirectional SPP branches, one in each gap, are present in this system ( Fig. 1(I), red). The SPP dispersion equation derived based on the nonlocal model [Eq. (2)] is shown in the supporting materials. For relatively small k values (e.g., those accessible in experiments), the SPP dispersion can be approximated by the simpler expression obtained at the local limit k max → ∞: Within the bulk gaps, the SPP unidirectionality is topologically protected and therefore immune to backscattering. One example of such effect is shown in Figs. 1(J)-1(L). In this setup ( Fig. 1(J)), the grating at the center is designed to be in resonance with two in-gap SPP modes traveling in opposite directions: (k 0 , ω + ) and (−k 0 , ω − ), which are indicated by blue dots in Fig. 1(I). When excited by light of frequency ω − , only the left-traveling mode is allowed at the interface ( Fig. 1(K)). In comparison, when tuning the light frequency to ω + , the SPP wave launched only travels toward the right ( Fig. 1(I)). In both cases, the propagating SPP experiences no scattering at the sharp bending points placed along its paths. We note that SPP modes outside the bulk gaps can also be unidirectional within selected frequency ranges ( Fig. 1(I)), though these out-of-gap unidirectional waves are not topologically protected and subject to scattering with the bulk modes.

EXPERIMENTAL RESULTS
To realize the gyrotropic/isotropic heterostructure in experiment, Au films are deposited on the surfaces of undoped InSb single crystals. Here, magnetized InSb serves as the gyrotropic conducting medium. Its high electron mobility and small effective mass allows the generation of strong magneto-optical effects with weak magnetic fields [41][42][43][44][45][46][47][48][49][50][51]. Grating structures are fabricated in the Au layer to provide the momentum matching and allow the excitation of SPPs from free-space light ( Fig. 2(A)). In experiment, the magnetic field is kept parallel to the grating grooves and the incident polarization is perpendicular to the grooves. The SPPs launched need to satisfy the momentum conservation rule where d is the grating constant and θ is the incident angle. Equation (5) (Fig. 2(C)) and the other with an Au grating structure on top ( Fig. 2(D)). In this InSb sample, the electron mobility maximizes at 50 K (µ = 22 m 2 /Vs), where the carrier density n = 1 × 10 20 m −3 and the carrier scattering rate = e /µm * = 5.7 × 10 11 s −1 (m * = 0.014m 0 is the effective electron mass) are relatively low, corresponding to an effective plasma frequency of On the unpatterned region, free-space light cannot directly couple to the SPPs, and the most noticeable feature observed is the field-induced reflectance modulations near ω c = e B/m * , which blueshift as the magnetic field increases ( Fig. 2(C)). In comparison, the reflectance peaks associated with SPPs, as well as their anticrossing with the cyclotron resonances, are clearly visible in the data acquired on the region covered by Au grating (Fig. 2(D)). At zero field, since Eqs. (4) and (5) are both symmetric in the k-space, a pair of SPP modes are simultaneously generated, which travel in opposite directions but have the same frequency and propagation length (Fig. 2(F)). This pair of SPPs are represented by a single peak in the reflectance spectrum (Fig. 2(D)). Applying a nonzero field, the SPP branches below ω c are no longer k-symmetric (Figs. 2(G) and 2(H)). When B = 0.45 T, ω c overlaps with the SPP frequency and causes the SPP propagation in both directions to be significantly damped (Fig. 2(G)). When B further increases to 0.7 T, a pair of SPP modes are again excited with approximately the same frequency, though the left-traveling wave has a substantially longer propagation length compared to the right-traveling one (Fig. 2(H)). Figure 2(E) plots the SPP resonance frequencies calculated for different field values, which are highly consistent with the experimental observations. In general, at 50 K, left-traveling and right-traveling SPPs excited by the normalincidence light are degenerate in energy. They are either generated in pairs or both suppressed when varying the magnetic field.
The situation is different when θ = 0. As shown in Fig. 3(A), mounting the sample next to a silver mirror and forming a normal angle between the two surfaces, a THz beam focused on the sample-mirror boundary hits the two surfaces consecutively with an angle and gets retroreflected in the end. One complication associated with this setup is that the reflectance measured is a combined result of θ = ±45 • incidences: θ for the beam portion that hits the sample first is 45 • , whereas the other portion that gets reflected by the mirror first hits the sample surface with a -−45 • angle. The sample reflectances corresponding to the two incident angles are linked by symmetry. For simplicity, we first focus on the case of θ = 45 • , and the other case of θ = −45 • can be directly inferred by the relation of r (−θ, B) = r (θ, −B).
For θ = 45 • , Eq. (5) results in two slanted lines in the (k, ω) plane that are no longer k-symmetric (solid lines, Figs. 3(E)-3(G)). Such asymmetry lifts the energy degeneracy between the SPPs excited in opposite directions and thus allows them to be distinguished experimentally as shown by the two separate SPP peaks in Fig. 3(C). As the external field varies, the cyclotron resonance intercepts with the left-and right-traveling SPP modes one at a time (Figs. 3(C) and 3(D)), allowing them to be manipulated individually. This effect produces multiple magnetic field windows in which the SPP launched is unidirectional (Fig. 3(D), green shaded regions). Near B = 0.35 T, the left-traveling mode is suppressed by the field, and only the right-traveling SPP is allowed at the interface (Fig. 3(F)). The situation is reversed near B = 0.57 T, where only the left-traveling SPP is allowed (Fig. 3(G)).
Comparing the SPP dispersions shown in Figs. 2(G), 2(H) and Fig. 3(G) to the ideal lossless case shown in Fig. 1(I), a big difference is that the SPP modes between the two bulk gaps are no longer unidirectional for magnetic fields larger than 0.4 T. Since the SPPs outside the bulk gaps are not topologically protected, their directionality is sensitive to the changes in symmetry-unrelated sample parameters. When the carrier density of InSb is low (e.g., at 50 K), the SPP dispersion between the bulk gaps is strongly affected by the surface quality of the Au film. In this work, a literature-reported Au film permittivity of ε m = −800 + i500 obtained from SPP-based measurements [52,53] is found to be most consistent with our experimental observations [54]. If the bulk permittivity value (ε Drude m = −2.3 × 10 5 + i8.6 × 10 5 ) [55] can be realized at the surface of the Au films, the SPP dispersion at the interface will become much more similar to the ideal case (Fig. S1, Supplement  1).
At higher temperatures, thermal activation causes the electron density in InSb to rise, changing the bulk and surface band structures significantly (Fig. S2 of Supplement 1, Figs. 4(C), 4(G), and 4(K)). Figures 4(A), 4(E), and 4(I) compare the normal-incidence reflectances measured at 50 K, 180 K (n = 4.5 × 10 20 m −3 , Here, instead of extracting the absolute reflectance r (B), which requires a reference sample to be repeatedly positioned to precisely match the different sample positions at varied temperatures, the reflectance data is processed in a self-referenced manner. That is, the raw reflectance signal detected at a nonzero field is divided by its zero-field value (r (B)/r (0)), which captures the field-induced reflectance change without the need for separate reference measurements.
On bare InSb (Figs. 4(A), 4(E), 4(I), top), the most outstanding features of r (B)/r (0) are a peak near ω * p and a plateau that broadens as B increases. The right edge of the plateau traces the bottom of the upper bulk band, which approximates ω c when ω * p ω c . With the Au grating, the broadening of the plateau appears to be intercepted at the zero-field SPP frequency (Figs. 4(A), 4(E), 4(I), bottom), and the peak near the plateau edge indicates the SPP frequency under the influence of the nonzero field. The single SPP peak detected at θ = 0 • splits into two when θ changes to ±45 • (Figs. 4(B), 4(F), and 4(J)). At 50 K, the higher frequency peak represents the forward-propagating (as compared to the in-plane direction of the incidence light) modes generated by the ±45 • incidences [ω  (Fig. 4(C)). At 180 K and 250 K, the two peaks come from the forward-traveling mode generated by the +45 • incidence [ω , the raised ω * p makes the bulk gaps (grey shaded areas) and the unidirectional magnetic field windows (green shaded areas) both more sizable at higher temperatures. As a result, it is much easier to generate topologically protected unidirectional SPPs at 250 K (modes in regions where the two shaded areas overlap). Nonetheless, due to the reduced electron mobility, the propagation lengths of these modes are short.
An important feature of the SPPs at the gyrotropic/isotropic interface is that they are nonreciprocal. Since the operation of B → −B mirrors the k-asymmetric SPP dispersion curve with respect to k = 0, for any single θ value, flipping the field orientation will lead to the excitation of different SPP modes. The θ = ±45 • dual incident angle setup weakens such effect significantly, but due to the nonperfect beam shape in the experiment and the weak imbalance between the two incidence angles, the nonreciprocal SPP excitations can still be detected. The insets of Figs. 4(B) and 4(F) plot the differences between the data measured at ±0.7 T. The absolute value of the difference signal maximizes near the two SPP resonances, but the sign of the signal changes in between, indicating a spectral weight shift between the two modes as the field orientation flips. In comparison, the ±0.7 T signals measured on the bare InSb surface are almost identical (Fig. S3, Supplement 1). Notably, the sign of the difference signal also flips when raising the temperature from 50 K to 180 K. This observation is consistent with the different SPP asymmetry found at the two temperatures (Figs. 4(C) and 4(G)). At 50 K, the k < 0 SPP branch below BG2 is stronger than the k > 0 branch in the sense that it has much longer propagation lengths. At 180 K, the k > 0 branch becomes the stronger one since the k < 0 branch is no longer present. , bulk gaps are marked by grey and unidirectional field windows are marked by green. The unidirectionality of the modes residing in the overlapping regions is topologically protected. The modes residing inside the green region but outside the grey region are also unidirectional but are not topologically protected.

DISCUSSION
In summary, we have successfully excited unidirectional THz surface magnetoplasmons at the gyrotropic-metal/isotropic-metal interface formed between magnetized InSb single crystal and Au film. Varying the external magnetic field, the directionality of the interfacial SPPs that can propagate is flexibly tunable. Controlling the carrier density in InSb, we are able to change the bulk gap sizes and observe different nonreciprocal interface dispersions. These results demonstrate the possibility of realizing topologically protected active nonreciprocal plasmonics at THz frequencies using heterostructures formed from materials with distinct optical band topologies.
For practical applications, we find it desirable to work with a gyrotropic conductor with a relatively larger effective plasma frequency ω * p . On one hand, it helps widen the bulk optical bandgaps and thus extend the bandwidth of the topologically protected interfacial SPPs. On the other, it makes the unidirectionality of the out-of-gap SPPs, which are not topologically protected, more robust against the parameter fluctuations in the isotropic metal layer. Since a larger ω * p also means that a larger magnetic field is required to generate a sizable gyrotropic permittivity element, there exists an optimal window of ω * p for unidirectional SPP applications, the value of which is determined by the effective carrier mass in the gyrotropic medium. For InSb, such optimal ω * p window occurs when the carrier density is at the level of 10 21 −10 22 m −3 . With such electron densities, however, the propagation lengths of the unidirectional interfacial SPPs are significantly limited by both the interface quality of the deposited Au film and the electron mobility in InSb. The interfacial properties of Au films are often affected by the presence of an additional adhesion layer (e.g., Ti as used in this work). By depositing directly on substrates cooled to cryogenic temperatures without an adhesion layer [56] or using unconventional adhesion materials [57,58], ultrasmooth Au films can be fabricated with greatly improved plasmonic performances. For undoped InSb single crystals as used in this experiment, the 10 21 − 10 22 m −3 electron density values are found near room temperature. Since in this case the dominating mobility limiting factor is the carrier scattering with optical phonons [59][60][61], there is little room for improvement. Alternatively, similar electron densities can also be found in lightly doped n-type InSb near liquid nitrogen temperature but with much higher mobility values [61,62]. Through these future material optimizations, the long-living unidirectional SPPs that can be excited at gyrotropic-metal/isotropic-metal interfaces will provide a unique venue for exploring nonreciprocal plasmonic functionalities on chip.