Abstract
Plasmon polaritons in topological insulators attract attention from a fundamental perspective and for potential THz photonic applications. Although polaritons have been observed by THz farfield spectroscopy on topological insulator microstructures, realspace imaging of propagating THz polaritons has been elusive so far. Here, we show spectroscopic THz nearfield images of thin Bi_{2}Se_{3} layers (prototypical topological insulators) revealing polaritons with up to 12 times increased momenta as compared to photons of the same energy and decay times of about 0.48 ps, yet short propagation lengths. From the images we determine and analyze the polariton dispersion, showing that the polaritons can be explained by the coupling of THz radiation to various combinations of Dirac and massive carriers at the Bi_{2}Se_{3} surfaces, massive bulk carriers and optical phonons. Our work provides critical insights into the nature of THz polaritons in topological insulators and establishes instrumentation and methodology for imaging of THz polaritons.
Introduction
Plasmon polaritons in metals, doped semiconductors and twodimensional (2D) materials have wide application potential for fieldenhanced spectroscopies, sensing, imaging, and photodetection^{1,2,3,4,5,6,7}. Recently, topological insulators (TIs) have been attracting large attention as an alternative class of plasmonic materials, as they can support plasmon polaritons that are formed not only by massive but also by Dirac carriers^{8,9}. Dirac plasmon polaritons (DPPs) are electromagnetic modes that can be formed when the massless Dirac carriers at the surfaces of a TI collectively couple to electromagnetic radiation^{8,10,11,12,13}. Due to the 2D nature of these collective excitations, the polariton momentum – and thus the field confinement – is much larger than that of free space photons of the same energy^{8,9}, similar to plasmons in 2D materials such as graphene^{14,15}. In addition, spinmomentum locking of the electrons in the TI surface states promises additional unique phenomena, such as spinpolarized plasmon waves^{16}. For these reasons, DPPs in TIs have attracted significant interest from both a fundamental and applied perspective^{17,18,19,20,21}.
DPPs have been reported experimentally by terahertz (THz) farfield spectroscopy of TI microresonator structures^{8,9,22}. Due to the unavoidable presence of massive bulk carriers in TI thin films and crystals^{9,13,23,24}, however, the analysis and interpretation of the observed THz resonances has been challenging and controversial. The presence of THz bulk phonon polaritons in the prototypical TI Bi_{2}Se_{3} further complicates the observation of DPPs^{24,25}. Although farfield spectroscopy of TI resonators has provided various fundamental insights, it does not allow for imaging of polariton propagation or mode profiles. Imaging of polaritons – often performed by scatteringtype scanning nearfield optical microscopy (sSNOM)^{26,27}  has proven to be of great importance in the infrared spectral range to distinguish between propagating and localized modes in thin layers and resonator structures, for measuring polariton propagation lengths, phase and group velocities, lifetimes and modal field distributions^{6,7,14,15,27,28,29,30}. However, due to the lack of THz nearfield imaging instrumentation offering high spatial and spectral resolution, as well as a large signaltonoise ratio, the realspace imaging of THz polaritons is still a challenging task^{31,32,33}.
Here, we demonstrate that THz polaritons in TIs can be imaged spectroscopically by sSNOM employing the tunable monochromatic radiation from a powerful THz gas laser and interferometric detection. Specifically, we performed THz polariton interferometry on epitaxially grown Bi_{2}Se_{3} films of different thicknesses \(d\). Challenged by the short polariton propagation lengths, we determine the polariton wavevector (and thus dispersion) by complexvalued nearfield analysis of our experimental data. Further, using an analytical model, we show that the experimental polariton dispersion can be reproduced when Dirac and massive carriers at the surfaces, massive bulk carriers and optical phonon are taken into account. From propagation length measurements and group velocities determined from the experimental polariton dispersion, we finally determine the decay times of the THz polaritons, amounting to ~0.48 ps and thus being comparable or even better than that of typical plasmon decay times in standard (nonencapsulated) graphene.
Results
THz nanoimaging
For real space imaging of polaritons in thin Bi_{2}Se_{3} films grown by molecular beam epitaxy (Methods section) on sapphire (Al_{2}O_{3}), we used a THz sSNOM (based on a commercial setup from Neaspec, Germany; sketched in Fig. 1a; for details see Supplementary Note 1 and Supplementary Fig. 1), where a metallized atomic force microscope (AFM) tip acts as a THz nearfield probe. The tip is illuminated with monochromatic THz radiation from a gas laser (SIFIR50, Coherent Inc., USA), which is focused with a parabolic mirror. Via the lightning rod effect, the tip concentrates the THz radiation into a nanoscale nearfield spot at the tip apex^{34}. The momenta of the near fields are large enough to launch polaritons in Bi_{2}Se_{3}. The tiplaunched polaritons that propagate to the edge are reflected at the edge and propagate back to the tip (illustrated in Fig. 1b by the red sine waves). Consequently, by recording the tipscattered field as function of tip position, we map the interference of forward and backward propagating polaritons. Collection and detection of the tipscattered field is done with the same parabolic mirror and a GaAsbased Schottky diode (WR0.4ZBD, Virgina Diodes Inc. USA). To obtain backgroundfree nearfield signals, the tip is oscillated at a frequency Ω (tapping mode) and the detector signal is demodulated at higher harmonics n of the oscillation frequency, \(n\Omega\). Demodulated nearfield amplitude and phase signals, \({s}_{n}\) and \({\varphi }_{n}\), were obtained by synthetic optical holography (SOH), which is based on a Michelson interferometer where the reference mirror (mounted on a delay stage) is translated at a constant velocity along the reference beam path^{35,36}. The interferometric detection is key to improve the background suppression and to enable a complexvalued analysis of nearfield profiles, which is critical for a reliable measurement of the wavelength of polaritons with short propagation lengths. To increase the signaltonoise ratio, we used commercial gold tips with a large apex radius^{37} of ~500 nm (Team Nanotec LRCH). They were oscillated at a frequency of about \(\Omega\) ≈ 300 kHz with an amplitude of ~200 nm.
Representative THz nearfield amplitude and phase images, \({s}_{3}\) and \({\varphi }_{3}\), of a d = 25nmthick Bi_{2}Se_{3} film are shown in Fig. 1d. The phase image reveals a dark fringe on the Bi_{2}Se_{3}, which is oriented parallel to the film edge (obtained by scratching the film) and resembles sSNOM images of shortrange plasmon and phonon polaritons observed at midIR frequencies on graphene and hBN, respectively^{14,38,39}. In contrast, the simultaneously recorded topography image (Fig. 1c) reveals a homogenous thickness of the Bi_{2}Se_{3} film, from which we can exclude that the fringe in the THz image is caused by a thicknessdependent dielectric material contrast.
To verify that the dark fringe in Fig. 1d can be attributed to polaritons, we recorded nearfield amplitude and phase line profiles perpendicular to the Bi_{2}Se_{3} edge, \({s}_{3}(x)\) and \({\varphi }_{3}(x)\), at different THz frequencies \(\omega\). The phase profiles \({\varphi }_{3}(x)\) are shown in Fig. 1e. We find that the minimum of the nearfield phase signal (marked by an arrow, corresponding to the dark fringe of Fig. 1d) shifts towards the Bi_{2}Se_{3} edge with increasing frequency, supporting our assumption that the nearfield signal reveals polaritons of several micrometer wavelength. However, the lack of multiple signal oscillators prevents a straightforward measurement of the polariton wavelength.
Complexvalued analysis of THz line profiles
To establish a procedure for measuring the polariton wavelengths \(\lambda\)_{p} and corresponding wavevector \({k}_{{{{{{\rm{p}}}}}}}^{\prime{}}=2\pi /{\lambda}_{{{{{{\rm{p}}}}}}}\), we performed a complexvalued analysis of the THz nearfield amplitude and phase line profiles, as illustrated in Fig. 2 with data obtained on a 25nmthick Bi_{2}Se_{3} film that were recorded at 2.52 THz. We first constructed complexvalued line profiles \({\sigma }_{3}\left(x\right)={s}_{3}{\left(x\right)e}^{i{\varphi }_{3}\left(x\right)}\) and plotted the corresponding trajectories in the complex plane, i.e. as a polar plot where the polar amplitude and phase represent the nearfield amplitude \({s}_{3}\left(x\right)\,\)and the nearfield phase \({\varphi }_{3}(x)\), respectively. We find that the nearfield signal describes a spiral (red data in Fig. 2e, based on the line profiles shown in Fig. 2c) around a complexvalued offset C that corresponds to the nearfield signal at large tipedge distances \(x\). The spiral stems from a harmonic oscillation (describing a circle) whose amplitude decays with increasing \(x\), indicating a single propagating mode that is strongly damped. Indeed, after removing the offset (blue data in Fig. 2e), we obtain a monotonically decaying amplitude and a linearly increasing phase signal (Fig. 2d). To verify that the spiral reveals a damped propagating wave, we fitted the complexvalued experimental line profile (red data in Fig. 2e) by
which describes the electric field of a backreflected, radially (i.e. tiplaunched) propagating damped wave (black curve in Fig. 2e). The fitting parameters are A, \({k}_{{{{{{\rm{p}}}}}}}\) and C. \({k}_{{{{{{\rm{p}}}}}}}\)is the complexvalued polariton wavevector \({k}_{{{{{{\rm{p}}}}}}}={k}_{{{{{{\rm{p}}}}}}}^{\prime }+i{k}_{{{{{{\rm{p}}}}}}}^{\prime \prime},\) where \({k}_{{{{{{\rm{p}}}}}}}^{ \prime{} }=2\pi /{\lambda }_{{{{{{\rm{p}}}}}}}\) and 1/\({k}_{{{{{{\rm{p}}}}}}\,}^{\prime\prime}\)is the propagation length. The complexvalued offset \(C\) corresponds to the tipsample nearfield interaction in absence of polaritons that are backreflected from the Bi_{2}Se_{3} edge, i.e. when the tip is far away from the edge. A is a complexvalued factor. This offset is present in all polariton maps obtained by sSNOM and described for example in refs. ^{40,41}. It can be also seen in our numerical simulations discussed in Fig. 2f–i. After removing the offset C, we obtain \(A{e}^{2{k}_{{{{{{\rm{p}}}}}}}^{\prime\prime}x}{e}^{i4\pi x/{\lambda }_{{{{{{\rm{p}}}}}}}}/\sqrt{2x}\), where the term \(A{e}^{2{k}_{{{{{{\rm{p}}}}}}}^{\prime\prime}x}/\sqrt{2x}\) describes a decaying amplitude and the term \({e}^{i4\pi x/{\lambda }_{{{{{{\rm{p}}}}}}}}\) a linearly increasing phase \(\varphi \,=\,4\pi x/{\lambda }_{{{{{{\rm{p}}}}}}}\) when the distance x to the Bi_{2}Se_{3} edge increases. The linear relation between distance \(x\) and phase \(\varphi\) thus reveals directly the polariton wavelength according to \({\lambda }_{{{{{{\rm{p}}}}}}}\,=\,4\pi {{{{{\rm{\cdot }}}}}}\Delta x/\Delta \varphi\). The fits (black curves in Fig. 2c–e) match well the experimental data, in particular the linear increase of the phase (Fig. 2d) when C is removed, which is the key characteristics of a propagating mode. The fit yields a wavevector of \({k}_{{{{{{\rm{p}}}}}}}\) = 0.55 + 0.17i μm^{−1}, corresponding to a normalized wavevector \(q\) = \({k}_{{{{{{\rm{p}}}}}}}/{k}_{0}\) = 10.4 + 3.2i, where \({k}_{0}\) is the photon wavevector. Note that for fitting we excluded the first 200 nm from the edge, in order to avoid a potential influence of tipedge nearfield interaction and edge modes (see discussion below). In the Supplementary Figs. 2 and 3 of the Supplementary Note 2 we show all recorded line profiles and fittings reported in this work.
To verify our analysis of the experimental sSNOM profiles and the determination of the polariton wavevector \({k}_{{{{{{\rm{p}}}}}}}\), we performed wellestablished numerical model simulations^{30,42}. As illustrated in Fig. 2f, the sSNOM tip is described by a vertically orientated dipole source and the Bi_{2}Se_{3} layer by a 2D sheet of an optical conductivity \(\sigma\) (blue layer in Fig. 2f). The electric field \({E}_{{{{{{\rm{z}}}}}}}=\left{E}_{{{{{{\rm{z}}}}}}}\right{e}^{i{\varphi }_{z}}\) (describing the sSNOM signal) below the dipole is calculated and plotted as function of the distance x between the dipole source and the sheet edge (mimicking the scanning of the tip, Fig. 2g). To obtain the sheet conductivity \(\sigma\), we assume that optical polariton modes are probed in our experiment (where the polariton fields normal to the film have opposite sign at the top and bottom surface; for further discussion see below). In this case, \(\sigma\) can be obtained from the dispersion relation of polaritons in a 2D sheet within the large momentum approximation^{43,44}:
where q is the normalized complexvalued polariton wavevector along the film, \(\omega\) the frequency, and \({\varepsilon }_{{{{{{\rm{sub}}}}}}}\) = 10 the permittivity of the Al_{2}O_{3} substrate^{9,24,45} (see Supplementary Note 3 and Supplementary Fig. 4). We note that the approximation of the layer by a 2D sheet conductivity is justified, despite Bi_{2}Se_{3} being an anisotropic material hosting hyperbolic polaritons, as the layers are much thinner than the polariton wavelength and the damping of the polaritons is rather high (for further details see Supplementary Fig. 5 and Supplementary Notes 3 and 4). For \({k}_{{{{{{\rm{p}}}}}}}\) = 0.55 + 0.17i μm^{−1} (according to our analysis of the experimental sSNOM line profiles in Fig. 2c–e), we obtain the simulated nearfield line profiles shown in Fig. 2g–i. An excellent agreement with the experimental sSNOM line profiles is found. Particularly, complexvalued fitting of the simulated line profiles by a radially decaying wave (according to Eq. (1)) yields a polariton wavevector \({k}_{{{{{{\rm{p}}}}}}}\) = 0.63 + 0.15i μm^{−1}, which closely matches the value determined from the experimental sSNOM line profiles. The simulations thus confirm that the experimental sSNOM line profiles reveal a polariton mode that (i) is launched by the tip, (ii) propagates as a damped wave radially along the Bi_{2}Se_{3} film, (iii) reflects at the edge of the film back to the tip, and (iv) is scattered by the tip.
For a demonstration of our conclusions, we show the electric nearfield distribution around the dipole source placed above the conductivity sheet, \({{{{{\rm{Re}}}}}}\left[{E}_{z}\left(x,y\right)\right].\) When the dipole is placed inside the sheet, i.e. far away from any edge, we clearly observe radially propagating wavefronts (Fig. 3a, upper panel). Most important, fitting of the field distribution along the horizontal dashed red line by \({{{{{\rm{Re}}}}}}\left[A{e}^{i{k}_{{{{{{\rm{p}}}}}}}x}/\sqrt{x}\right]\) yields \({k}_{{{{{{\rm{p}}}}}}}\) = 0.60 + 0.15i μm^{−1} (Fig. 3a, lower panel), which agrees well with the wavevectors \({k}_{{{{{{\rm{p}}}}}}}\) obtained from the experimental and simulated sSNOM line profiles. The slight discrepancies between the various wavevectors (<15%) may be attributed to the excitation of an edge mode when the tip comes into close proximity of the sheet edge. The edge mode can be actually recognized in the simulations when the dipole is located at the sheet edge (Fig. 3b, upper panel). Its wavelength is slightly reduced compared to that of the sheet mode, and its fields are strongly confined to the edge, similarly to what has been observed for plasmon and phonon polariton modes in graphene and hBN flakes^{30,46,47}. Most important, since the edge mode propagates exclusively along the edge and its field is strongly confined to the edge, its contribution in the experimental and simulated sSNOM line profiles perpendicular to the edge (shown in Fig. 2) is minor, when we allow small uncertainties below 15%.
Analysis of the THz polariton dispersion
In Fig. 4a we show the phase images of two Bi_{2}Se_{3} films of different thicknesses recorded at different frequencies, which were used to determine the polariton dispersions according to the procedure described in Fig. 2 (red symbols in Fig. 4b; \(q\) = \({k}_{{{{{{\rm{p}}}}}}}/{k}_{0}\) is the polariton wavevector \({k}_{{{{{{\rm{p}}}}}}}\) normalized to the photon wavevector \({k}_{0}\)). As is typical for polaritons, we find that \({{{{{\rm{Re}}}}}}[q]\) increases with increasing frequency \(\omega\) and with decreasing film thickness d. To understand the physical origin of the polaritons, we compare the experimental results with analytical calculations of the polariton dispersion employing Eq. (2) and various conductivity models describing the Bi_{2}Se_{3} film (see Supplementary Note 4, Supplementary Fig. 6 and Supplementary Table 1). For modeling of the conductivity, we performed Hall measurements of Bi_{2}Se_{3} films of different thicknesses (see Supplementary Fig. 7 and Supplementary Notes 5 and 6), yielding an effective 2D carrier concentration of about \({n}_{2{{{{{\rm{D}}}}}},{{{{{\rm{Hall}}}}}}}\) = 2.5 × 10^{13} cm^{−2} for layers with a thickness of ~25 nm (Supplementary Fig. 8).
First, we assume that only optical phonons (OP) and Dirac carriers (DC) located on both film surfaces contribute to the conductivity, yielding \({\sigma =\sigma }_{{{{{{\rm{OP}}}}}}+{{{{{\rm{DC}}}}}}}^{{{{{{\rm{model}}}}}}}=\frac{\omega d}{4\pi i}{\varepsilon }_{{{{{{\rm{phonon}}}}}}}+\,2{\sigma }_{{{{{{\rm{Dirac}}}}}}},\) where \({\varepsilon }_{{{{{{\rm{phonon}}}}}}}\) is the bulk dielectric function of Bi_{2}Se_{3} including optical phonons. \({\sigma }_{{{{{{\rm{Dirac}}}}}}}\) is the sheet conductivity of one Bi_{2}Se_{3} surface (see Supplementary Eq. (8))^{11,12,25}, assuming that the sheet carrier concentration at one surface is \({n}_{{{{{{\rm{Dirac}}}}}}}={n}_{2{{{{{\rm{D}}}}}},{{{{{\rm{Hall}}}}}}}/2\) = 1.25 × 10^{13} cm^{−2} (independent of the thickness). The resulting dispersions are shown by the orange curves in Fig. 4b. We find that the calculated wavevectors are significantly larger than the experimental values, from which we conclude that pure Dirac plasmon polaritons coupled to phonon polaritons cannot explain the experimental dispersion. As a second case, we assume that all carriers are bulk carriers (BC), yielding \({\sigma =\sigma }_{{{{{{\rm{OP}}}}}}+{{{{{\rm{BC}}}}}}}^{{{{{{\rm{model}}}}}}}=\frac{\omega d}{4\pi i}({\varepsilon }_{{{{{{\rm{phonon}}}}}}}+{\varepsilon }_{{{{{{\rm{Drude}}}}}}}),\) where \({\varepsilon }_{{{{{{\rm{Drude}}}}}}}\) is the Drude contribution to the bulk dielectric function (see Supplementary Note 4). In this case, we use an effective threedimensional (3D) concentration of the massive carriers according to \({n}_{{{{{{\rm{bulk}}}}}}}\) = \({n}_{2{{{{{\rm{D}}}}}},{{{{{\rm{Hall}}}}}}}\)/25 nm = 1 × 10^{19} cm^{−3}. We obtain the dispersions shown by the light blue curves in Fig. 4b. For the 25 nm thick film a reasonable match of the experimental polariton wavevectors is found, however, not for the 60 nm thick film, revealing that polaritons comprising only bulk carriers (BC) and optical phonons (OP) cannot explain the polariton dispersions either. A similar observation is made (green curves in Fig. 4b) when we assume that all carriers stem from a massive 2D electron gas (2DEG)  which is known to exist in TIs due to surface band bending^{18,48,49}  yielding \({\sigma =\sigma }_{{{{{{\rm{OP}}}}}}+2{{{{{\rm{DEG}}}}}}}^{{{{{{\rm{model}}}}}}}=\frac{\omega d}{4\pi i}{\varepsilon }_{{{{{{\rm{phonon}}}}}}}+2{\sigma }_{2{{{{{\rm{DEG}}}}}}}\), where \({\sigma }_{2{{{{{\rm{DEG}}}}}}}\) (see Supplementary Eq. (9))^{11,12,49} is the sheet conductivity of one Bi_{2}Se_{3} surface with \({n}_{2{{{{{\rm{DEG}}}}}}}={n}_{2{{{{{\rm{D}}}}}},{{{{{\rm{Hall}}}}}}}/2\) = 1.25 × 10^{13} cm^{−2}.
We next assume that both massive bulk carriers and surface carriers (either Dirac or massive 2DEG carriers) contribute to the conductivity,\(\,{\sigma =\sigma }_{{{{{{\rm{OP}}}}}}+{{{{{\rm{BC}}}}}}+{{{{{\rm{DC}}}}}}}^{{{{{{\rm{model}}}}}}}=\frac{\omega d}{4\pi i}({\varepsilon }_{{{{{{\rm{phonon}}}}}}}+{\varepsilon }_{{{{{{\rm{Drude}}}}}}})+2{\sigma }_{{{{{{\rm{Dirac}}}}}}}\) and \({\sigma =\sigma }_{{{{{{\rm{OP}}}}}}+{{{{{\rm{BC}}}}}}+2{{{{{\rm{DEG}}}}}}}^{{{{{{\rm{model}}}}}}}=\frac{\omega d}{4\pi i}({\varepsilon }_{{{{{{\rm{phonon}}}}}}}+{\varepsilon }_{{{{{{\rm{Drude}}}}}}})+2{\sigma }_{2{{{{{\rm{DEG}}}}}}}\), respectively. Note that massive bulk carriers in thin films are barely captured by the Hall measurements (due to their supposedly smaller mobility compared to that of the Dirac carriers; see Supplementary Note 6). We thus assign the total Hallmeasured 2D concentration (\({n}_{2{{{{{\rm{D}}}}}},{{{{{\rm{Hall}}}}}}}\) = 2.5 × 10^{13} cm^{−2}) fully to either Dirac or massive 2DEG carriers for both the 25 nm and 60 nm thick film, and consider an additional massive bulk carrier concentration \({n}_{{{{{{\rm{bulk}}}}}}}\). From Hall measurements of thick Bi_{2}Se_{3} films we estimate \({n}_{{{{{{\rm{bulk}}}}}}}\) = 2.15 × 10^{18} cm^{−3} (Supplementary Note 6), yielding the bulk Drude contribution \({\varepsilon }_{{{{{{\rm{Drude}}}}}}}\). The calculated dispersions (black and purple lines Fig. 4c, labeled OP+BC+2DEG and OP+BC+DC, respectively) again do not match the experimental results (red symbols).
In the following, we attempt to fit the experimental dispersions by various parameter variations. We first added an additional 2DEG contribution, such that \({\sigma =\sigma }_{{{{{{\rm{OP}}}}}}+{{{{{\rm{BC}}}}}}+{{{{{\rm{DC}}}}}}+2{{{{{\rm{DEG}}}}}}}^{{{{{{\rm{model}}}}}}}=\frac{\omega d}{4\pi i}({\varepsilon }_{{{{{{\rm{phonon}}}}}}}+{\varepsilon }_{{{{{{\rm{Drude}}}}}}})+{2\sigma }_{{{{{{\rm{Dirac}}}}}}}+2{\sigma }_{2{{{{{\rm{DEG}}}}}}}\) (red lines Fig. 4d). Using for each surface the carrier concentrations \({n}_{{{{{{\rm{bulk}}}}}}}\) = 2.15 × 10^{18} cm^{−3} and \({n}_{{{{{{\rm{Dirac}}}}}}}=\) 1.25 × 10^{13} cm^{−2} (from the Hall measurements), we obtain the fitting parameter \({n}_{2{{{{{\rm{DEG}}}}}}}=\) 0.375 × 10^{13} cm^{−2} for each surface. We note that in Hall measurements we cannot separate Dirac and massive carriers directly (Supplementary Note 6) to confirm these carrier concentrations, but they are close to the numbers reported in literature^{9,23,50,51,52}. Interestingly, the experimental dispersions can be also fitted without considering a 2DEG (employing the conductivity model \({\sigma =\sigma }_{{{{{{\rm{OP}}}}}}+{{{{{\rm{BC}}}}}}+{{{{{\rm{DC}}}}}}}^{{{{{{\rm{model}}}}}}}\), light purple line in Fig. 4d). However, we have to assume an increased bulk carrier concentration of \({n}_{{{{{{\rm{bulk}}}}}}}=\) 3.72 × 10^{18} cm^{−3} (fitting parameter; \({n}_{{{{{{\rm{Dirac}}}}}}}\,\)as before). Although the required bulk carrier concentration is nearly twice as high as the one estimated from our Hall measurements, it represents a reasonable value reported in literature^{9,23,24,49,50}, which may not be fully revealed by Hall measurements (see discussion in Supplementary Note 6). We also fitted the experimental dispersions without considering Dirac carriers employing the conductivity model \({\sigma =\sigma }_{{{{{{\rm{OP}}}}}}+{{{{{\rm{BC}}}}}}+2{{{{{\rm{DEG}}}}}}}^{{{{{{\rm{model}}}}}}}\,\)(gray line in Fig. 4d) with \({n}_{{{{{{\rm{bulk}}}}}}}\,\)= 2.15 × 10^{18} cm^{−3} and \({n}_{2{{{{{\rm{DEG}}}}}}}\) being the fit parameter. A good matching of the experimental dispersion is achieved for \({n}_{2{{{{{\rm{DEG}}}}}}}=\) 0.95 × 10^{13} cm^{−2} for each surface. However, this value of \({n}_{{{{{{\rm{total}}}}}},2{{{{{\rm{DEG}}}}}}}=\) 1.9 × 10^{13} cm^{−2} is significantly higher than what has been reported in literature^{18,49,51}. Altogether, we conclude from our systematic dispersion analysis that an unambiguous clarification of the nature and concentration of carriers forming the polaritons is difficult without additional experiments where the concentrations of the different carriers can be measured separately. However, such measurements are challenging to carry out at room temperature due to thermal smearing, and lowtemperature measurements are unlikely to be accurate at room temperature due to thermal excitations. On the other hand, considering that Bi_{2}Se_{3} growth is highly reproducible and that Dirac carriers have been verified in samples like ours^{53}, these carriers may contribute to the signal.
Polariton propagation length and lifetime
From the complexvalued fitting of the sSNOM line profiles we also obtain the propagation length of the polaritons, \({L}={1}/{k}_{{{{{{\rm{p}}}}}}\,}^{\prime\prime}\). For the 25 nm thick film we find \(L\) = 6 μm and accordingly the amplitude decay time \(\tau =L/{v}_{{{{{{\rm{g}}}}}}}\) = 0.48 ps, which is similar to the decay times measured by farfield extinction spectroscopy of Bi_{2}Se_{3} ribbons^{9}. The group velocities \({v}_{{{{{{\rm{g}}}}}}}\) were obtained from the polariton dispersion according to \({v}_{{{{{{\rm{g}}}}}}}=d\omega /d{k}_{{{{{{\rm{p}}}}}}\,}^{\prime}\) = 0.042c.
Interestingly, the polariton decay time in Bi_{2}Se_{3} is comparable or even larger than that of graphene plasmons at infrared frequencies^{14,15,28}. On the other hand, the inverse damping ratio of the Bi_{2}Se_{3} polaritons, \({\gamma }^{1}\) = \({k}_{{{{{{\rm{p}}}}}}\,}^{\prime}/{k}_{{{{{{\rm{p}}}}}}\,}^{\prime\prime}\) = 3.2, and their relative propagation length, \(L{/\lambda }_{{{{{{\rm{p}}}}}}}=\frac{1}{2\pi \gamma }=\) 0.5, is significantly smaller than that of the infrared graphene plasmons (\({\gamma }^{1}\) = 5)^{15}. To understand the small relative propagation length of the Bi_{2}Se_{3} polaritons, we express it as a function of the polariton wavevector \({k}_{{{{{{\rm{p}}}}}}}=q\omega /c\), group velocity \({v}_{{{{{{\rm{g}}}}}}}\,\)and decay time \(\tau\):
From Eq. (3) it becomes clear that for a given \(\tau\), \({v}_{{{{{{\rm{g}}}}}}}\) and \(q\), the relative propagation length decreases with decreasing frequency; that is simply because the temporal oscillation period becomes longer. Since typical THz frequencies are more than one order of magnitude smaller than infrared frequencies, one can expect, generally, that the relative propagation length of THz polaritons in thin layers (including 2D materials) is significantly smaller than that of infrared polaritons. For an illustration, we show in Fig. 5a the calculated relative propagation length \(L{/\lambda }_{{{{{{\rm{p}}}}}}}\,\)of 2.52 THz polaritons of 0.48 ps amplitude decay time as function of \({{{{{\rm{Re}}}}}}[q]\) and \({v}_{{{{{{\rm{g}}}}}}}\). The white symbol marks the relative propagation length of the THz polaritons observed in the 25nmthick Bi_{2}Se_{3} film. We find that propagation lengths of more than wavelength, \(L{/\lambda }_{{{{{{\rm{p}}}}}}} \, > \, 1\), are possible only for large group velocities (>0.1c) when the normalized polariton wavevector (i.e. polariton confinement) is moderate (\({{{{{\rm{Re}}}}}}[q]\) > 10). To achieve \(L{/\lambda }_{{{{{{\rm{p}}}}}}} \, > \, 1\) for group velocities below 0.05c, large polariton wavevectors with \({{{{{\rm{Re}}}}}}[q]\) > 20 are required. For comparison, we show in Fig. 5b the relative propagation length for polaritons of 0.6 ps amplitude decay time. Plasmon polaritons of such rather exceptionally large amplitude decay time were observed experimentally in highquality graphene encapsulated in hBN (marked by black symbol). Only because of their very large confinement (\({{{{{\rm{Re}}}}}}[q]\) = 70; owing to coupling with adjacent metallic gate electrodes), these polaritons possess a large relative propagation length of \(L/{\lambda }_{{{{{{\rm{p}}}}}}}=1\).5, although their group velocity is rather small (\({v}_{g}\) = 0.014c). Generally, we conclude from Eq. (3) and Fig. 5 that the relative propagation lengths of THz polaritons are generally short, unless THz polaritons of extraordinary long decay times^{33}, large wavevectors or large group velocities are studied.
Discussion
We note that in our experiments we could only observe the optical polariton modes, although sSNOM in principle can map acoustic polariton modes as well^{31}. We explain the absence of acoustic polariton modes (where the sign of the effective surface charges is opposite on both surfaces) by their extremely short wavelengths, which may prevent efficient coupling with the probing tip. Further, the acoustic modes might be strongly overdamped due to the presence of bulk carriers. In the future, sharper tips and TIs of lower bulk carrier concentration which can be grown using a buffer layer technique^{54} and may allow the study of the ultraconfined acoustic modes in real space.
In summary, we demonstrated an instrumentation for sSNOM that allows for spectroscopic nanoimaging of thinfilm polaritons around 2.5 THz, even in case of weak polaritonic image contrasts. We applied it to record realspace images of THz polaritons in the TI Bi_{2}Se_{3}. Despite the short polariton propagation, we could measure the polariton dispersion and propagation length, owing to complexvalued analysis of nearfield line profiles. In the future, the highly specific signature of polaritonic spatial signal oscillations in the complex plane – representing a spiral – could be also applied to distinguish them from nonpolaritonic spatial signal oscillations that, for example, are caused by spatial variations of dielectric material properties or by laser intensity fluctuations. Using dispersion calculations based on various optical conductivity models, we found that the polaritons can be explained by simultaneous coupling of THz radiation to various combination of Dirac carriers, massive 2DEG carriers, massive bulk carriers and optical phonons. During the revision of our manuscript, another THz sSNOM study of polaritons in TIs was published^{55}, reporting that massive 2DEGs need to be considered when interpreting THz polaritons in Bi_{2}Se_{3}. We note, however, that the contribution of massive carriers may be strongly reduced or even absent^{2,13} by growing the Bi_{2}Se_{3} by alternative methods, for example by using a buffer layer technique^{52,56}. Beyond sSNOMbased dispersion analysis as demonstrated here, our work paves the way for studying THz polaritons on other TI materials, 2D materials or 2DEGs, such as the mapping of modal field patterns in resonator structures^{30} and moiré superlattices^{57}, or the directional propagation on inplane anisotropic natural materials^{42} and metasurfaces^{58}.
Methods
Sample preparation
Films of Bi_{2}Se_{3} are grown via molecular beam epitaxy (Veeco GenXplor MBE system) on singleside polished sapphire substrate (0001) plane (10 mm × 10 mm × 0.5 mm, MTI Corp., U.S.A.). All films are grown at the same substrate temperature as measured by a noncontact thermocouple (325 °C), growth rate (0.8 nm/min), and selenium: bismuth flux ratio as measured by an ion gauge (≈50), and the selenium cell has a hightemperature cracker zone set at 900 °C^{52}. Film thicknesses are determined via xray reflection (XRR) measurement. Xray diffraction further confirmed that only a single phase and one orientation (0001) of Bi_{2}Se_{3} has been epitaxially grown on cdirection on the substrate. The sheet concentration (~3.0 × 10^{13} cm^{−2}) for 120nmthick Bi_{2}Se_{3} are obtained via Hall effect measurement in a van der Pauw configuration at room temperature (see Supplementary Note 5), with error bars ~6%.
Data availability
Data that support the results of this work are available upon reasonable request from the corresponding author.
References
Maier, S. A. Plasmonics: Fundamentals and Applications (Springer US, 2007).
LaL, S., Link, S. & Halas, N. J. Nanooptics from sensing to waveguiding. Nat. Photon. 1, 641–648 (2007).
Schuller, J. A. et al. Plasmonics for extreme light concentration and manipulation. Nat. Mater. 9, 193–204 (2010).
Naik, G. V., Shalaev, V. M. & Boltasseva, A. Alternative plasmonic materials: beyond gold and silver. Adv. Mater. 25, 3264–3294 (2013).
Zhang, X. et al. Terahertz surface plasmonic waves: a review. Adv. Photon. 2, 014001 (2020).
Basov, D. N., Fogler, M. M. & Garcia de Abajo, F. J. Polaritons in van der Waals materials. Science 354, agg1992 (2016).
Low, T. et al. Polaritons in layered twodimensional materials. Nat. Mater. 16, 182–194 (2017).
Di Pietro, P. et al. Observation of Dirac plasmons in a topological insulator. Nat. Nanotechnol. 8, 556–560 (2013).
Ginley, T. P. & Law, S. Coupled Dirac plasmons in topological insulators. Adv. Opt. Mater. 6, 1800113 (2018).
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Stauber, T., GómezSantos, G. & Brey, L. Spincharge separation of plasmonic excitations in thin topological insulators. Phys. Rev. B 88, 205427 (2013).
Stauber, T. Plasmonics in Dirac systems: from graphene to topological insulators. J. Phys. Condens. Matter 26, 123201 (2014).
Politano, A. et al. Interplay of surface and Dirac plasmons in topological insulators: the case of Bi_{2}Se_{3}. Phys. Rev. Lett. 115, 216802 (2015).
Chen, J. et al. Optical nanoimaging of gatetunable graphene plasmons. Nature 487, 77–81 (2012).
Fei, Z. et al. Gatetuning of graphene plasmons revealed by infrared nanoimaging. Nature 487, 82–85 (2012).
Kung, H. H. et al. Chiral spin mode on the surface of a topological insulator. Phys. Rev. Lett. 119, 136802 (2017).
Pesin, D. & MacDonald, A. H. Spintronics and pseudospintronics in graphene and topological insulators. Nat. Mater. 11, 409–416 (2012).
Sim, S. et al. Ultrahigh modulation depth exceeding 2,400% in optically controlled topological surface plasmons. Nat. Commun. 6, 8814 (2015).
Ginley, T., Wang, Y., Wang, Z. & Law, S. Dirac plasmons and beyond: the past, present, and future of plasmonics in 3D topological insulators. MRS Commun. 8, 782–794 (2018).
Di Pietro, P. et al. Terahertz tuning of Dirac plasmons in Bi_{2}Se_{3} topological insulator. Phys. Rev. Lett. 124, 226403 (2020).
Wang, Z. et al. Plasmon coupling in topological insulator multilayers. Phys. Rev. Mater. 4, 115202 (2020).
Autore, M. et al. Terahertz plasmonic excitations in Bi_{2}Se_{3} topological insulator. J. Phys. Condens. Matter 29, 183002 (2017).
Dordevic, S. V., Wolf, M. S., Stojilovic, N., Lei, H. & Petrovic, C. Signatures of charge inhomogeneities in the infrared spectra of topological insulators Bi_{2}Se_{3}, Bi_{2}Te_{3} and Sb_{2}Te_{3}. J. Phys. Condens. Matter 25, 075501 (2013).
Deshko, Y., KrusinElbaum, L., Menon, V., Khanikaev, A. & Trevino, J. Surface plasmon polaritons in topological insulator nanofilms and superlattices. Opt. Express 24, 7398–7410 (2016).
Wu, J.S., Basov, D. N. & Fogler, M. M. Topological insulators are tunable waveguides for hyperbolic polaritons. Phys. Rev. B 92, 205430 (2015).
Keilmann, F. & Hillenbrand, R. Nearfield microscopy by elastic light scattering from a tip. Philos. Trans. R. Soc. Lond. 362, 785–805 (2004).
Chen, X. et al. Modern scatteringtype scanning nearfield optical microscopy for advanced material research. Adv. Mater. 31, e1804774 (2019).
Woessner, A. et al. Highly confined lowloss plasmons in grapheneboron nitride heterostructures. Nat. Mater. 14, 421–425 (2015).
Yoxall, E. et al. Direct observation of ultraslow hyperbolic polariton propagation with negative phase velocity. Nat. Photon. 9, 674–678 (2015).
Nikitin, A. Y. et al. Realspace mapping of tailored sheet and edge plasmons in graphene nanoresonators. Nat. Photon. 10, 239–243 (2016).
AlonsoGonzalez, P. et al. Acoustic terahertz graphene plasmons revealed by photocurrent nanoscopy. Nat. Nanotechnol. 12, 31–35 (2017).
Soltani, A. et al. Direct nanoscopic observation of plasma waves in the channel of a graphene fieldeffect transistor. Light Sci. Appl. 9, 97 (2020).
de Oliveira, T. et al. Nanoscaleconfined Terahertz polaritons in a van der Waals crystal. Adv. Mater. 33, e2005777 (2021).
Huber, A. J., Keilmann, F., Wittborn, J., Aizpurua, J. & Hillenbrand, R. Terahertz nearfield nanoscopy of mobile carriers in single semiconductor nanodevices. Nano Lett. 8, 3766–3770 (2008).
Schnell, M., Carney, P. S. & Hillenbrand, R. Synthetic optical holography for rapid nanoimaging. Nat. Commun. 5, 3499 (2014).
Chen, C. et al. Terahertz nanoimaging and nanospectroscopy of chalcogenide phasechange materials. ACS Photon. 7, 3499–3506 (2020).
Maissen, C., Chen, S., Nikulina, E., Govyadinov, A. & Hillenbrand, R. Probes forultrasensitive THz nanoscopy. ACS Photon. 6, 1279–1288 (2019).
Dai, S. et al. Phonon polaritons in monolayers of hexagonal boron nitride. Adv. Mater. 31, e1806603 (2019).
Menges, F. et al. Substrateenhanced photothermal nanoimaging of surface polaritons in monolayer graphene. APL Photon. 6, 041301 (2021).
Huber, A., Ocelic, N., Kazantsev, D. & Hillenbrand, R. Nearfield imaging of midinfrared surface phonon polariton propagation. Appl. Phys. Lett. 87, 81103 (2005).
Gerber, J. A., Berweger, S., O’Callahan, B. T. & Raschke, M. B. Phaseresolved surface plasmon interferometry of graphene. Phys. Rev. Lett. 113, 055502 (2014).
Ma, W. et al. Inplane anisotropic and ultralowloss polaritons in a natural van der Waals crystal. Nature 562, 557–562 (2018).
GomezDiaz, J. S., Tymchenko, M. & Alu, A. Hyperbolic plasmons and topological transitions over uniaxial metasurfaces. Phys. Rev. Lett. 114, 233901 (2015).
Nikitin, A. Y. in World Scientific Handbook of Metamaterials and Plasmonics. Vol 4: Recent progress in the field of nanoplasmonics (ed. Aizpurua, J.) (World Scientific, 2017).
Rajab, K. Z. et al. Broadband dielectric characterization of aluminum oxide (Al_{2}O_{3}). J. Micro Elect. Pack. 5, 101–106 (2008).
Fei, Z. et al. Infrared nanoscopy of Dirac plasmons at the graphene–SiO_{2} interface. Nano Lett. 11, 4701–4705 (2011).
Li, P. et al. Optical nanoimaging of hyperbolic surface polaritons at the edges of van der Waals materials. Nano Lett. 17, 228–235 (2017).
Bianchi, M. et al. Coexistence of the topological state and a twodimensional electron gas on the surface of Bi_{2}Se_{3}. Nat. Commun. 1, 128 (2010).
Mooshammer, F. et al. Nanoscale nearfield tomography of surface states on (Bi_{0.5}Sb_{0.5})_{2}Te_{3}. Nano Lett. 18, 7515–7523 (2018).
Brahlek, M., Kim, Y. S., Bansal, N., Edrey, E. & Oh, S. Surface versus bulk state in topological insulator Bi_{2}Se_{3} under environmental disorder. Appl. Phys. Lett. 99, 012109 (2011).
Bansal, N., Kim, Y. S., Brahlek, M., Edrey, E. & Oh, S. Thicknessindependent transport channels in topological insulator Bi_{2}Se_{3} thin films. Phys. Rev. Lett. 109, 116804 (2012).
Ginley, T. P. & Law, S. Growth of Bi_{2}Se_{3} topological insulator films using a selenium cracker source. J. Vac. Sci. Technol. B 34, 02l105 (2016).
Ginley, T., Wang, Y. & Law, S. Topological insulator film growth by molecular beam epitaxy: a review. Crystals 6, 154 (2016).
Wang, Y., Ginley, T. P. & Law, S. Growth of highquality Bi_{2}Se_{3} topological insulators using (Bi1xInx)_{2}Se_{3} buffer layers. J. Vac. Sci. Technol. B 36, 02d101 (2018).
Pogna, E. A. A. et al. Mapping propagation of collective modes in Bi_{2}Se_{3} and Bi_{2}Te_{2.2}Se_{0.8} topological insulators by nearfield terahertz nanoscopy. Nat. Commun. 12, 6672 (2021).
Koirala, N. et al. Record surface state mobility and quantum Hall effect in topological insulator thin films via interface engineering. Nano Lett. 15, 8245–8249 (2015).
Sunku, S. S. et al. Photonic crystals for nanolight in moiré graphene superlattices. Science 362, 1153–1156 (2018).
Li, P. N. et al. Infrared hyperbolic metasurface based on nanostructured van der Waals materials. Science 359, 892–896 (2018).
Acknowledgements
We thank Curdin Maissen for help with the THz sSNOM setup and initial work. R.H. acknowledges financial support from the Spanish Ministry of Science, Innovation and Universities (national project RTI2018094830B100 and the project MDM20160618 of the Marie de Maeztu Units of Excellence Program), and the Basque Government (grant No. IT116419). A.Y.N. acknowledges the Spanish Ministry of Science and Innovation (national projects No. PID2020115221GBC42 and MAT201788358C33R) and the Basque Department of Education (PIBA202010014). M.S. acknowledge the Spanish Ministry of Science and Innovation (grand No. PID2020115221GAC44). Z.W. and S.L. acknowledge support from the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award DESC0017801. We acknowledge the use of the Materials Growth Facility (MGF) at the University of Delaware, which is partially supported by the National Science Foundation Major Research Instrumentation Grant No. 1828141.
Author information
Authors and Affiliations
Contributions
R.H. and S.C. conceived the study with the help of A.Y.N. Samples were grown by Z.W. and G.C., supervised by S.L. S.C. performed the experiments, data analysis, and calculations. A.B. derived the analytical solutions, supervised by A.Y.N. A.B., P.L. and M.S. participated in the data analysis. R.H. supervised the work. R.H., S.C. and A.B. wrote the manuscript with input from all authors. All authors contributed to scientific discussion and manuscript revisions.
Corresponding author
Ethics declarations
Competing interests
R.H. is cofounder of Neaspec GmbH, a company producing scattering type scanning nearfield optical microscope systems, such as the one used in this study. The remaining authors declare no competing financial interests.
Peer review
Peer review information
Nature Communications thanks the anonymous, reviewer(s) for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chen, S., Bylinkin, A., Wang, Z. et al. Realspace nanoimaging of THz polaritons in the topological insulator Bi_{2}Se_{3}. Nat Commun 13, 1374 (2022). https://doi.org/10.1038/s4146702228791x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s4146702228791x
This article is cited by

Realspace observation of ultraconfined inplane anisotropic acoustic terahertz plasmon polaritons
Nature Materials (2023)

Twodimensional Dirac plasmonpolaritons in graphene, 3D topological insulator and hybrid systems
Light: Science & Applications (2022)

Closing the THz gap with Dirac semimetals
Light: Science & Applications (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.