Correlated plasmons in the topological insulator Bi2Se3 induced by long-range electron correlations

Recently, electron correlation has been shown to play an important role in unconventional plasmon generation in highly correlated electron systems. Electrons in topological insulators, on the other hand, are massless and insensitive to nonmagnetic scattering due to their protection by time-reversal symmetry, which makes these materials appealing platforms for hosting exotic plasmonic excitations. Here, using a combination of angle-dependent spectroscopic ellipsometry and angle-resolved photoemission spectroscopy as a function of temperature supported by first-principles calculations, we reveal a new pair of correlated plasmonic excitations at 1.04 and 1.52 eV and a significant Fermi level shift of 0.12 eV accompanied by spectral weight transfer in the topological insulator bismuth selenide (Bi2Se3). Interestingly, such a spectral weight transfer over a broad energy range causes a drastic change in the charge carrier density whereby the contribution of charge carriers in the bulk starts to rival those in the surface states and Bi2Se3 becomes more uniformly conducting. Our results show the importance of electronic correlations in determining the electronic structure and appearance of correlated plasmons in topological insulators and their potential applications in plasmonics. The temperature dependence of plasmons, exotic particles of light and matter, in topological insulators has been mapped by researchers in Singapore, France and the US. Topological insulators can conduct an electrical current on their surface even though they are insulators on the inside. This gives rise to unusual properties, such as effectively massless electrons. Andrivo Rusydi and Thomas Whitcher from the National University of Singapore and co-workers have investigated how the long-range interactions between these electrons change with temperature. They used numerous state-of-the-art experimental techniques to map the possible energy levels of the electrons in the topological insulator bismuth selenide. The results, supported by numerical calculations, provided evidence of the existence of plasmons: particles created when electrons strongly couple to light. Long-range plasmon interactions could be useful in plasmonics, a form of information processing. Using a combination of angle-dependent spectroscopic ellipsometry and angle-resolved photoemission spectroscopy as a function of temperature and supported by first-principles calculations, we reveal a new pair of correlated plasmonic excitations at 1.04 and 1.52 eV and a significant Fermi level shift of 0.12 eV, accompanied by spectral weight transfer in the topological insulator, Bismuth Selenide (Bi2Se3). Interestingly, such a spectral weight transfer over a broad energy range causes a drastic change in the charge-carrier density whereby the contribution of charge-carriers in the bulk starts to rival those in the surface states and Bi2Se3 becomes more uniformly conducting.

In this study, we focus on Bi 2 Se 3 because some of the advantages of this material are that the suppression of Se vacancies favors the stability of Bi 2 Se 3 -based devices 33,34 and that it can be easily generated as bulk crystals for use in devices by the Bridgman-Stockbarger method 35 and by molecular beam epitaxy 36,37 . Although the electronic properties of Bi 2 Se 3 and the other topological insulators have been thoroughly investigated, there has been very limited, if any, study on the electronic correlation and how the electronic correlation influences the electronic structure, in particular as a function of temperature.
Recently, electronic correlation has been shown to play an important role in unconventional plasmon generation in correlated electron systems [38][39][40][41][42] . This has generated a new idea of studying optical and electronic structures and searching for multiplasmon generation in topological insulators.
Electron spectroscopies, such as angle-resolved photoemission spectroscopy (ARPES), on the one hand, have already been extensively used to study these materials, focusing on the Dirac cone and electronic dispersion closest to the bulk band gap and surface states 43,44 . On the other hand, a direct way to probe the electronic correlation is to measure spectral weight transfer using spectroscopic ellipsometry, which can be used to simultaneously obtain the complex dielectric function and the loss function 38,45,46 . However, until now, no temperature-dependent spectroscopic ellipsometry has been performed on this system, and therefore, our understanding of the electronic correlations is very limited 47,48 . In this article, using a combination of temperature-and angle-dependent spectroscopic ellipsometry and angle-resolved photoelectron spectroscopy (ARPES) supported by first-principles calculations, we reveal that upon cooling, Bi 2 Se 3 shows unusual spectral weight transfer over a broad energy range accompanied by the occurrence of a new type of correlated plasmons and a shift in the Fermi level. The complex dielectric and loss functions of the topological insulator are simultaneously determined as a function of temperature, and from these, we deduce that the cause of the temperature-dependent shift is that the longrange electron-electron correlation affects the Fermi level.

Sample preparation
Epitaxial growth of Bi 2 Se 3 thin films on 5 × 5 mm 2 GaAs (111) substrates was performed using an MBE reactor, where source fluxes were provided by conventional standard effusion cells. Epitaxial growth of Bi 2 Se 3 epilayers was then carried out at T = 300°C with a Se/Bi beam equivalent pressure ratio greater than 6.5. The Bi 2 Se 3 thin films are composed of many micron-sized grains covering the whole substrate. Samples were capped with protective amorphous Se layers to prevent contamination and oxidation prior to air exposure, depending on the kind of measurement planned.

Angle-resolved photoemission spectroscopy
For angle-resolved photoemission spectroscopy, a specific procedure implying the use of a Se cap layer was used, allowing us to recover a clean surface under UHV after a mild annealing step (with a pressure not exceeding 5 × 10 −10 mbar). ARPES measurements were carried out using a highresolution Scienta SES 2002 photoelectron energy analyzer equipped with a delay line detector at the TEMPO beamline of Synchrotron SOLEIL 49 and a high-resolution Scienta DA30 electron analyzer equipped with a 3-dimensional spin detector at the SUV Beamline of the Singapore Synchrotron Light Source 50 . The beam diameter for ARPES in both cases was on the order of a few hundred μm 2 . This is much smaller than the sample surface, and therefore, the signal is from the thin film.

Spectroscopic ellipsometry
Spectroscopic ellipsometry measurements were carried out using a variable-angle spectroscopic ellipsometer (V-VASE, J. A. Woollam Co.) with a rotating analyzer and a compensator at the Singapore Synchrotron Light Source (SSLS). The measurements were taken in the energy range of 0.6−6.0 eV while the sample was inside a UHV cryostat with a base pressure of 10 −8 mbar. The sample was heated to 475 K and annealed for 12 h to remove the α-Se cap. The measurements were taken at angles of 68°, 70°and 72°, which are limited in range by the UHV windows. Measurements were taken over a range of temperatures from 77 K to 475 K at an angle of 70°. The stabilization accuracy of the temperature was ±0.5 K, and the sample was cooled/heated slowly and evenly across the sample. When at the target temperature, the sample was left to stabilize for 10 min. Each measurement at the specified temperature took approximately 2 h to complete. The output data, ψ and Δ, are shown in Supplementary Fig. S1. Analysis of the spectroscopic ellipsometry data was performed using the W-VASE analysis program, where the complex dielectric function was obtained using a fitting procedure that models the data and takes into account the anisotropy of the sample 51 . Further details are provided in the Supplementary Material.

First-principles calculations
First-principles calculations were performed within the framework of density-functional theory (DFT) using the generalized gradient approximation (GGA) based on the Perdew-Burke Ernzerhof (PBE) functional and projector augmented wave (PAW) potentials, as implemented in the Vienna ab initio Simulation Package (VASP5.4.4.18) 52,53 . The cutoff energy for the plane wave expansion was set to 500 eV. In all the calculations, spin-orbital interactions were included, and the criterion of electronic convergence was set to 1.0 × 10 −6 e. The experimental lattice constants (a = b = 4.14 Å and c = 28.64 Å) of the Bi 2 Se 3 crystal were adopted, in which the atoms were optimized until the force was less than 0.01 eV/Å, including the van der Waals correction (DFT-D3) 54 . A Γ-centered 6 × 6 × 6 k-point mesh was used to sample the first Brillouin zone (BZ) of the Bi 2 Se 3 primitive cell. Figure 1a, b show the real, ε 1 , and imaginary, ε 2 , parts of the complex dielectric function, respectively, measured using spectroscopic ellipsometry over a range of temperatures from 475 K down to 77 K. There is a clear difference in both parts of the dielectric function as Bi 2 Se 3 is cooled, with the largest changes occurring at and around 2.0 eV. There is also an edge that occurs in the ε 2 spectra at 1.0 eV for all temperatures that becomes sharper as the sample is cooled. As Bi 2 Se 3 is a topological insulator, the electronic structure is highly dependent on the thickness of the sample. Fortunately, another advantage of spectroscopic ellipsometry is that we can use this technique to simultaneously measure the thickness of Bi 2 Se 3 , and in this case, it is 30.2 nm (or 29 quintuple layers), which is also later confirmed by atomic force microscopy (AFM) measurements.

Results and discussion
Since spectroscopic ellipsometry is a photon-in photonout technique, it can be used to search for plasmonic activity in correlated electron systems 41,55 . For sub-X-ray photons, the momentum transfer, q, is finite but approaches zero because it is much less than the crystal momentum. In this limit, the distinction between the longitudinal and transverse ε(ω) vanishes, i.e., which allows sub-X-ray spectroscopic ellipsometry to probe both optical and plasmonic properties of materials in the low-q limit 43,44 . Figure 1c shows a comparison of ε 2 and the loss function given by for all temperatures measured between 475 and 77 K in the spectral region of 0.6-0.6 eV. The peaks in the loss function at 1.04 and 1.52 eV indicate the presence of a plasmon at these energies below 250 K, and they are closely linked to the peaks in ε 2 at 1.00 and 1.48 eV, which are optical excitations. These peaks are identified by fitting the data to a Gaussian-Lorentzian model and applying background subtraction to reveal the positions of the peaks, as shown in Supplementary Fig. S2. The existence of both peaks within 40 meV of the peaks in ε 2 means that there is a strong coupling between the optical and plasmonic excitations of Bi 2 Se 3 and is indicative of correlated plasmons 38,39 . The correlated plasmons appear in both the ε 2 and loss function spectra, which means that the correlated plasmons can readily be excited by, decay radiatively into and thus couple with free space photons without any external mechanisms. This is one of the specific properties that identify the peaks in the loss function as corresponding to correlated plasmons. The other properties are as follows. First, unlike conventional plasmons, correlated plasmons occur in materials with a low free charge density, such as the bulk of Bi 2 Se 3 so do not arise from the collective oscillation of free charges but from the collective oscillation of quasilocal (or correlated) electrons 38,39 . This is directly linked to the second property in that they therefore have more than one plasmon energy, in this case at 1.04 and 1.52 eV. The energies of the correlated plasmons are determined not only by the charge carrier density of the free electrons but also by the potential energy of the long-range correlation between electrons. As there are multiple long-range correlations present between quasi-local electrons, there can be multiple correlated plasmon energies. Finally, correlated plasmons also have a significantly lower loss than conventional plasmons and a positive real part of the dielectric function, ε 1 , both of which can be seen in Fig.  1 38,39 . The limit for observable correlated plasmons in Sr 1 −x NbO 3+d is 20 nm; while this may be different between materials, we are still able to observe correlated plasmons in a 30 nm thin film of Bi 2 Se 3 38,39 . Although a plasmonic resonance has been detected in another topological insulator, Bi 1.5 Sb 0.5 Te 1.8 Se 1.2 , within the optical-UV range 56 , this is the first time that correlated plasmons have been detected in topological insulators. We note that high-energy plasmonic resonances have been detected in Bi 2 Se 3 above 7 eV using energy −loss spectroscopy and scanning transmission electron microscopy (STEM) 57 .
The difference in the complex dielectric function with temperature (shown in Supplementary Fig. 1) highlights the increasing change that can be seen in the topological insulator's electronic response to external electromagnetic fields with temperature. There is a significant shift in the spectral weight of ε 2 as Bi 2 Se 3 is cooled, and to explore this, we need to look at the optical conductance, which is related to ε 2 by where ω is the photon frequency and σ 1 is the real part of the complex conductivity. Figure 2a shows the optical conductivity of the Bi 2 Se 3 thin film for each of the temperatures measured. For the spectral weight transfer analysis, the conductivity was divided into three spectral regions as follows: I: 0.60-1.65 eV, II: 1.65-2.90 eV and III: 2.90-6.00 eV. Figure 2b shows the change in the conductivity in region III as the sample is cooled from 475 K to each of the measured temperatures, and there is a clear and consistent decrease in the spectral weight starting from 250 K down to 77 K. This is converse to what is seen in region II of Fig.  2a, where the spectral weight appears to increase. The integration of the optical conductivity is related to the free charge carrier density, n e , via the charge conserving f-sum rule: where e is the elementary charge and m e is the electron mass 38,45,51,58 . Therefore, the integration of the part of the spectral region between E 1 and E 2 , given by is proportional to the number of free charge carriers within that spectral region. By analyzing the change in W with temperature, we can gain insight into the behavior of the free charge carriers as the Bi 2 Se 3 sample is cooled 38,45,58 .
The total spectral weight, W, of the Bi 2 Se 3 thin film across the measured spectral range of 0.6-6.0 eV is shown in Fig. 2c as a function of temperature. There is an increase in W as the sample is cooled from 475 to 250 K, followed by a decrease down to roughly the same level as that at 475 K as the sample is further cooled to 77 K. This result indicates that there are more electrons with energies between 0.6 and 6.0 eV at 250 K than at any other temperature. This means that above and below 250 K, the number of electrons with energies outside of the spectral range measured is increasing. This shift in spectral weight cannot be explained by thermal activation, as the energies b Difference in the optical conductivity between 475 K and the colder measurements in the third region between 2.9 and 6.0 eV. c Optical conductivity of Bi 2 Se 3 integrated over the whole spectrum as a function of temperature. d Change in the integrated optical conductivity of the three different spectral regions as the sample is cooled from 475 K. associated with temperatures below 500 K are too small (<43 meV); therefore, the extra energy gained or lost must come from the potential energy electron-electron correlations. Both of the correlated plasmons seen in Fig. 1c disappear at temperatures of 250 K and above, which also coincides with the drop in electron conductivity and thus electron density seen in Fig. 2c. Figure 2d shows the change in W for each of the three spectral regions from Fig. 2a as the sample is cooled from 475 K to 77 K. While there is little change in the lowenergy and mid-energy regions below 250 K, the loss in the spectral weight in the high-energy region represents a significant shift in the electron density from this spectral range to higher energies (above 6.0 eV). The increase in electron energy on the order of several eV comes from the long-range electron-electron correlations, which are now prominent due to decreased electronic screening 38 . This in turn gives rise to plasmonic activity, which has been seen in conductive materials 38,45,59 .
One of the important effects of the appearance of highenergy plasmons is that electron scattering may be able to occur within the system. The scattering of free charge carriers in the surface states of topological insulators is extremely limited, as back-scattering from nonmagnetic impurities is prohibited by time-reversal symmetry 2 , phonons are too weak for electron scattering 60 and the conventional Dirac plasmons reported in the surface states have energies in the THz regime (on the order of 10 meV) and thus cannot cause significant scattering 20 . However, since the correlated plasmons have energies on the order of 1 eV, they can cause electron scattering from the surface states to the bulk, as seen in other 2D materials such as graphene 61 . To determine if this is the case, we use ARPES to detect changes in the Fermi level as a function of temperature. Figure 3 presents the electronic band structure of the Bi 2 Se 3 thin film along the Γ-M direction measured using ARPES with an incident photon energy of 60 eV. Panel (a) shows the spectrum taken at room temperature (300 K), while panel (b) shows the spectrum taken at liquid nitrogen temperature (100 K). The Dirac fermion surface state is clearly observed at both room temperature and low temperature. Other band features at lower binding energies are also very sharp, indicating the good quality surface of our thin film. The most striking change in the low temperature spectrum compared to the room temperature spectrum is the location of the Dirac point. The Dirac point at room temperature is observed at a binding energy of 0.30 eV, while the Dirac point has shifted to the higher binding energy of 0.42 eV at 100 K (the energies are measured with respect to the Fermi energy). At lower temperatures, the shift in the Dirac point by 0.12 eV towards a higher binding energy shows that the electronic structure of the Bi 2 Se 3 film has changed. Although initial charge transfer might occur between the substrate and thin film, this does not affect the changes in the electronic structure measured as a function of temperature 62 .
Although surface effects such as residual moisture contamination in the vacuum chamber 63 and subsequent deselenization can also alter the electronic structure, in the time frame of the measurements, these effects are minimized (see the Supplementary Material for further details). One possible mechanism that might occur is charge build-up in the GaAs substrate due to the substrate's pyroelectric properties 64,65 ; however, this is minimized by the slow heating/cooling of the sample. The substrate is also grounded, and the sample is given time to stabilize at the required temperature. The measurements at a given temperature take approximately 2 h to complete, so any charge build-up within the substrate would most certainly dissipate over time. Another possible mechanism is the effect of the GaAs substrate band gap, where photons with energies higher than it may charge the substrate. However, because the energy gap for GaAs is 1.45 eV, the films cannot be charged because there is no state below the band gap of GaAs. Furthermore, above 1.45 eV, the photon penetration depth dramatically drops and only reaches 0.03 microns (30 nm) in the film; therefore, the signal only comes from the film (Supplementary Fig. 2). All these results clarify that the correlated plasmons are truly intrinsic properties of Bi 2 Se 3 and not from other effects, such as the substrate.
The shift in the spectrum can also be seen in the energy distribution curves (EDCs), as shown in panels (c) and (d) of Fig. 3. Apart from the downward shift in the spectra, the spectral features of Bi 2 Se 3 at low and high temperatures are similar. This suggests an increase in n-doping 66 , which would have profound effects on the surface and transport properties of the material 43 . The consequences of this shall now be explored in terms of carrier contributions from the bulk and surface states.
The free charge carrier densities of both the bulk and surface states can be calculated from the optical conductivity in Fig. 2 and the charge carrier mobility by using the following equation: where μ e is the electron mobility. The electron mobility of Bi 2 Se 3 is given as 880 cm 2 /Vs at room temperature and 1380 cm 2 /Vs at low temperatures in ref. 5. From this, the electron density is calculated to be 1.83 × 10 20 cm −3 at room temperature and 1.15 × 10 20 cm −3 at 100 K, with a decrease of 6.8 × 10 19 cm -3 as the sample is cooled. The ARPES images in Fig. 3 at 300 and 100 K show that although the Fermi level with respect to the Dirac point has increased at 100 K, there are actually fewer free carriers within the measured spectral range up to 6.0 eV than there are at 300 K. This matches what is shown in Fig. 2a. We can also use the energy of the conduction band minimum from Figs. 3a and 3b to calculate the charge density in the conduction band within the bulk of Bi 2 Se 3 using the following equation: 67,68 where n BD is the bulk carrier density, m* is the effective electron mass, which is usually ≈0.15m e in Bi 2 Se 3 , and E CB is the difference between the Fermi energy and the conduction band minimum. For 100 K, E CB is 330 meV, which gives a bulk carrier density of 5.00 × 10 19 cm −3 , and for 300 K, E CB is 220 meV, which yields a bulk carrier density of 2.72 × 10 19 cm −3 . Note that these values are very high for what is supposed to be an insulating bulk, but in reality, few Bi 2 Se 3 samples created achieve a truly insulating bulk due to the Mott criterion and Ioffe-Regel criterion 67 . The bulk states can be considered as a bad conductor in most cases or more accurately as a weaker conductor than the surface states. We also note that both the bulk and surface carrier densities in our thin film extracted from the room temperature measurements are similar to those found in Sample B of ref. 69. The electron densities calculated from ARPES are smaller than those calculated from the spectroscopic ellipsometry data because the values from the latter are the electron densities for the entire sample-both the bulk states and the conducting surface states. Therefore, with these values, we can calculate the percentage of carriers from the surface states that contribute to the overall conduction for each temperature. At 300 K, approximately 85% of carriers come from the surface states, whereas at 100 K, only 57% of the carriers come from the surface states. This result agrees with our earlier hypothesis that the introduction of high-energy correlated plasmons below 250 K would cause electron scattering from the surface to bulk states. However, the n e calculated from ellipsometry may be underestimated because the spectral range shown in Figs. 2 and 3 is from 0.6 to 6.0 eV, so electrons outside of this energy range are not considered; therefore, the fraction of charge carriers from the surface states at both temperatures may be higher. Figure 4 shows the first-principles-calculated band structures of Bi 2 Se 3 surface slabs near the Fermi level.  Fig. 4 exhibit a remarkable similarity to the ARPES data in Fig. 3, in which the ndoped simulations and the 100 K data both have a higher Fermi level compared with the high temperature/undoped results. The major difference is that the simulations for undoped Bi 2 Se 3 show the Fermi level at the Dirac point, whereas the room temperature ARPES results show the Fermi level to be 0.29 eV above the Dirac point. This difference occurs because the sample is not a perfect Bi 2 Se 3 single crystal but a polycrystalline thin film, and as discussed in the Supplementary Material, effects such as moisture contamination (Supplementary Fig. 3a) and deselenization will also shift the Fermi level 73,74 ; however, the changes due to temperature are shown to be reversible ( Supplementary Fig. 3b).
The calculated optical spectra of Bi 2 Se 3 with electron doping are shown in Fig. 4d, e, in which the electronelectron correlation effects were included at the G 0 W 0 -RPA level. The contributions of p x and p y to the dielectric function and loss function are equal, which is why the blue lines of the xx component and dashed black lines of the yy component overlap in both Fig. 4d, e. Two peaks are found at the energy positions of 1.02 and 1.38 eV in the imaginary part of the dielectric function and electron energy loss function, which we believe are plasmon peaks because they are only due to electron-electron interactions. Together, these plasmons could contribute to the shift in the conduction carrier density from the surface to the bulk through electron scattering.
Because electronic correlation is important for the generation of correlated plasmons, one may consider applying correlated plasmons as a gating material in submicron devices and using temperature to control the on and off states. As Bi 2 Se 3 is insulating within the bulk at room temperature, this would keep the device in the 'off' position. When the temperature is lowered, correlated plasmons appear, and the bulk of the material starts to become conducting, which would allow currents to pass through; thus, the device would be in the 'on' position.

Conclusions
In summary, by simultaneously determining the complex dielectric function, loss function and electronic structure and dispersion of Bi 2 Se 3 as a function of temperature using a combination of spectroscopic ellipsometry and ARPES supported by theoretical calculations, we observe unusual spectral weight transfer yielding a significant shift in the Fermi level and two correlated plasmons, a new type of plasmons seen in correlated systems. From the spectral weight transfer analysis, we find that upon cooling below 250 K, the decrease in electronic screening leads to an increase in long-range electron correlations, which increase the potential energy of the system and result in the formation of correlated plasmons. Electron scattering from the high-energy correlated plasmons results in an increase in the bulk carrier density and, subsequently, a reversible shift in the Fermi energy. Our results show new correlated plasmons in Bi 2 Se 3, and the methodology introduced here can be used to probe plasmons in topological insulators.