## Introduction

Strongly correlated insulators are materials that are expected to be metals according to conventional band theory but are actually insulators with half-filled orbitals or unpaired electrons1,2,3. On-site Coulomb interactions (U) must be considered when describing the electronic band structures of correlated insulators as they are the cause of the unexpected bandgaps3. Based on the Mott-Hubbard model, the energy bands near Fermi level (EF) are renormalized because of U, resulting in an upper Hubbard band (UHB) and a lower Hubbard band (LHB) (Supplementary Fig. 1)4,5,6. Depending on the difference between U and the charge-transfer energy (Δ), strongly correlated oxides are classified as either charge-transfer insulators (U > Δ) or Mott insulators (U < Δ). Several fascinating properties have been observed in strongly correlated oxides, including correlated topological phases7,8, metal-insulator transitions (MIT)3, and unconventional superconductivity9, among others. As collective motions of electrons, plasmons arise from long-range Coulomb interaction. In strongly correlated electron systems, plasmonic behavior can be drastically altered due to the strong correlation effects, leading to novel properties and unprecedented functionalities10,11,12. For example, correlation effect with long-range Coulomb interactions could induce unconventional correlated plasmons with multiple plasmon frequencies and low-loss11,12.

As a general rule, the frequency of plasmons, $${\omega }_{p}$$, strongly depends on the wave vector q. In traditional three-dimensional (3D) metals, the relationship is $${\omega }_{p}={\omega }_{0}+(3{v}_{F}^{2}/10{\omega }_{0}){q}^{2}$$ in the long-wavelength limit13, where $${\omega }_{0}$$ is the plasmon frequency for $$q\to 0$$, and $${v}_{F}$$ is the Fermi velocity. In a two-dimensional (2D) system such as graphene, it follows that $${\omega }_{p}\propto \sqrt{q}$$ when $$q\to 0$$14, as in an ideal 2D electron gas. Similar plasmon behavior ($${\omega }_{p}\propto \sqrt{q}$$) has been reported in 2D metallic monolayers and quasi-2D metals at small q. However, the plasmons become dispersionless (referred hereafter as flat plasmons) over a relatively large range of wave vectors (0.1 Å−1−0.3 Å−1)15,16,17 due to the screening effects arising from the interband transitions18,19,20. Additionally, flat plasmons have been reported in twisted bilayer graphene21,22,23. Interestingly, flat plasmons can transition to localized plasmon wave packets in real-space. By tracking these plasmon wave packets, novel time-resolved plasmonic imaging technique could be realized18. However, flat plasmons have been reported to date only over a limited range of wave vectors, q <~ 0.7 Å−1 15,16,17, and exclusively in 2D or quasi-2D systems18,21,22.

In this work, we report on flat plasmons that can propagate up to an ultra-large wave vector, q > 1.2 Å−1 (beyond the first Brillouin zone), with a small energy fluctuation of less than 40 meV in a strongly correlated 3D oxide, α-Ti2O3. α-Ti2O3 is a typical Mott-insulator with strongly correlated 3d1 electrons, exhibiting a broad MIT above 400 K24,25,26. As a consequence, α-Ti2O3 has a narrow bandgap of ~0.1 eV at room temperature which in turn gives rise to fascinating physical properties and applications27,28,29, such as high-performance mid-infrared photodetection27 and photothermal conversion28,29. Moreover, novel superconductivity30 and interesting catalytic properties31 have been reported in newly epitaxial stabilized Ti2O3 polymorphs, and these are closely related to the electronic correlations therein. Because of strong electronic correlation effects, the energy bands near EF are renormalized and become relatively flat, leading to flat plasmons in α-Ti2O3. Due to the crystal symmetry and negligible absorption at the plasmonic frequency, these plasmons can propagate beyond the first Brillouin zone. Additionally, we present evidence for a hyperbolic property arising from the anisotropic electronic structures of α-Ti2O3.

## Results

α-Ti2O3 has a corundum structure with the space group $$R\bar{3}c$$. The conventional and primitive cells of Ti2O3 are shown in Fig. 1a, b, respectively. The conventional cell has a hexagonal representation with lattice parameters a = b = 5.15 Å and c = 13.64 Å32. The primitive cell is a rhombohedra with a1 = a2 = a3 = 5.517 Å and an angle of 55.2° between lattice vectors. Each unit has a titanium atom surrounded by six oxygen atoms, constituting a distorted octahedral configuration. The four titanium atoms in the primitive cell lie adjacent along the c-axis, forming Ti-Ti dimers with face-shared octahedra33. Figure 1c shows high-resolution X-ray diffraction (HR-XRD) scans for the α-Ti2O3 single crystal with (0006) and (11$$\bar{2}$$0)-oriented surface planes; these scans confirm the hexagonal characteristics of α-Ti2O3. (More structural details are provided in Supplementary Fig. 2).

Within the hexagonal structure, there are face-sharing octahedra along the c-axis and edge-sharing octahedra in the ab-plane. Strong d-d orbital interaction arises from the Ti atoms with face-sharing and edge-sharing octahedra due to the short Ti-Ti distances which in turn form bonding and antibonding molecular orbitals24,33. The proposed molecular orbital diagram for α-Ti2O3, based on Goodenough’s model, is shown as an inset in Fig. 1d. The states near EF are dominated by Ti 3d orbitals whereas the O 2p orbitals are far below EF, consistent with our theoretical calculations (Supplementary Fig. 3). The bonding $${a}_{1g}$$ and antibonding $${a}_{1g}^{*}$$ molecular orbitals are directed along the c-axis with the bonding $${e}_{g}^{\pi }$$ and antibonding $${e}_{g}^{\pi*}$$ molecular orbitals lying in the ab-plane, leading to an anisotropic band structure near EF25,33. The $${a}_{1{{{{{\rm{g}}}}}}}$$ band is fully occupied with Ti 3d1 electrons and is well separated from the $${a}_{1g}^{*}$$ band. (The splitting between the $${e}_{g}^{\pi }$$ and $${e}_{g}^{\pi*}$$ bands is negligible33). Due to the U value for the 3d1 electrons, the $${a}_{1g}$$ band is separated from the $${e}_{g}^{\pi }$$ band, resulting in a gap of ~0.1 eV at room temperature. Figure 1d shows the temperature-dependent resistivity for α-Ti2O3 with I | | xy-plane and I | | z. Clearly, its resistivity is increased with decreasing temperature, which attests to the correlated insulating behavior of α-Ti2O324,25,26. Moreover, the anisotropic behavior for the two directions of current flow is consistent with the anisotropic band structure near EF.

In order to further study the anisotropic behavior of α-Ti2O3, we investigated its optical properties by theoretical calculations and experimental measurements. The optical properties of crystalline solids can be described by a complex permittivity $${\varepsilon (\omega )}_{\alpha \beta }={{{{{\rm{Re}}}}}}[{\varepsilon (\omega )}_{\alpha \beta }]+{{{{{\rm{Im}}}}}}[{\varepsilon (\omega )}_{\alpha \beta }]$$, where α, β represent the different Cartesian directions. The permittivity is the sum of the interband and intraband transition contributions. For the interband contributed part, the imaginary term $${{{{{\rm{Im}}}}}}[\varepsilon (\omega)_{\alpha \beta }^{{{\mbox{inter}}}}]$$ can be calculated from the interband transitions, while the real term $${{{{{\rm{Re}}}}}}[\varepsilon (\omega)_{\alpha \beta }^{{{\mbox{inter}}}}]$$ is determined from $${{{{{\rm{Im}}}}}}[\varepsilon (\omega)_{\alpha \beta }^{{{\mbox{inter}}}}]$$according to the Kramers-Kronig relation34. For the intraband contributed part, the Drude model is used to describe transitions within the partially occupied bands of the material35,

$$\varepsilon (\omega)_{\alpha \beta }^{{{\mbox{intra}}}}=1-\frac{{\omega }_{p,\alpha \beta }^{2}}{{\omega }^{2}+i\gamma \omega }$$
(1)

here, ωp is the plasma frequency and $$\gamma$$ is the lifetime broadening which is the reciprocal of the excited-state lifetime.

For a non-magnetic bulk material which is anisotropic along the in-plane and out-of-plane directions, the isofrequency surface for transverse magnetic (TM) polarized waves (kx, ky, kz) is given by

$$\frac{{k}_{x}^{2}+{k}_{y}^{2}}{{\varepsilon }_{\parallel }}+\frac{{k}_{z}^{2}}{{\varepsilon }_{\perp }}={(\frac{\omega }{c})}^{2}$$
(2)

where c is the speed of light. ε|| and ε denote the components of permittivity tensor parallel and perpendicular to the anisotropy axis, respectively. If the real parts of ε|| and ε have opposite signs in a medium, the isofrequency surface will be a hyperboloid and this class of materials is known as “hyperbolic materials” (HMs)36. Furthermore, HMs can be classified into type I (ε > 0 and ε|| < 0) and type II (ε < 0 and ε|| > 0) HMs37. Schematic diagrams of the dispersion relations for these two types of HMs are shown in Fig. 2a. Type I HMs usually have fewer reflections and possess lower losses than do type II HMs38.

For α-Ti2O3, the permittivity is isotropic in the xy-plane, and anisotropic along the z direction (c-axis) (i.e. ε = εx and ε|| = εz). Figure 2b presents the real part and imaginary part of permittivity for α-Ti2O3 measured by ellipsometry at room temperature with E | | xy-plane and E | | z. Two type-I hyperbolic regions at 1.06−1.21 eV and 1.88−1.90 eV with Reεx > 0 and Reεz < 0 are observed experimentally. Moreover, both Imεx and Imεz in the second hyperbolic region present small values where Re(ε) changes sign, which is well consistent with the theoretical results (Fig. 2c, d). The only differences are the absolute frequency values. In Fig. 2c, without considering the local field effect (LFE) (dashed lines), there are two hyperbolic windows at the energy range of 0−5 eV. In the first hyperbolic window at 1.52−1.79 eV, the imaginary part of permittivity Imε(ω) has a relatively large value, indicating a large energy loss due to the electron transitions. But in the second hyperbolic region of 2.67−3.26 eV, Imε(ω) is greatly suppressed, making α-Ti2O3 an ideal type I hyperbolic material within the corresponding photon energy range. When considering the LFE (solid lines), consistent with the experimental results, the second hyperbolic window shrinks to almost disappear because of the red-shifted frequency of Reεz(ω) = 0 from 3.26 eV to 2.69 eV. The large correction effect of LFE on εz(ω) can be attributed to the unhomogeneous distribution of the wave function along the z direction33, which causes the off-diagonal terms of $${\varepsilon }_{{{{{{\boldsymbol{G}}}}}}={{{{{\boldsymbol{G}}}}}}{{{\prime} }}}({{{{{\boldsymbol{q}}}}}},\omega )$$.

Multiple Reε= 0 points are observed in the experimental39 and theoretical results40, and this unusual predominance making them applicable as functional hyperbolic metamaterials39. Figures 3a, b present the plasmon properties of α-Ti2O3 in the energy range of 0–5 eV, obtained from the calculated electron energy loss spectra (EELS) along the in-plane (Γ−S0) and out-of-plane (Γ−T) directions. The plasmon dispersion is shown, as extracted from the peak values of the EELS. Surprisingly, the plasmons along the in-plane and out-of-plane directions are both nearly dispersionless, with a small energy fluctuation Δωp < 40 meV. For the Γ−S0 direction (Fig. 3a), the plasmon mode starts at ~2.60 eV, close to the frequency where Reεx(ω) = 0 (Fig. 2c). Importantly, the plasmons can propagate over a large momentum range and remain visible well beyond the first Brillouin zone (q < ~1.2 Å−1). Notably, the nearly dispersionless behavior can be observed from the excitation spectrum and the maximum change of the plasmon energy is only Δωp(Γ−S0) = 40 meV within the first Brillouin zone. To verify the collective excitation and long-lived features of the plasmon modes, we plot the dielectric function and loss function at selected momenta, q = 0.077 Å−1 (Fig. 3c) and q = 0.77 Å−1 (Fig. 3d). For an undamped plasmon, the dielectric function should fulfill the condition Reε = 0 with $${Im}\varepsilon /{\partial }_{\omega }{Re}\varepsilon > 0$$ at the peak energy in the loss function41,42. At the same time, $${Im}\varepsilon$$ should have a vanishing value, indicating the plasmon is free of Landau damping42. For q = 0.077 Å−1 along the Γ−S0 direction (Fig. 3c), Reε crosses zero from negative to positive values at ωp = ~2.60 eV with a vanishing Imε, which corresponds to the energy of the sharp peak in the loss function. For q = 0.77 Å−1 (Fig. 3d), those conditions are also satisfied, verifying the robustness of the undamped plasmon along the Γ−S0 direction. Additionally, the energy of the in-plane plasmon (Γ−S0) is blue shifted from 2.37 eV to 2.61 eV (solid lines in Fig. 3c, d) at large q = 0.77 Å−1, while that at small q = 0.077 Å−1 is negligible, as the LFE is considered.

For the Γ−T direction (Fig. 3b), the plasmon exhibits some behavior similar to that along the Γ−S0 direction. The out-of-plane plasmon propagates from ~2.58 eV without dispersion and also persists beyond the first BZ. In contrast to the EELS along the Γ−S0 direction, there are some additional broad and weak peaks at higher energies in the excitation spectrum (Fig. 3b). Nevertheless, these peaks are damped modes, which originate from single particle excitations41. According to the dielectric function and loss function at selected momenta (Fig. 3e, f), the LFE has large effects on the out-of-plane plasmon mode. Without the LFE, there is only one peak in the loss function, located at 3.08 eV and 3.02 eV for q = 0.044 Å−1 and q = 0.44 Å−1, respectively. When the LFE is taken into account, however, the peak splits into two features, a sharper one at a lower energy of 2.58 eV (2.59 eV) and a broader one at higher energy at 3.95 eV (3.46 eV) for q = 0.044 Å−1 (q = 0.44 Å−1). The lower energy mode is stronger and meets the conditions for the undamped plasmons, contributing to the long-lived and well-defined out-of-plane plasmon (Fig. 3b). For the higher energy mode, although the definition of the plasmon Reε = 0 is satisfied, the continuum character of Imε and the small derivative of Reε in the nearby region indicates the rapid decay of this mode into electron-hole pairs42.

It is noteworthy that the ultra-flat dispersion of plasmons is rare in most materials, and this fact motivates us to further study its physical origins in this 3D oxide. Since the ultra-flat plasmon comes mainly from the absorption peak centered at ~1.2 eV in the $${Im}\varepsilon (\omega,q\to 0)$$ (Fig. 2c), we first analyze the origin of the absorption peak from the electronic transition processes. The orbital-resolved electronic band structure of α-Ti2O3 is plotted in Fig. 4a. The two bands below EF are largely derived from the Ti $$3{d}_{{z}^{2}}$$ orbital, whereas the conduction bands near EF originate mainly from the Ti$${3d}_{{xy}}$$ and $${3d}_{{x}^{2}-{y}^{2}}$$ orbitals. We analyzed the symmetry of the wave functions at the Γ point using the Irvsp43 code. The point group of α-Ti2O3 is D3d, which contains the space inversion operation $$P$$ respect to O point (Fig. 4b). The degenerate states at Γ can be used as the basis functions for the construction of the irreducible representations of the D3d point group. The corresponding irreducible representations for states 1~12 at the Γ point are listed in Table 1. Due to the space inversion symmetry of α-Ti2O3, the wave functions for Γ point have certain parities of space inversion operation P. The parities of the wave functions are also presented, and these are the eigenvalues of the space inversion operation $$P$$. The transition dipole moment matrix associated with the transition between an initial states m and n is defined as $$\left\langle {\psi }_{m\Gamma }\right|{{{{{\boldsymbol{r}}}}}}\left|{\psi }_{n\Gamma }\right\rangle$$, where $${\psi }_{m\Gamma }$$ and $${\psi }_{n\Gamma }$$ are the electron wave functions, and $${{{{{\boldsymbol{r}}}}}}$$ is the position operator. For α-Ti2O3, it shows a space inversion symmetry with respect to coordinate O point (Fig. 4b). If $${\psi }_{m\Gamma }$$ has the same parity as $${\psi }_{n\Gamma }$$ with respect to O point, we will have $$\left\langle {\psi }_{m\Gamma }\right|{{{{{\boldsymbol{r}}}}}}\left|{\psi }_{n\Gamma }\right\rangle=0$$. Thus, only the interband transitions between the two states with opposite parities are allowed and have contributions to the Imε. In Table 1, we summarize the allowed and forbidden electric-dipole transitions at the Γ point according to the optical selection rule. We further calculated the matrix element $$\left\langle {\psi }_{m\varGamma }\right|{{{{{\boldsymbol{r}}}}}}\left|{\psi }_{n\varGamma }\right\rangle$$ for transitions between different states at the Γ point. According to our calculations, the transitions from state 2 below EF to the degenerate states 3 and 4 exhibit the largest matrix elements, whereas the matrix elements of other transitions are almost negligible (Supplementary Table 1). It should be mentioned that the flat plasmons in α-Ti2O3 along both directions are located at ~2.6 eV. Thus, its Landau damping should be arisen from the interband transitions with the transition energy close to 2.6 eV. As shown in Fig. 4a, only the transition between state 1 and state 12 has the corresponding energy difference. However, the wave functions of state 1 and state 12 both exhibit an even parity, so this transition is forbidden by the selection rule (Table 1). Hence, we get vanished values for Im(ε) at the plasmon energy, which preserves the plasmons are long-lived. Finite-temperature effect on the damping of the plasmons in α-Ti2O3 is discussed in the supplementary materials (Supplementary Fig. 46, Supplementary Note 1).

In the long-wavelength limit, the imaginary part of the interband dielectric function is proportional to the joint density of states (JDOS) and the transition matrix elements44. The JDOS is defined as $${D}_{{{\mbox{JDOS}}}}(E)=\frac{1}{{\left(2\pi \right)}^{3}}{\sum }_{c,v}{\int }_{E}\frac{d{S}_{{{{{{\boldsymbol{k}}}}}}}}{|{\nabla }_{{{{{{\boldsymbol{k}}}}}}}({E}_{{{{{{\boldsymbol{k}}}}}},c}{-}{E}_{{{{{{\boldsymbol{k}}}}}},v})|}$$, where $${E}_{k,c}$$ and $${E}_{{{{{{\boldsymbol{k}}}}}},v}$$ are energies in the conduction and valence band, respectively, and $${S}_{{{{{{\boldsymbol{k}}}}}}}$$ is the constant-energy surface defined by $${E}_{{{{{{\boldsymbol{k}}}}}},c}{-}{E}_{{{{{{\boldsymbol{k}}}}}},v}=E$$. In Fig. 4c, there is one peak $$\alpha$$ below EF and three main peaks ($$\beta,\gamma$$and $$\xi$$) above EF in the energy range −2 eV to 2 eV. Obviously, $$\alpha \to \beta$$, $$\alpha \to \gamma$$ and $$\alpha \to \xi$$ are the three possible transitions that can contribute significantly to the JDOS. Among the three transitions, only $$\alpha \to \beta$$ has a transition energy close to 1.2 eV. Based on the electronic band structure (Fig. 4a) and the parities of the Bloch electron wave functions (Table 1), we attribute the absorption peak to the transitions between state 2 and states 3 & 4, corresponding to the transition from $${a}_{1{{{{{\rm{g}}}}}}}$$ to $${e}_{g}^{\pi }$$ (inset of Fig. 1d)25.

Having discussed the electronic transitions corresponding to the plasmons, we now consider the origin for their ultra-flat behavior. In general, the plasmon modes can be obtained by solving $${\det }|{\varepsilon }_{{{{{{\boldsymbol{G}}}}}},{{{{{{\boldsymbol{G}}}}}}}^{{{{\prime} }}}}({{{{{\boldsymbol{q}}}}}},\omega ) \vert=0$$, with the dielectric function matrix element $${\varepsilon }_{{{{{{\boldsymbol{G}}}}}}{{{{{{\boldsymbol{G}}}}}}}^{{{{\prime} }}}}({{{{{\boldsymbol{q}}}}}},\omega )={\delta }_{{{{{{\boldsymbol{G}}}}}},{{{{{\boldsymbol{G}}}}}}{{{\prime} }}}{-}v({{{{{\boldsymbol{q}}}}}}+{{{{{\boldsymbol{G}}}}}}){\chi }_{{{{{{\boldsymbol{GG}}}}}}{{{\prime} }}}^{0}({{{{{\boldsymbol{q}}}}}},\omega )$$45,46. The flat plasmon is attributed to a weak dependence of the dielectric function $${\varepsilon }_{{{{{{\boldsymbol{GG}}}}}}{{{\prime} }}}({{{{{\boldsymbol{q}}}}}},\omega )$$ on $$|{{{{{\boldsymbol{q}}}}}} \vert=q$$ along a particular direction. Although LFE shows large effect on the plasmon behavior along the out-of-plane direction (Fig. 3e, f). However, from qΓ-T = 0.044 Å−1 to qΓ-T = 0.44 Å−1, the variation of the plasmon energy without LFE is only 62 meV (3.079 eV−3.017 eV), which indicates the plasmon mode already exhibits a dispersionless behavior without considering LFE. Hence, when we study the origin of the flat behavior for the plasmons, we neglect the LFE and reduce the dielectric function matrix to $${\varepsilon }_{00}({{{{{\boldsymbol{q}}}}}},\omega )$$,

$${\varepsilon }_{00}({{{{{\boldsymbol{q}}}}}},\omega )=1-v({{{{{\boldsymbol{q}}}}}})\frac{1}{V}\mathop{\sum }\limits_{{{{{{\boldsymbol{k}}}}}}}^{BZ}\mathop{\sum }\limits_{n,n{\prime} }\frac{{f}_{n,{{{{{\boldsymbol{k}}}}}}}-{f}_{n{\prime},{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}}}}{\hslash \omega+{E}_{n,{{{{{\boldsymbol{k}}}}}}}-{E}_{n{\prime},{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}}}+i\eta }{|\langle {\psi }_{n,{{{{{\boldsymbol{k}}}}}}}|{e}^{-i{{{{{\boldsymbol{q}}}}}}\cdot {{{{{\boldsymbol{r}}}}}}}|{\psi }_{n{\prime},{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}}}\rangle|}^{2}$$
(3)

here, $$v({{{{{\boldsymbol{q}}}}}})=4\pi {e}^{2}/{\varepsilon }_{r}{q}^{2}$$ is the Fourier component of the three-dimensional Coulomb potential. For the interband transitions in the long-wavelength limit, we get $$\langle {\psi }_{n,{{{{{\boldsymbol{k}}}}}}}\vert{e}^{{-}i{{{{{\boldsymbol{q}}}}}}\cdot {{{{{\boldsymbol{r}}}}}}}\vert{\psi }_{n^{{{\prime} }},{{{{{\boldsymbol{k}}}}}}{{{{{\boldsymbol{+}}}}}}{{{{{\boldsymbol{q}}}}}}}\rangle \approx {-}i{{{{{\boldsymbol{q}}}}}}\cdot \left\langle {\psi }_{n,{{{{{\boldsymbol{k}}}}}}}\vert{{{{{\boldsymbol{r}}}}}}\vert{\psi }_{n{{{\prime} }},{{{{{\boldsymbol{k}}}}}}}\right\rangle$$47, and thus$${|\langle {\psi }_{n,{{{{{\boldsymbol{k}}}}}}}\vert{e}^{{-}i{{{{{\boldsymbol{q}}}}}}\cdot {{{{{\boldsymbol{r}}}}}}}\vert{\psi }_{n{{{\prime} }}{{{{{\boldsymbol{k}}}}}}{{{{{\boldsymbol{+}}}}}}{{{{{\boldsymbol{q}}}}}}}\rangle|}^{2} \sim {q}^{2}$$. Considering that $$v({{{{{\boldsymbol{q}}}}}})\sim {q}^{{-}2}$$, we thus expect that the dependence of the dielectric function $${\varepsilon }_{00}({{{{{\boldsymbol{q}}}}}},\omega )$$ on $$q$$ to be dominated by the dependence of $${E}_{n{{{\prime} }},{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}}}$$ on $$q$$. Therefore, the flat behaviors of the plasmon can be attributed to the relatively flat band $${E}_{n{{{\prime}}},{{{{{\boldsymbol{k}}}}}}}$$. That is, the dielectric function $${\varepsilon }_{00}({{{{{\boldsymbol{q}}}}}},\omega )$$ is independent of q, as long as the band $${E}_{n{{{\prime} }},{{{{{\boldsymbol{k}}}}}}}$$ is flat, which leads to the flat plasmons. Notably, the flat plasmon can also exist when a series of flat bands coexists in an energy window, due to the sum over the band index $$n{{{\prime} }}$$ in Eq. (3). To explore the flatness of the bands, we plotted the k-ratio in Fig. 4c, which is defined as follows. For a certain energy $$E$$, there are a series of electronic states $${\psi }_{n,{{{{{\boldsymbol{k}}}}}}}$$ in the energy window ranging from $$E{-}\Delta E$$ to $$E+\Delta E$$. We define the number of the wave vectors $${{{{{\boldsymbol{k}}}}}}$$ corresponds to these electronic states $${\psi }_{n,{{{{{\boldsymbol{k}}}}}}}$$ as $${N}_{1}$$, and the number of the wave vector over the entire Brillouin zone as $$N$$. The k-ratio is defined by the ratio of $${N}_{1}$$ and $$N$$. According to this definition, if the k-ratio at the energy $$E$$ is equal to 1, there is at least one band in the energy window from $$E{-}\Delta E$$ to $$E+\Delta E$$ in the entire first Brillouin zone (not restricted to the highly-symmetric directions). For relatively small values of $$\Delta E$$, k-ratio = 1 indicates that the band at energy $$E$$ is relatively flat. In this work, we choose $$\Delta E=0.3{{\mbox{eV}}}$$, because the minimum bandwidth is nearly 0.3 eV. In Fig. 4c, the k-ratio of peaks $$\beta,\gamma$$ and $$\xi$$ are all very close to 1, indicating the existence of flat bands in the energy window from ~0.3 to 1.3 eV above EF in α-Ti2O3. This result is consistent with the proposed narrow bands in α-Ti2O3, which are renormalized by the Hubbard U48,49. Thus, the electronic transitions from the valence bands to the nearly flat conduction bands lead to the flat behaviors of the plasmon. Moreover, the LFE can further reduce the plasmons dispersion, leading to the ultra-flat plasmons in α-Ti2O3 with the energy corrugation less than 40 meV. Same conclusion can be made beyond the long-wavelength limit, which is presented in the supplementary materials (Supplementary Fig. 7, Supplementary Note 2).

To further explore the origin of the nearly flat bands, the real parts of the wave functions for state 1, 2, 3, 4 at the Γ point are plotted in Fig. 4b. Obviously, state 1, 2 consists largely of Ti $$3{d}_{{z}^{2}}$$ orbital (the $${a}_{1g}$$ bonding molecular orbitals) character. State 1 has even parity with respect to O point, whereas state 2 has odd parity with respect to O point. States 3, 4 derives from Ti $${3d}_{{xy}}$$ and $${3d}_{{x}^{2}{-}{y}^{2}}$$ orbitals (the $${e}_{g}^{\pi }$$ bonding molecular orbitals) and has even parity, consistent with our previous discussion. Notably, for state 1, 2 at the Γ point, there is considerable overlap of orbitals centered on different atoms, whereas the orbitals from different atoms in state 3, 4 overlap less. The relatively weak interaction between the in-plane orbitals of Ti atoms further contributes to the flatness of the conduction band ($${e}_{g}^{\pi }$$ and $${e}_{g}^{\pi*}$$ bands) near EF.

Finally, we compare the flat plasmon modes in α-Ti2O3 with some representative materials that have dispersionless plasmon modes18,50,51,52,53,54,55,56,57. We limit our comparison to materials for which the maximum change of plasmon energy (Δωp) is less than 0.1 eV. The start-stop momentum and the flatness of the plasmons in these materials are shown in Fig. 5. (More details are shown in Supplementary Table 2.) Clearly, both in-plane and out-of-plane plasmons in α-Ti2O3 can propagate through a larger momentum space and maintain a higher degree of localization than that of the other low dimensional materials. Noteworthily, for the other materials considered in Fig. 5, they, except VSe2, are not strongly correlated systems, thus the renormalized flat energy bands do not exist in these systems. VSe2 is also a 3d1 electron system, same as α-Ti2O3. Correlation effect should also exist among those V 3d1 electrons, thus renormalized energy bands can be expected in VSe2. And there are indeed some relatively flat unoccupied V 3d bands near EF57. However, its correlation is not so strong to open a band gap at EF, which makes it a metallic system. Thus, its flat plasmon originates from the intraband transition with the screening effect of the interband transitions18,57, which is similar to those in the other non-correlated systems but different from that in α-Ti2O3. As we discussed above, α-Ti2O3 is a strongly correlated insulator, and its long-lived flat plasmons originate from the interband transitions between relatively flat occupied and unoccupied bands.

Based on our analysis, at least one flat occupied band and one flat unoccupied band are needed to generate flat plasmons in α-Ti2O3. Adding more flat bands near EF, more absorption can be achieved, which would further change the plasmon energy and intensity. As for tuning the plasmon behaviors in α-Ti2O3, we believe chemical doping (i.e. V-doping) could be an efficient way. In strongly correlated systems, chemical doping can tune the strength of correlation effect, which further tunes the band renormalization that leads to varied bandwidth, band position, band gap, and thus absorption25. Since the flat plasmons in α-Ti2O3 originate from the interband transitions between those correlation-effect renormalized occupied and unoccupied flat bands, their behaviors (including plasmon energy, flatness and intensity) could be tuned by chemical doping.

## Discussion

In summary, we systematically investigated the electronic structure and plasmonic properties of the strongly correlated 3D oxide α-Ti2O3. Our results show that α-Ti2O3 possesses long-lived plasmons in both in-plane and out-of-plane directions with propagation momentum over the first Brillouin zone. Moreover, plasmon modes in α-Ti2O3 exhibit ultra-flat behavior, with an energy fluctuation of less than 40 meV. We correlate these intriguing plasmons in α-Ti2O3 to the relatively flat conductive bands that are renormalized by the strong electron-electron interaction U and present a general physical mechanism for this effect. Significantly, the ultra-flat feature in the plasmon dispersion in α-Ti2O3 is superior to that of other low dimensional materials; the resulting highly localized and low-velocity plasmon wave packets can have considerable potential for fine electronic structure detection and electric field enhancement18,58,59,60. Notably, the mechanism presented here is universal as long as the plasmon originates from interband transitions from a relatively flat occupied band to a flat unoccupied band. Our work extends the study of flat plasmons in 3D systems and highlights the interplay of correlation effects, electronic bandwidth and plasmon dispersion in strongly correlated systems. This study will stimulate searches for and investigations of flat plasmons in other correlated systems.

## Methods

### Sample preparation and experimental characterizations

The α-Ti2O3 single crystals were synthesized by mixing high-purity TiO2 and TiH4, and then calcining the mixture at 1000 °C in vacuum61,62. Prior to structural characterization and electrical and optical measurements, the single crystals were cut parallel to the (0001) and (11$$\bar{2}$$0) surfaces and then polished. For the XRD measurements, the sample was characterized using a Bruker D8 DISCONVER high-resolution diffractometer, which is equipped with Cu Kα radiation source and LynxEye detector. The X-ray source is operated at 40 kV and 60 mA. The resistivity vs, temperature data were taken using the standard four-probe method in a commercial Quantum Design physical property measurement system (PPMS). A commercial spectroscopic ellipsometer (M2000DI and IR-VASE Mark II; J.A. Woollam Co.) was used to measure the optical response in the xy-plane and along the z-axis of the α-Ti2O3 single crystals. The measurement was operated in an ultra-high vacuum cryostat at room temperature.

### Theoretical calculations

Our first-principles calculations were performed using density functional theory, as implemented in the Vienna ab simulation package (VASP)63 and GPAW codes64, both of which employ the projected augmented-wave method to model interactions between electrons and ions65. The exchange-correlation functional was treated self-consistently within the generalized gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) functional66. The cutoff energy was set to 500 eV. The GGA + U method67 accounting for strong Coulomb interaction between the partially filled 3d-shells of Ti was also employed. The Hubbard interaction parameter Ueff (U - J, where J = 0) was set to 3.0 eV, to bring the calculated band gap closer to the experimental value68,69. Structure relaxation and electronic properties of Ti2O3 were calculated using VASP with the 8 × 8 × 8 (11 × 11 × 4) k-point mesh for primitive cell (conventional cell). The lattice constants and the atomic positions were fully relaxed until the atomic forces on the atoms were less than 0.01 eV/Å and the total energy change was less than 10−5 eV.

Calculations of the dynamic dielectric function and loss function were performed using linear response theory70 implemented in the GPAW code. The conventional cell for Ti2O3 was used to calculate the q → 0 limited dielectric function for different directions along the principal axis. A denser k mesh of 32 × 32 × 10 was adopted to converge the optical calculations. In order to conserve computing resources, the primitive cell was used in calculating the q-dependent loss function. Two orthogonal directions along Γ−S0 and Γ−T were chosen with a dense k-point grid of 31 × 31 × 31. Under the random phase approximation (RPA), the dielectric matrix for wave vector q was represented as:

$${\varepsilon }_{G,G^{{{\prime} }}}^{{{\mbox{RPA}}}}({{{{{\boldsymbol{q}}}}}},\omega )={\delta }_{G,G^{{{\prime} }}}{-}\frac{4\pi }{{{{{{\rm{|}}}}}}{{{{{\boldsymbol{q}}}}}}+{{{{{\boldsymbol{G}}}}}}{{{{{{\rm{|}}}}}}}^{2}}{\chi }_{G,G^{{{\prime} }}}^{0}({{{{{\boldsymbol{q}}}}}},\omega )$$

where $${\chi }_{G,G{{{\prime} }}}^{0}$$ is the non-interacting density response function in reciprocal space, written as45,46

$${\chi }_{{GG}^{{{\prime} }}}^{0}({{{{{\boldsymbol{q}}}}}},\omega )=\frac{1}{\Omega }\mathop{\sum }\limits_{{{{{{\boldsymbol{k}}}}}}}^{{{\mbox{BZ}}}}\mathop{\sum }\limits_{n,n^{{{\prime} }}}\frac{{f}_{n{{{{{\boldsymbol{k}}}}}}}{-}{f}_{n^{{{\prime} }}{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}}}}{\omega+{\varepsilon }_{n{{{{{\boldsymbol{k}}}}}}}{-}{\varepsilon }_{n^{{{\prime} }}{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}}}+i\eta }\times \left\langle {\psi }_{n{{{{{\boldsymbol{k}}}}}}}\left|{e}^{{-}i({{{{{\boldsymbol{q}}}}}}+{{{{{\boldsymbol{G}}}}}})\cdot {{{{{\boldsymbol{r}}}}}}}\right|{\psi }_{n^{{{\prime} }}{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}}}\right\rangle {\Omega }_{{{\mbox{cell}}}}$$
$$\times \left\langle {\psi }_{n{{{{{\boldsymbol{k}}}}}}}\left|{e}^{i({{{{{\boldsymbol{q}}}}}}+{{{{{\boldsymbol{G}}}}}}^{{{\prime} }})\cdot {{{{{\boldsymbol{r}}}}}}^{{{\prime} }}}\right|{\psi }_{n^{{{\prime} }}{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}}}\right\rangle {\Omega }_{{{\mbox{cell}}}}$$

where G and q are the reciprocal lattice vector and wave vector, respectively. $$f$$ is the Fermi distribution function calculated by the following formula

$$f(E)=\frac{1}{1+{\exp }[(E{-}{E}_{F})/{k}_{B}T]}$$

The Kohn-Sham energy eigenvalues $${\varepsilon }_{n{{{{{\boldsymbol{k}}}}}}}$$, the wave function $${\psi }_{n{{{{{\boldsymbol{k}}}}}}}$$ and the Fermi distribution function $${f}_{n{{{{{\boldsymbol{k}}}}}}}$$ for the nth band at wave vector k were obtained from the ground-state calculations. The electron energy loss spectrum (EELS) can be calculated from the inverse of the macroscopic dielectric matrix $${\varepsilon }_{M}({{{{{\boldsymbol{q}}}}}},\omega )=1/{\varepsilon }_{{{{{{\boldsymbol{G}}}}}}={{{{{\boldsymbol{G}}}}}}^{{{\prime} }}=0}^{{-}1}({{{{{\boldsymbol{q}}}}}},\omega )$$

$$L(q,\omega )={-}{{{{{\rm{Im}}}}}}[1/{\varepsilon }_{M}(q,\omega )]$$

The plasmon energy was then extracted from local maxima in the EELS. In our calculations, 84 empty bands were considered to describe the response function. The broadening parameter η was taken to be 0.05 eV. A cut-off of 50 eV was used to account for local field effects. The irreducible representations and parity of electronic states was computed using the Irvsp43 code.