Ultra-flat and long-lived plasmons in a strongly correlated oxide

Plasmons in strongly correlated systems are attracting considerable attention due to their unconventional behavior caused by electronic correlation effects. Recently, flat plasmons with nearly dispersionless frequency-wave vector relations have drawn significant interest because of their intriguing physical origin and promising applications. However, these flat plasmons exist primarily in low-dimensional materials with limited wave vector magnitudes (q < ~0.7 Å−1). Here, we show that long-lived flat plasmons can propagate up to ~1.2 Å−1 in α-Ti2O3, a strongly correlated three-dimensional Mott-insulator, with an ultra-small energy fluctuation (<40 meV). The strong correlation effect renormalizes the electronic bands near Fermi level with a small bandwidth, which is responsible for the flat plasmons in α-Ti2O3. Moreover, these flat plasmons are not affected by Landau damping over a wide range of wave vectors (q < ~1.2 Å−1) due to symmetry constrains on the electron wavefunctions. Our work provides a strategy for exploring flat plasmons in strongly correlated systems, which in turn may give rise to novel plasmonic devices in which flat and long-lived plasmons are desirable.


Theoretical calculations
It is known that the damping of plasmons is determined by the imaginary part of dielectric function ( , ) ε ω ′ GG q , which can be written as The finite temperature can affect the imaginary part of dielectric function through the where is the Fermi energy, kB and T are Boltzmann constant and absolute temperature, respectively. Then, we examined the effect of finite temperature by recalculating the Im(ε) and EELS via setting the temperature values from 0 to 400 K.
As shown in Supplementary Fig. 4, the Im(ε) nearly keeps unchanged with temperature increased to 400 K, indicating the damping of plasmons is nearly not influenced by the temperature. Moreover, the difference for plasmon energy, intensity and the maximum propagating wave vector at different temperatures ( Supplementary Fig. 5) is very small (~ 1%). Therefore, the finite temperature has negligible effects on the damping of plasmons in α-Ti2O3, based on the calculations.
It should be noted that temperature can affect the plasmon behavior via affecting electron-phonon interactions and thermal excitations of electrons, which would introduce additional dissipation channels for plasmons S1,S2 . However, these higherorder decay processes can only affect the plasmon modes that have low energy and close to the phonon energy.

Experimental measurements
In fact, the temperature can change the lattice constants and correlation strength in α-Ti2O3, which can affect the band structure near Fermi level (indicated by the temperature-dependent resistivity result (Fig. 1d)). However, band changing near Fermi level can only influence the dielectric function at low energy range, which cannot affect the plasmon modes much since the plasmon energy at the much higher energy (~ 2.6 eV). As shown in Supplementary Fig. 6, the plasmon energy only changed by ~ 63 meV and ~ 50 meV for in-plane and out-of-plane directions, respectively. And the Im (ε) keeps nearly vanished at the plasmon energy for all temperatures, and does not change much with temperature. Therefore, the damping of plasmons in α-Ti2O3 would not be changed much by finite temperature.
Hence, since the plasmon energy in α-Ti2O3 is much larger than the kBT thermal energy and phonon-related coupling energy, the impact of finite temperature on the damping of plasmons can be neglected S1,S3 . Thus, the long-lived feature of plasmons in α-Ti2O3 is robust for finite temperatures.

Beyond the long-wavelength limit
We made an approximation when we discuss the origin for the ultra-flat behavior of the plasmons in α-Ti2O3 using The approximation .
According to the Kramers-Kronig relation, the real part of ( , ) ε ω q can be evaluated from the imaginary part, (2) Therefore, The plasmon mode is determined by Re ( , ) 0 ε ω = q . The flat plasmon means that the plasmon dispersion is independent of q. Thus, it should satisfy Re ( , for ω  at the plasmon energy. That is, the flat plasmon requires that . Notably, the supplementary Eq. (3) where , c v are the band indexes of valence and conductive bands, respectively.
where ( ) J ω q  is the joint density of states (JDOS) for finite wave vector q , which is defined as 3 , ,