Compact localized states in magnonic Lieb lattices

Lieb lattice is one of the simplest bipartite lattices, where compact localized states (CLS) are observed. This type of localization is induced by the peculiar topology of the unit cell, where the modes are localized only on selected sublattices due to the destructive interference of partial waves. We demonstrate the possibility of magnonic Lieb lattice realization, where flat bands and CLS can be observed in the planar structure of sub-micron in-plane sizes. Using forward volume configuration, the Ga-doped YIG layer with cylindrical inclusions (without Ga content) arranged in a Lieb lattice with 250 nm period was investigated numerically (finite-element method). The structure was tailored to observe, for a lowest magnonic bands, the oscillatory and evanescent spin waves in inclusions and matrix, respectively. Such a design reproduces the Lieb lattice of nodes (inclusions) coupled to each other by the matrix with the CLS in flat bands.


Supplementary Note 1. Doubly-extended Lieb lattice
We can generate further extensions of the magnonic Lieb lattice [1] by adding more inclusions B, i.e. by introducing additional majority sublattices. We consider here a doublyextended Lieb lattice  to check to what extent the magnonic system corresponds to the tight-binding model. The mentioned lattice consists of seven nodes; six belong to majority sublattices B and one belongs to minority sublattice A (Supplementary Fig. 1). The magnetic parameters were kept as for basic and Lieb-5 lattices, considered in the manuscript.
The geometrical parameters have changed only as a result of the introduction of additional inclusions B. Therefore, the unit cell has increased to the size of 500x500 nm. that the bands will be symmetric with respect to the fourth band, exhibiting particle-hole symmetry. However, due to the dipolar interaction, we did not expect such symmetry.
Another feature that one may deduce from the tight-binding model is that bands no. 2, 4, and, 6 should be flat while bands no. 1, 3, 5, and, 7 are considered dispersive. Moreover, band no. 3 and, 5 suppose to form a Dirac cone intersecting flat band no. 4 at the Γ point.
We calculated the dispersion relation for magnonic Lieb-7 lattice (Supplementary Fig. 2(a)), which share many properties with those characteristic for the tight-binding model

Supplementary Note 2. Realization of Lieb lattice by shaping demagnetizing field
We have considered also an alternative realization method for a magnonic Lieb lattice in a ferromagnetic layer. This approach is based on shaping the internal demagnetizing field.
The structure under consideration is presented in Supplementary Fig. 4. It consists of a thin (14.75 nm) and infinite CoFeB layer on which a Py antidot lattice (ADL), of 14.75 nm thickness, is deposited.
The cylindrical holes in ADL are arranged in the shape of the basic Lieb lattice. The size of the unit cell and diameter of holes remains the same as for the basic Lieb lattice proposed in the main part of the manuscript (see Fig. 1(a)). Due to the absence of perpendicular magnetic anisotropy (PMA), we decided to apply a much larger external magnetic field The deposition of the ADL made of Py (material of lower M S ) above the CoFeB layer (material of higher M S ) is critical for spin-wave localization in CoFeB below the exposed parts (holes) of the ADL. The demagnetization field produced on CoFeB/air interface creates wells partially confining the spin waves. However, this pattern of internal demagnetizing field becomes smoother with increasing distance from the ADL.
The obtained dispersion relation is shown in Supplementary Fig. 5. It is worth noting that the lowest band is very dispersive, while the highest band is flattened more than in the case of the structure presented in the main part of the manuscript (see Fig. 2). The middle band, which suppose to support CLS, varies in extent similar to the third band. For this structure, Dirac cones in the M point cannot be clearly identified.

Supplementary Note 3. Lieb lattice formed by YIG inclusions in non-magnetic matrix
The periodic arrangement of ferromagnetic cylinders surrounded by non-magnetic material (e.g. air) seems to be the simplest realization of the Lieb lattice. To refer this structure to the bi-component system investigated in the main part of the manuscript, we assumed the same material and geometrical parameters for inclusions as for the structure presented in Fig. 1(a).
The advantage of this system is that the confinement of spin waves within the areas of inclusions is ensured for arbitrarily high frequency. We are not limited here by the FMR Therefore, the interaction between inclusions is much smaller in general, which leads to a significant narrowing of all magnonic bands ( Supplementary Fig. 5). The widths of the second and third bands can be even smaller than the gap separating from the first bands - Supplementary Fig. 5(b). Such strong modification of the spectrum makes the applicability of the considered system for the studies of magnonic CLS questionable.

Supplementary Note 4. Demagnetizing field in YIG|Ga:YIG Lieb lattice
The difficulty in designing the magnonic system is not only due to the adjustment of geometrical parameters of the system but also due to the shaping of the internal magnetic field H eff .
The components of the effective magnetic field can be divided into long-range and short- In the literature, this phenomenon has been described for the tight-binding model and is called node dimerization of the lattice [6].