Cover of Physical Review Letters (Volume 137, Number 17)
Haifeng Ding’s Group Advances Precise Energy Control in Two-Dimensional Flat-Band Systems
Researchers led by Professor Haifeng Ding at Nanjing University’s School of Physics have developed a practical approach to tuning the energies of two-dimensional flat bands. Combining tight-binding theory, atomic manipulation, and scanning tunneling spectroscopy, the team designed a cluster-decorated Lieb lattice that enables continuous, predictable adjustment of flat-band energies and precise alignment with the Fermi level.
The study, titled “Cluster-Decorated Lieb Lattice: A Convenient and Reliable Platform for Tuning Flat-Band Energy,” was published online in Physical Review Letters on October 20, 2026 [Phys. Rev. Lett. 137, 176401 (2026)]. It was selected as an Editors’ Suggestion and featured on the cover of the journal’s October 23 issue. The American Physical Society’s Physics Magazine also highlighted the work in a synopsis titled “Tuning Flat Bands,” written by Senior Editor Charles Day.
Figure 1. The cover of Physical Review Letters and the accompanying synopsis by Physics Magazine Senior Editor Charles Day.
Flat bands are characterized by strongly suppressed electron kinetic energy and a high density of states. When a flat band lies near the Fermi level, electron–electron interactions can become particularly significant, making these systems promising platforms for investigating superconductivity, magnetism, and other unconventional electronic states.
Researchers have realized a range of two-dimensional artificial flat-band lattices, including directly assembled lattices, flat bands induced by periodically patterned substrates, and twisted graphene structures. Much of this work has focused on whether flat bands exist. A convenient and reliable experimental method for continuously tuning their energies—especially for aligning them precisely with the Fermi level—has remained an important challenge.
The team began with the Lieb lattice, a well-established flat-band geometry previously investigated by several groups at Nanjing University, including researchers Rui Shen and Academician Dingyu Xing [Phys. Rev. B 81, 014410(R) (2010)]. By replacing each edge site of the conventional Lieb lattice with a geometrically tunable, rhombus-shaped four-site cluster, the researchers constructed a cluster-decorated Lieb lattice.
Theoretical analysis revealed five flat bands at four distinct energies, with two bands degenerate at E1(d). These flat-band energies correspond directly to the eigenenergies of an isolated rhombus-shaped cluster with the same geometry. Changing the cluster geometry therefore shifts its energy levels and, in turn, tunes the flat-band energies of the extended lattice.
To test this approach, the researchers used atomic manipulation to assemble cluster-decorated Lieb lattices from 57 iron atoms on a silver Ag(111) surface. They then performed scanning tunneling spectroscopy at different lattice sites. The measured spectral peaks matched the flat-band energies predicted by tight-binding calculations of the local density of states, confirming the feasibility of the proposed method.
The team further investigated how the E1(d) flat-band energy changes with three geometric parameters of the rhombus-shaped clusters. By developing a model linking cluster geometry to flat-band energy, they identified the configuration required to place this band at the Fermi level and subsequently achieved the alignment experimentally.
These results establish a direct connection between the eigenenergies of individual clusters and the flat-band energies of a cluster-decorated Lieb lattice. This relationship provides a practical route to continuous, predictable energy tuning and precise alignment with the Fermi level. The strategy can also be extended to other cluster geometries and parent flat-band lattices, offering a broader platform for exploring flat-band physics. In his Physics Magazine synopsis, Charles Day highlighted the platform’s potential for further investigations.
Figure 2. Design principle and experimental verification of the cluster-decorated Lieb lattice. (a), (b) A conventional Lieb lattice and its band structure. (c), (d) A cluster-decorated Lieb lattice and its band structure. (e) An artificial lattice assembled from 57 Fe atoms on an Ag(111) surface. (f) Experimentally measured normalized scanning tunneling spectra and calculated local densities of states, showing a flat band aligned with the Fermi level.
This work builds on the group’s 2025 proposal of a general method for constructing two-dimensional flat-band lattices [Phys. Rev. Lett. 134, 076402 (2025)], marking another advance in its research on flat-band systems.
Nanjing University is the sole institutional affiliation for both studies. Haotian Li, a doctoral student in the School of Physics, is the first author of both papers. Doctoral student Wenlin Fan is a co-first author of the present study.
The research was supported by the Collaborative Innovation Center of Advanced Microstructures, the National Laboratory of Solid State Microstructures, the Jiangsu Key Laboratory of Quantum Information Science and Technology, and the Shishan laboratory for frontier research in quantum computing and quantum sensing. Funding was provided by the National Key Research and Development Program of China, the National Natural Science Foundation of China, and the National Science and Technology Major Project for Quantum Communication and Quantum Computing.
Haotian Li acknowledges support from Nanjing University’s “Zhujian” Program for Doctoral Students. Gong Chen acknowledges support from the Jiangsu Provincial Innovation and Entrepreneurship Talent Program. The theoretical calculations were performed using the e-Science facilities of the Collaborative Innovation Center of Advanced Microstructures.
Several faculty members involved in the study have extensive experience teaching solid-state physics. They express their sincere gratitude to An Hu and Weiyi Zhang for their guidance, which helped lay the foundations for this research.
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