Introduction

Flat band (FB) characterized dispersionless electronic band with nearly zero bandwidth gives rise to an infinitely large effective mass for Bloch electrons, nearly zero Fermi velocity, and negligible kinetic energy. Such the suppression of electron kinetic energy effectively enhances electrons’ Coulomb correlations, which makes FBs a fertile ground for realizing strongly correlated phenomena, including superconductivity, fractional quantum Hall state1,2, Excitonic insulator3,4,5, Wigner crystallization6,7,8, and unconventional magnetism9,10,11. Various routes have been explored to realize FBs experimentally, including Landau levels induced by strong magnetic fields12,13,14, heavy fermion formation in Kondo lattice systems15,16,17, and moiré-induced localization in twisted bilayer van der Waals materials18,19,20,21. Despite conceptual appeal, these approaches face critical limitations such as extreme field requirements, ultralow temperatures, or intricate fabrication procedures. The other approach to realize FBs is the exploration of low-dimensional materials in which FBs would be generated through electronic quantum interference22,23. Nevertheless, their intrinsic geometric frustration renders them structurally unstable and experimentally elusive.

Recently, frustrated lattices, such as kagome24,25, Lieb26,27, and coloring-triangle lattices28,29, have emerged as promising platforms for hosting FB-driven correlated states. The FBs in such the frustrated lattices arise from destructive interference of electron wavefunctions in a frustrated geometry. Representative progresses in these systems include the discovery of charge density wave, superconductivity or magnetism in AV3Sb5 (A = Cs, Rb, K)30,31,32, Co3Sn2S233,34, Fe3Sn235, and Mn3Sn36. Despite these advances, a critical challenge remains, that is, the precise regulation of FBs in energy scale. The central objective is to modulate the Fermi level to the FB in order to further tune the electron filling fraction. Although electrostatic gating is commonly employed in two-dimensional (2D) moiré systems to tune carrier densities (~1013 cm-2), it is fundamentally insufficient for 2D frustrated lattices, which demand orders-of-magnitude higher densities (~1016 cm-2) to effectively modulate flat-band filling. To overcome this limitation, introducing charged defects among frustrated lattice offers an alternative pathway to design and modulate FBs in atomic length scale. 2D phosphorene and phosphides epitaxially grown on noble metal substrates37,38 have attracted extensive interest recently due to their unique electronic properties. In these systems, the Au(111) surface is an ideal substrate for synthesizing 2D lattices with significant electronic doping capability39. Previous studies indicated that phosphorus deposition on Au(111) typically forms Au9P18, a structure where P9 units are bridged by Au atoms40. Deposition of potassium (K) atoms on the Au9P18 yields 2D frustrated lattices in addition to on-site doping effect41. It indicates that FBs with tunable energy position would be realized in 2D K-associated phosphide, however, such the approach remains unexplored experimentally.

Here, we report the synthesis of a crystalline K3P thin film via thermal substitution of Au atoms in Au9P18 by K. Combined with atomically resolved scanning tunneling microscopy/spectroscopy (STM/STS) and density functional theory (DFT) calculations, 2D K3P is identified as a double atomic layer Lieb-lattice system. Three highly anisotropic FBs have been observed in its electronic structure, which are given rise to next-nearest-neighbor hopping in Lieb lattice and the surface K atoms. Furthermore, charged defects in K3P generate spatially resolved band bending, mimicking electrostatic potentials of point charges. This system provides a highly controllable pathway to design FB materials and manipulate their electronic states through defect engineering in atomic scale.

Results and discussion

A clean Au(111) substrate was prepared prior to growing the K3P thin film. Figure 1b shows the large-area morphology of the Au(111) substrate, where the characteristic (22×√3) herringbone reconstruction is clearly visible. The atomic periodicity was measured to be 2.9 Å, as shown in Fig. 1a. Subsequently, P4 molecules were deposited onto the surface while maintaining the substrate temperature at 530 K. The evaporated P₄ molecules self-assemble into P9 units interconnected via three Au atoms between adjacent units, resulting in a stoichiometric ratio of Au9P18 per unit cell with a lattice constant of 1.45 nm, in agreement with previous reports42. Each P9 unit exhibits a buckled configuration and the atomically resolved STM image in Fig. 1c reveals the top three P atoms of this unit. Our prior work demonstrated that the Au9P18 structure possesses abundant active sites41,43, which offers a favorable platform for facilitating surface reactions. To explore the substitution of Au atoms, K, an element with higher reducibility, was deposited onto the Au9P18 surface. When the substrate is kept at room temperature (RT), two types of frustrated K lattices—a distorted coloring-triangle lattice and a kagome lattice—adsorbed at distinct sites on the Au9P18 surface were observed41. This indicates that K deposition at RT is solely a physical adsorption process, with no chemical reaction occurring between K and Au9P18. Therefore, to investigate potential substitution reactions, the sample was post-annealed at elevated temperatures. Upon annealing the K-deposited Au9P18 at 570 K, square lattices in large areas are observed, as shown in Fig. 1f. The high-resolution STM image in Fig. 1e reveals that the square lattice has a periodicity of 6.2 Å. It should be noted that the typical bond lengths of K-P, Au-P, Au-Au, and P-P are 3.2 Å, 2.4 Å, 2.9 Å, and 2.17 Å, respectively. These values are much smaller than the observed lattice constant of the square phase (6.2 Å). This discrepancy implies that the square lattice likely possesses a substructure formed by a suitable arrangement of K-P-Au bonds. The 6.2 Å lattice constant is approximately twice the K-P bond length, suggesting a structural motif based on K-P bonding, which will be discussed thoroughly in later case. Furthermore, the angle between the K-P layers and the Au(111) substrate can be determined from regions exhibiting small K-P islands alongside a (2×2) superlattice relative to the Au(111) substrate periodicity. Supplementary Note 1 and Supplementary Fig. S1 present the measured angles between the K-P layers and this (2×2) phase. The variation in these angle values suggests that the interaction between the K-P layers and the Au(111) substrate is relatively weak.

Fig. 1: Growth process of K3P films.
Fig. 1: Growth process of K3P films.The alternative text for this image may have been generated using AI.
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a, b Close-up (V = −50 mV, I = 1.5 nA) and large-area (V = 1 V, I = 0.1 nA) scanning tunneling microscopy (STM) images of the clean Au(111) substrate. The phosphorus (P) source material is evaporated on the surface in the form of P4 molecules. c, d Close-up (V = 50 mV, I = 1 nA) and large-area (V = 0.1 V, I = 50 pA) STM images of the Au9P18 film. The film periodicity of 1.45 nm is marked by a white rhombus, and its atomic structure is shown in the inset of panel (c). Potassium (K) atoms were deposited onto the Au9P18 film followed by annealing to synthesize K3P. e, f Close-up (V = 100 mV, I = 0.8 nA) and large-area (V = −1.2 V, I = 0.3 nA) STM images of the K3P lattice. K3P exhibits a square unit cell with a lattice constant of 6.2 Å, as marked by the white square. The color bar codes for the topographic height (L: low, H: high). Scale bar: a 0.6 nm; b 15 nm; c 1 nm; d 14 nm; e 0.6 nm; f 3 nm.

To determine the atomic configuration of K-P compound on Au(111), we systematically perform DFT calculations to match the experimental results. Considering the realistic lattice constant 6.2 Å is equal to 2 times K-P bond, the most direct atomic model is the K2P compound consisted of a Lieb lattice (see Supplementary Note 2 and Supplementary Fig. S2). However, the optimized lattice constant of K2P layers is 5.54 Å, deviating a lot from the observed value. From the point of view in coordination environment, the P atoms can accept three electrons to achieve the full shell electron structure, which means that one P atom should be connected with three K atoms. Consequently, we add one K atom at the top of each P atom in current K2P Lieb lattice model. This eventually gives rise to a triple layer structure consisting of an Au(111) substrate, an intermediate K2P layer, and surface KI atoms, where the intermediate layer still forms a long-range ordered Lieb lattice through the arrangement of KII and P atoms. Three-dimensional atomic structure of K3P is exhibited in the up panel in Fig. 2a and the down panel shows the side view of this structure (Supplementary Data 1). Moreover, the calculation of lattice relaxation indicates the K3P has an optimized lattice constant of 6.2 Å (Supplementary Fig. S2), which is consistent with the experimental results. The simulated STM image is in a good agreement with experimental observations, further corroborating the K3P structure model (Fig. 2b). Consequently, we have demonstrated that a thermally stabilized K3P Lieb lattice can be achieved through annealing K-adsorbed Au9P18. In principle, the tight-binding (TB) model for Lieb lattices (considering only nearest-neighbor hopping) exhibits characteristic FBs close to Dirac points (Supplementary Note 3 and Supplementary Fig. S3). While the KII-P staggered lattice in K3P introduces next-nearest-neighbor hopping between KII atoms (Fig. 2c). This leads to partial dispersion of the middle FB while giving rise to a FB character in the bottom band (Fig. 2d), effectively realizing controllable modulation of FB energy positions. Nevertheless, DFT calculations in Fig. 2e reveal an FB (labeled as band 4) that is not explained by destructive interference between hopping pathways in the Lieb lattice. Partial charge distribution demonstrates that this FB originates from strong electron localization of the top surface KI (region 4 in Fig. 2f). Consequently, three FBs (labeled as FB-1, FB-2 and FB-3, respectively) eventually form and give rise to three resonances in the calculated density of states (DOS) result, as shown in the right panel in Fig. 2e.

Fig. 2: Atomic structures and calculated bands of K3P Lieb lattice.
Fig. 2: Atomic structures and calculated bands of K3P Lieb lattice.The alternative text for this image may have been generated using AI.
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a Schematic of two-dimensional K3P on Au(111) (top and side views), showing KI (light blue), KII (dark blue), P (purple), and Au (white) atoms. b Experimental (left) (V = 50 mV, I = 1 nA) and simulated (right) scanning tunneling microscopy images of K3P, accompanied by its top-view atomic structure. c Lieb lattice model featuring nearest-neighbor (t1) and next-nearest-neighbor (t2) hopping, with labeled sites 1–3. d Lieb lattice tight-binding bands with hopping parameters t1 = 1 and t2 = 0.5. e The band structure and density of states obtained from density functional theory calculations including four characteristic bands and three flat band (FB) resonances (FB-1 ~ FB-3). f Partial charge density distributions (1–4) corresponding to the four energy bands in (e).

In order to verify the theoretical prediction, we performed STS measurements to probe the electronic structure of K3P layers. Figure 3a shows the sampling positions for point STS, selected in regions free of impurities or vacancies to ensure observation of the intrinsic DOS. The band structure mapping acquired along the white arrow in Fig. 3a is presented in Fig. 3b. We found that the dI/dV spectra collected at different sites within the K3P lattice share common features, characterized by three DOS resonances located at ~0.3 eV, −0.1 eV, and −0.7 eV, respectively. To quantitatively compare the intensities of these three DOS resonances, the average STS spectrum was calculated and is shown in Fig. 3c. These spectral features are in excellent agreement with our calculated band structure for the K3P lattice. Additional theoretical calculations (Supplementary Note 4 and Supplementary Fig. S4) indicate that K3P possesses a lower work function than the Au(111) surface. This implies that the K3P layers donate electrons to the substrate, resulting in a downshift of the Fermi level relative to the pristine material. Therefore, the DOS resonance at 0.3 eV can be attributed to the FB labeled FB-1 in Fig. 2e, while the resonances at −0.1 eV and −0.7 eV correspond to FB-2 and FB-3, respectively. STS measurements on the two differently oriented islands reveal a similar set of features, including the characteristic three FB peaks (Supplementary Note 5 and Supplementary Fig. S5). It is noted that the van Hove singularities (vHs) predicted in DFT is too weak to be detected in the STS results, as indicated in Supplementary Note 6 and Supplementary Fig. S6. We also performed comparative STS measurements and lattice analysis (Supplementary Note 7 and Supplementary Fig. S7) to distinguish the electronic states of the K3P monolayer from those of the Au(111) substrate. The absence of the Au(111) Shockley surface state at its characteristic energy and the lack of its periodicity in fast Fourier transform (FFT)/ inverse FFT images confirm that the spectroscopic features discussed above originate from the K3P layer, with negligible contribution from the substrate. In addition to the pristine surface, some defects can be found in the surface of K3P lattice, which is manifested as a bright spot in the STM image as marked by a red circle in Fig. 3d. Besides the bright spot defects, the black hole features can also be found on the surface of K3P. The possible structural origin about these defects is discussed in Supplementary Note 8 and Supplementary Fig. S8. The STS collected at different distances (denoted as r) from the central bright spot defect exhibit a clear band bending behavior, as shown in Fig. 3e. The low-density part in the mapping results comes from the dip feature accompanying FB-3, as shown in Fig. 3f. Such band bending is a common phenomenon observed at ferroelectric domain walls and other interfaces44,45 characterized by strong electron transfer, all originating from the built-in electric fields within the materials. Consequently, the observed band bending indicates that the bright spot defect can act as a charge center (CC), generating a strong electric field in its vicinity. Given that the energy positions of FB-3 shift from −0.3 eV to −0.7 eV as the distance increases, this defect must act as a negative CC, with the resulting electric field directed inward toward the center. Theoretically, the electrostatic potential U generated by an ideal negative point charge is given by U = -q/(4πϵr), where q is the charge magnitude, ϵ is the dielectric constant of the material, and r is the distance from the CC. This Coulomb potential results in an energy gradient that decreases with increasing distance. The spatial gradient of this energy variation manifests directly as the slope of the band bending profile observed in Fig. 3e. It is evident that the band bending profile closer to the CC exhibits a steeper slope, which is qualitatively consistent with the behavior expected for the potential of a point charge. Although FB-1 and FB-2 are theoretically predicted to exhibit band bending similar to FB-3, our experiments show that their bending is too subtle to be detected (Supplementary Note 9 and Supplementary Fig. S9). This phenomenon, previously reported in MoSe2 and MoS2 grain boundary systems and attributed to strain46,47, leads us to propose that the minimal bending of FB-1 and FB-2 may also originate from lattice distortion around the charge defects. To rule out possible artifacts from the electric field of the metallic STM tip, we acquired I–V and dI/dV spectra at varying tip heights around the charge defect (Supplementary Note 10 and Supplementary Fig. S10). The dI/dV line shapes remain nearly identical across these measurements, ruling out a significant influence of the tip’s electric field on the energy position of FB-3.

Fig. 3: Electronic structure of K3P Lieb lattice.
Fig. 3: Electronic structure of K3P Lieb lattice.The alternative text for this image may have been generated using AI.
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a Atomically resolved scanning tunneling microscopy (STM) image (V = 100 mV, I = 0.8 nA) of K3P. Color-coded circles mark the dI/dV sampling points. b dI/dV spectra acquired along the white arrow in (a), presented as waterfall plots. Semi-transparent lines indicate the positions of the three flat band (FB) resonances (denoted as FB-1, FB-2, and FB-3). c Average dI/dV spectrum calculated from ten spectra in (b). d STM image (V = 50 mV, I = 1 nA) of a region containing a charged defect. Red circle marks the position of a charge center. e dI/dV spectra acquired at varying distances from the charge center in (d), presented as waterfall plots. A white dashed line traces the evolution of the FB-3 peak and the associated dip feature. f dI/dV spectrum extracted at position r3 in (e). The FB-3 peak and its accompanying dip feature are marked. The color bar in (a) and (d) codes for the topographic height (L: low, H: high). The color bar in (b) and (e) codes for the dI/dV intensity (L: low, H: high). Scale bar: (a) 1 nm; (d) 1 nm.

To demonstrate the designable atomic-scale energy modulation of the FB in K3P, we identified a region containing multiple CCs. Figure 4a shows the atomically resolved STM image containing three negative CCs, which are labeled as CC-1, CC-2, and CC-3, respectively. Using a grid-based STS mapping method, the spatial distribution of FB-3 at different energies can be obtained. Figure 4b–m presents the evolution of dI/dV maps at varying sample bias voltages, where dark regions correspond to the contour lines of the dip state associated with FB-3. From −380 mV to −840 mV, a CC located in the top-left corner is distinguishable, and the radius of its surrounding contour line gradually increases. At approximately −860 mV, the contour lines originating from this top-left CC and another CC located outside the observed field of view begin to touch. As the sample bias voltage further decreases, the contour lines separate at the contact point, forming two expanding arcs within the selected area, visible in the top-right and bottom regions of Fig. 4j–m. Another prominent feature is the emergence of a contour ring in the central region, which progressively shrinks as the sample bias decreases. This behavior is well explained by the distribution of equipotential surfaces generated by three point charges in vacuum. Figure 4n illustrates the equipotential contour lines generated by three negative point charges, each carrying a charge of −1 C, in an infinite vacuum medium. With CC-1 located at the coordinate origin (0, 0), the relative positions of the three charges are specified as (0, 0), (1, 0), and (0.53, –0.85), in units of centimeters. The red solid line in Fig. 4n outlines the characteristic region observed experimentally. In this simulation, key experimental observations—the expanding rings around individual CCs, the merging and subsequent separation of arcs beyond the touching point, and the emergence and contraction of the central contour ring—are accurately reproduced. This close agreement indicates that the idealized three-point-charge model captures the essential physical origin of the phenomena and, in principle, we can obtain any desired flat band position at specific atomic sites by further designing the multiple defect locations.

Fig. 4: dI/dV mapping on K3P Lieb lattice containing three charge centers.
Fig. 4: dI/dV mapping on K3P Lieb lattice containing three charge centers.The alternative text for this image may have been generated using AI.
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a Scanning tunneling microscopy image (V = 100 mV, I = 0.1 nA) of a region containing three charge centers (CCs), labeled CC-1, CC-2, and CC-3. b–m Series of dI/dV maps acquired at different sample bias voltages. The field of view encompasses one CC (top-left) and excludes two additional CCs. White dashed lines trace the evolution of the dark contour features. The color bar codes for the dI/dV intensity (L: low, H: high). n Simulated equipotential contours generated by three point charges in vacuum. The red square marks the characteristic region corresponding to the experimental features. Scale bar: a 4 nm; b 1 nm.

Conclusion

In summary, we have synthesized a thermally stabilized K3P Lieb lattice on Au(111) through K substitution of the intermediate Au9P18 phase. The K3P structure—comprising a K2P interlayer and surface KI atoms—exhibits a square lattice and hosts three FBs, as validated by DFT calculations and STS measurements. The native defects on K3P can act as negative CCs, creating radial band-bending gradients consistent with point-charge electrostatics. Further dI/dV mapping visualizes equipotential contours around multiple CCs, enabling precise spatial and energy scale modulation of FB. This work not only establishes K3P/Au(111) as a versatile platform for flat-band physics but also presents a strategy for electrostatic engineering in 2D materials via defect control. A direct extension would be to use an STM tip to precisely create or reposition these defects into pre-designed arrays. Such defect engineering would create a programmable potential, potentially modulating the flat bands into mini-bands or inducing correlated states like Wigner crystals or magnetic orders at specific fillings. This would transform our system into a fully tunable quantum simulator.

Methods

Sample preparation

The Au(111) substrate was first prepared by repeated cycles of 1 keV Ar⁺ ion sputtering at room temperature followed by annealing at 850 K. Subsequently, P4 molecules were deposited onto the surface under ultra-high vacuum (UHV) conditions using a heated, well-degassed Ta boat. During deposition, the substrate temperature was maintained at 530 K, with a deposition flux of ∼0.05 monolayer min⁻¹. Finally, K atoms were deposited onto the surface at room temperature under UHV conditions from a heated, well-degassed K source, followed by annealing the sample at 570 K for 30 min.

STM measurements

All STM images were acquired in constant current mode at 4.2 K (liquid helium temperature) using a low-temperature ultra-high vacuum scanning tunneling microscope (Unisoku USM 1500 & Quantum Scale, NanoTech STM). Differential conductance (dI/dV) spectra were acquired using standard lock-in detection by applying a 5 mV (rms), 973 Hz modulation to the bias voltage. The STM images were processed using WSxM48 and Gwyddion software.

DFT calculations

The atomic and electronic structures of K3P were determined using the DFT, as implemented in the Vienna Ab initio Simulation Package49. The calculations for electron-electron exchange interactions relied on the generalized gradient approximation, as parameterized by Perdew–Burke–Ernzerhof50. We performed electronic state calculations using a 6.24 Å ×6.24 Å unit cell, and a vacuum region of ≥15.0 Å was used to exclude periodic surface–surface interactions. The atom positions were fully relaxed until the forces acting on them were reduced to 0.01 eV Å−1. The plane wave basis was set with an energy cut-off of 330 eV. DOS calculations employed a 20×20×1 k-mesh sampling of the K3P Brillouin zone, while the band structure was calculated along the G-X-S-Y-G high-symmetry path. Spin-orbit coupling is not considered.