⇦Chinese Physics Letters, 2022, Vol. 39, No. 5, Article code 057301 Lieb Lattices Formed by Real Atoms on Ag(111) and Their Lattice Constant-Dependent Electronic Properties Xiaoxia Li (李小霞)1†, Qili Li (李启立)1†, Tongzhou Ji (吉同舟)1, Ruige Yan (闫睿戈)1, Wenlin Fan (范文琳)1, Bingfeng Miao (缪冰锋)1,2, Liang Sun (孙亮)1,2, Gong Chen (陈宫)1,2, Weiyi Zhang (章维益)1,2, and Haifeng Ding (丁海峰)1,2* Affiliations 1National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China 2Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China Received 10 February 2022; accepted 28 March 2022; published online 26 April 2022 †These authors contributed equally to this work.
*Corresponding author. Email: hfding@nju.edu.cn Citation Text: Li X X, Li Q L, Ji T Z et al. 2022 Chin. Phys. Lett. 39 057301    Abstract Scanning tunneling microscopy is a powerful tool to build artificial atomic structures that do not exist in nature but possess exotic properties. In this study, we constructed Lieb lattices with different lattice constants by real atoms, i.e., Fe atoms on Ag(111), and probed their electronic properties. We obtain a surprising long-range effective electron wavefunction overlap between Fe adatoms as it exhibits a $\frac{1}{r^{2}}$ dependence with the interatomic distance $r$ instead of the theoretically predicted exponential one. Combining control experiments, tight-binding modeling, and Green's function calculations, we attribute the observed long-range overlap to being enabled by the surface state. Our findings enrich the understanding of the electron wavefunction overlap and provide a convenient platform to design and explore artificial structures and future devices with real atoms.
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 DOI:10.1088/0256-307X/39/5/057301 © 2022 Chinese Physics Society Article Text Artificial atomic structures built by scanning tunneling microscopy (STM) are a fertile playground to investigate fundamentals and explore potential applications. For instance, logic gates are realized by CO molecule cascades on Cu(111)[1] and Fe elliptical quantum corrals on Ag(111).[2] Quantum holographic encoding of CO on Cu(111) achieves information densities in excess of 20 bits/nm$^{2}$.[3] A kilobyte rewritable atomic memory is presented for Cl vacancies on Cu(100).[4] Fundamentally, Au atomic chains on NiAl(110) demonstrate the development of a one-dimensional band structure.[5] Spin–spin interactions are probed for Mn chains on CuN on Cu(100).[6] Majorana bound states are observed at the ends of chains for Fe on Pb(110)[7–9] and Re(0001).[10] Two-dimensional structures have been widely studied for quantum size effect in nanocorrals,[11–13] Dirac fermions in molecular graphene of CO on Cu(111),[14] quasi-crystals in Penrose tiling of CO on Cu(111),[15] and fractals in Sierpiński triangle CO on Cu(111).[16] Lieb lattice,[17] a two-dimensional square lattice consisting of an atom at the corner and two atoms at the middle of each edge, has attracted significant attention because of its exotic electronic band structure and predicted unusual properties in ferromagnetism,[17–20] superconductivity[21–23] and topological states,[24–26] even though it does not exist in nature. Recently, Lieb lattices have been experimentally realized in optical[27–30] and electronic systems with artificial objects.[31–34] As previously mentioned, STM naturally has the advantage of studying this structure. Lieb lattices have been successfully assembled and investigated with Cl vacancies on Cu(100)[32] and artificial atoms formed by the quantum confinement of the surface state through the CO molecules on Cu(111).[31] It is worth noting that artificial objects are used to mimic the real atoms in the study of Lieb lattices realized by the Cl vacancies and quantum states confined by CO molecules. Therefore, it is essential to construct Lieb lattices with real atoms and investigate their properties. In the present study, we employ atomic manipulation[35] to construct Lieb lattices with real atoms, i.e., Fe adatoms on Ag(111), instead of the anti-Lieb lattices formed by Cl or CO molecules. We obtain typical features of Lieb lattice by measuring the spectroscopy and differential conductance map. Furthermore, we systematically performed lattice constant-dependent studies by tuning the interatomic distance $r$ through atomic manipulation. Note that the high internal pressure in solid makes it nearly impossible to vary $r$ with large amplitude, which is imperative for the $r$-dependence studies. Compared with tight-binding modeling, we obtain that the effective overlap energy $t$ shows a $\frac{1}{r^{2}}$ dependence, in sharp contrast with the theoretically predicted exponential one for the direct overlap between two $s$ states.[36] Combining control experiments and Green's function-based calculations,[37,38] we find that the observed long-range features of Lieb lattices are an indirect interaction between Fe atoms mediated by the surface state.
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Fig. 1. (a) Typical topographic image of a $4 \times 4$ Lieb lattice ($r = 2.25$ nm) constructed with Fe adatoms on Ag(111). (b) Normalized ($dI/dV$)/($I/V$) spectra at corner sites (red) and edge sites (blue) with different $r$ (curves are shifted for clarity). The dashed line indicates the surface state onset energy of Ag(111). Orange, green, and gray arrows represent the high-, middle-, and low-energy peaks, respectively. (c) Typical $dI/dV$ maps obtained at the square area marked in (a) at the low, middle, and high energy peaks.
The experiments were performed in a low-temperature STM system with the base pressure of $2\times 10^{-11}$ mbar. The Ag(111) single crystals were prepared with cycles of Ar$^+$ sputtering (1.5 kV) and annealing ($\sim$580 ℃). High-purity Fe was deposited on the Ag(111) surfaces with a typical rate of 0.002 monolayer equivalent per minute through electron beam evaporation at $\sim $6 K. The measurements and atomic manipulations are performed with the W tips at $\sim $4.7 K. We constructed a series of Lieb lattices by laterally manipulating Fe atoms on the Ag(111) surface.[2,35] Figure 1(a) shows a typical topographic image of an assembled $4\times 4$ Lieb lattice ($r = 2.25$ nm). To investigate its electronic properties, we acquire the tunneling current $I$ and the differential conductance ($dI/dV$) as functions of the bias voltage $V$ on top of the Fe adatoms at the corner and edge sites [Fig. S5(a) in the Supplementary Information] and on an isolated single adatom using the lock-in technique with a modulation of the sample voltage of 4 mV at a frequency of 6.3 kHz [Fig. S4(b)]. The tip stabilizes at 50 mV and 1 nA before the $dI/dV$ measurements. To exclude the tip effect, we normalized the raw data by dividing the ($dI/dV$)/($I/V$) curves of the Fe adatoms in the lattices by the one obtained on top of the isolated Fe adatom with the same tip. The normalization also minimizes the influence of the tunneling matrices. As shown in Fig. 1(b), at the edge sites (blue curves), the normalized spectra show three pronounced sets of peaks: low-energy peaks (marked by gray arrows) located close to the surface state onset energy ($-$65 mV; gray dashed line),[39] middle energy peaks (green arrows), and high energy peaks (orange arrows). At the corner sites (red curves), they exhibit only two sets of peaks that coincide with the peaks near the onset energy of the surface state and the high energy peaks obtained at the edge sites. When the $r$ value increases from 1.75 to 4.8 nm, both sets of the middle and high energy peaks shift to lower energy, whereas the low-energy peak remains nearly unchanged. The middle energy peak is located approximately at the energy in the middle of the high and low-energy peaks. It is worth noting that there are other peaks at the energies above these three peaks. They may originate from the scattering of the surface state since Fe adatoms also serve as the scattering centers for the surface state, as confirmed through Green's function calculation in the Supplementary Information. Figure 1(c) shows the measured $dI/dV$ maps for $r = 2.25$ nm at three energies. It shows that the electrons are localized at edges sites at the energy of $E_{\rm m}$, whereas they distribute at both the corner and edge sites with energies of $E_{\rm l}$ and $E_{\rm h}$, respectively. The features in the middle of the Lieb lattice at $E_{\rm m}$ and the ring-shaped features at $E_{\rm h}$ in Fig. 1(c) are the interference patterns caused by the scattering of the surface state, as discussed in the Supplementary Information. Note that the $dI/dV$ maps were obtained with a constant height mode. Because of the chemical difference, the actual height between the tip and the surface might be different for the tip placed on top of Ag and Fe. Here, we mainly focus on the spectra on top of the Fe adatoms.
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Fig. 2. (a) Peak positions of the normalized differential conductance spectra as a function of $r$ and $1/r^{2}$ (inset). (b) The comparison of $(E_{\rm h} -E_{\rm m})/2$ and $(E_{\rm m} -E_{\rm l})/2$ as a function of $r$ and $1/r^{2}$ (inset). The curves outside (inside) the insets are the fittings with the inverse parabolic (linear) function.
To analyze the variation of the peak position with the atomic separation $r$, we plot $E_{\rm l}$, $E_{\rm m}$, and $E_{\rm h}$ as a function of $r$ in Fig. 2(a). Consequently, we found that both datasets can be fitted with $E=E_{0} +\frac{C}{r^{2}}$, where the value of $E_{0}$ is close to the onset energy of the surface state (Table S1). As shown in Fig. 2(b), we also obtained the $r$-dependent values of $(E_{\rm h} -E_{\rm m})/2$ and $(E_{\rm m} -E_{\rm l})/2$. Interestingly, we obtained that they are very similar within our experimental error margin. As the peak positions $E_{\rm l}$, $E_{\rm m}$, and $E_{\rm h}$ show the $\frac{1}{r^{2}}$-dependence, we fitted them with $\Delta E=\Delta E_{0} +\frac{C_{\Delta } }{r^{2}}$. As shown in Fig. 2(b), the fitted curve with fitting parameters $\Delta E_{0} =1.0\pm 2.2$ meV and $C_{\Delta } =273.9\pm 14.5$ meV$\cdot $nm$^{2}$ reproduces the experimental data well, indicating that half of the peak interval is $\frac{1}{r^{2}}$-dependent. For clarity, we also plot them as a function of $1/r^{2}$ and show their linear relationship in the insets of Fig. 2. To understand the observed electronic properties of the lattices, we first calculated the band structure and the local density of states (LDOS) using the tight-binding method by only considering the $s$ orbitals of the real atoms, implying that no substrate is considered. We set the onsite energy to be zero for the corner and edge sites and the nearest neighboring overlap energy to $t$. As shown in Fig. 3(a), the band structure features two Dirac bands and a flat band.[40–42] Correspondingly, the LDOS curve shows two peaks at the corner site and three peaks at the edge site, with the middle peak located at the center of the other two peaks. The calculated LDOS maps show that the electrons are mainly localized at edge sites at the middle energy, whereas the electrons are distributed at the corner and edge sites at low and high energies. We only consider the overlap between $s$ orbitals in the above tight-binding calculations. However, the Hamiltonian should also be valid when the nearest neighboring sites have relatively localized states, which overlap with effective overlap energy. For Fe adatom on Ag(111), the 4$s$ state has a strong coupling with the surface state, resulting in relatively localized states. It can be anticipated that the band structure formed by these localized states is similar to the one formed by the $s$ orbitals, except that the onsite and overlap energies are different. Therefore, the tight-binding calculated results essentially reproduce the features on normalized ($dI/dV$)/($I/V$) curves and the $dI/dV$ maps [Figs. 1(b) and 1(c)] observed in our experiments. Thus, the experimentally obtained $E_{\rm l}$, $E_{\rm m}$, and $E_{\rm h}$ are attributed to the characteristic peaks of the Lieb lattice. As shown in Fig. 3(a), the peak interval is 2$t$ (4$t$) at the edge (corner) site; i.e., the overlap energy $t$ is $(E_{\rm h} -E_{\rm m})/2$ or $(E_{\rm m} -E_{\rm l})/2$. Thus, the peak intervals in the experimental $dI/dV$ curves correspond to twice the overlap energy, and it is of $\frac{1}{r^{2}}$ dependence. Note that we only consider the nearest-neighbor hopping in the tight-binding calculation shown in Fig. 3, and the band structure is symmetric around the onsite energy. When the second nearest-neighbor hopping is included, the band structure is no longer symmetric. However, the observed $\frac{1}{r^{2}}$-dependent overlap energy in our experiments is in sharp contrast with the exponential dependence predicted in the literature.[36] The prediction was made for the direct overlap between the $s$ states of two hydrogen atoms in free space. The direct overlap energy can be calculated as $\sim $10 meV when their separation is 0.55 nm. In our experiments, the measured overlap energy is about 17 meV, even when $r$ is 4.8 nm. Therefore, the long-range overlap energy we observed on Ag(111) is not the direct overlap between Fe atoms. We attribute the difference between the theoretical prediction and our experiments to the atomic environment as the prediction was made for atoms in free space, while our experiments were conducted with adatoms placed on top of an Ag(111) surface. The latter enables a surface state around the Fermi energy. Moreover, we performed similar measurements on an Ag(100) substrate, which has no surface state near the Fermi energy. Interestingly, the spectra of the Fe Lieb lattices on Ag(100) do not show any apparent feature of the Lieb lattice (Fig. S10). Instead, the spectra obtained at the corner or edge sites are very similar to the spectrum of the isolated single adatom, even when $r$ approaches only $\sim $1 nm. The significantly different $r$-dependences for Fe adatoms on both substrates highlight the crucial role of the surface state on the observed long-range overlap on Ag(111) surface. According to previous studies, the spectrum of the isolated Fe adatom [Fig. S4(b)] shows that the 4$s$ state of Fe strongly overlaps with the surface state, resulting in relatively localized states.[43,44] It can be anticipated that the band structure formed by these localized states is similar to the one formed by the $s$ orbitals, except that the onsite and overlap energies are different. Thus, the electronic features of Lieb lattice are observed at a large atomic separation. This is not too surprising as in the exploration of the quantum mirage, the Kondo resonance peak of Co adatom located at one focus point of elliptical corral can be projected to the other focus point around 10 nm apart on Cu(111) through the surface state.[12] In a similar study, the inversion state of an Fe adatom can also be transferred to another location more than 10 nm apart in Ag(111) through the surface state.[2]
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Fig. 3. (a) Calculated band structure of a Lieb lattice and the corresponding LDOS at corner (red) and edge (blue) sites with the effective nearest-neighbor overlap energy $t$ using the tight-binding method. Note: The energy differences among $E_{\rm h}$, $E_{\rm m}$, and $E_{\rm l}$ are 2$t$ as marked. (b) LDOS maps at $E_{\rm h}$, $E_{\rm m}$, and $E_{\rm l}$, respectively. (c) Calculated peak positions of a $4\times 4$ Lieb lattice on Ag(111) using Green's function method as a function of $r$ and $1/r^{2}$ (inset). (d) The corresponding value of $(E_{\rm h} -E_{\rm m})/2$. The curves outside (inside) the insets in (c) and (d) are fittings with the inverse parabolic (linear) functions.
To further verify the essential role of the surface state, we performed Green's function-based calculations for Fe Lieb lattices built on Ag(111). The calculated LDOS curves exhibit close similarity with the experimental $dI/dV$ [Fig. S5(c)]. Note that we mainly consider the surface state in Green's function calculations, which are valid only above the onset of the surface state. The experimentally obtained peak $E_{\rm l}$ is very close to the onset energy, corresponding to small wavevector and large wavelength. Thus, $E_{\rm l}$ may be influenced by the onset of the surface state and it is not accurately determined. Therefore, we focus our discussion on $E_{\rm m}$ and $E_{\rm h}$. Figure 3(c) shows the plots of $E_{\rm m}$ and $E_{\rm h}$ versus $r$; they can be fitted well with $E=E_{0} +\frac{C}{r^{2}}$. The value of $(E_{\rm h} -E_{\rm m})/2$ also follows the $\frac{1}{r^{2}}$ dependence. The fitting parameter $C_{\Delta } =249.3\pm 8.1$ meV$\cdot $nm$^{2}$ is consistent with our experimental result of $273.9\pm 14.5$ meV$\cdot $nm$^{2}$. The insets in Figs. 3(c) and 3(d) show the linear dependences of the peak positions and $(E_{\rm h} -E_{\rm m})/2$ versus $1/r^{2}$, respectively. Note that in Green's function calculation, we consider the scattering of the surface state by the Fe adatom and the coupling of the Fe 4$s$ state and the surface state of Ag(111) with the empirical description of the inversion effect. For more accurate analysis, first-principle calculations may be needed. In summary, we have successfully constructed the Fe Lieb lattice on Ag(111) surface with real atoms and tuned its electronic properties by varying its atomic separation. It is found that the effective overlap energy between lattice sites exhibits a $\frac{1}{r^{2}}$ dependence. Combining the control experiments on Ag(100), tight-binding modeling, and Green's function calculations, we attribute the long-range overlap energy to the effective overlap of Fe adatom mediated by the surface state. It is realized that the hybridization between adatoms and surface state must be sufficiently strong to observe this long-range effect between lattice adatoms. Moreover, it is known that Co, Ag, Cu, and Mn adatoms also show strong hybridization on Ag(111), Cu(111), or Au(111)[37,45–49] besides Fe adatoms. Our experiments provide a convenient platform for designing and exploring artificial structures and devices with various materials combinations and exotic properties, such as flat and topological bands. We note that similar approaches are successfully used to realize atomic systems (with direct bonding),[50,51] flat-band system, molecule graphene, and quasi-crystal (with artificial atoms).[14,15,52] Acknowledgments. This work was supported by the National Key R&D Program of China (Grant Nos. 2017YFA0303202 and 2018YFA0306004), the National Natural Science Foundation of China (Grant Nos. 11974165, 92165103, 51971110, and 11734006), the China Postdoctoral Science Foundation (Grant No. 2019M651766), and the Natural Science Foundation of Jiangsu Province (Grant No. BK20190057). 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