Introduction

In the low-temperature limit, Kondo effect originates from the collective interaction between an atomic spin entity and surrounding conduction electrons of non-magnetic metal1,2,3,4,5. In the strong coupling regime, localized atomic spin is antiferromagnetically (AFM) coupled, i.e., spin-spin exchange interaction J < 0, with the itinerant spin bath from conduction-band electrons of host metal. Below the characteristic Kondo temperature (TK), such atomic spin is then effectively screened by the conduction electron spin cloud, resulting in the formation of a spin singlet ground state at the Fermi level (EF)6. Besides an increase of electrical resistivity, a pronounced electronic resonance at EF represents the distinctive hallmark of this highly-correlated many-body state, commonly referred to as the Kondo or Abrikosov–Suhl resonance7,8,9. In particular, zero-bias anomaly as a manifestation of Kondo resonance in the tunneling conductance (dI/dU) spectra can be accessed by utilizing scanning tunneling spectroscopy (STS)8,10,11,12,13,14,15, offering an ideal approach not only to resolve individual Kondo adsorbates at the atomic scale, but also to explore magnetic field and temperature dependence of Kondo effect with a high energy resolution.

Stanene, a two-dimensional (2D) honeycomb lattice composed of Sn atoms, i.e., tin (Sn) analogue of graphene, has received extensive attention because of its intriguing topological properties. For example, nontrivial band topology from in-plane s-p band inversion and the emergence of topological edge states have been investigated by Deng et al. in the epitaxial growth of ultraflat monolayer (ML) stanene16. Owing to a large atomic mass of Sn, a prominent spin-orbit-coupling (SOC) gap opening up to about 0.3 eV has been uncovered from angle-resolved photoemission spectroscopy (ARPES) studies16,17,18, contributing a key ingredient to fulfill quantum spin Hall effect (QSHE) and topological phase transition at room temperature (RT)19,20,21,22. Recently, introducing magnetism to 2D topological insulators arises as a spotlight issue in the pursuit of magnetic topological insulators (MTIs)23,24,25,26,27,28,29,30 with reduced dimensionality, which exhibit quantum anomalous Hall effect (QAHE) and extraordinary quantum electronic transport without a need of applying external magnetic field19,20,31. Despite several exceptional aspects supporting stanene for a promising candidate of 2D topological insulator, research efforts relevant to the development of low-dimensional magnetic topological materials by incorporating magnetic ingredients, e.g., considerable magnetic moment and effective magnetic anisotropy, to single-atomic-layer stanene are still lacking and have not been reported yet.

In this work, we have carried out systematic studies on the Kondo effect of magnetic Fe atom assembly on stanene/Cu(111) by using low-temperature STM/STS together with theoretical DFT and NRG calculations. By depositing Fe atoms onto the stanene monolayer at 80 K, thermally energetic Fe atoms are able to replace Sn atoms and spontaneously arrange into monomer, dimer and trimer structures. The zero-bias Kondo resonance in the dI/dU spectra as well as corresponding Zeeman splitting under external magnetic field have been resolved, where an effective spin Seff = 3/2 with an easy-plane magnetic anisotropy has been further characterized on the self-assembled Fe atomic dopants. Besides, Kondo temperature and Kondo peak amplitude continuously decreasing from Fe monomer to trimer are also conceivable from numerically simulated NRG results by taking substantial RKKY coupling between Sn-separated Fe atoms into account. After achieving the atomic-scale Fe doping and revealing the Kondo physics, one might have an opportunity to engineer topological band features and establish RKKY-induced magnetism in the magnetically-doped stanene monolayer.

Results and discussion

Growth of magnetic Fe atom Assembly/Stanene/Cu(111)

Figure 1a represents a topographic STM overview of as-grown Fe/stanene/Cu(111) sample prepared at 80 K, in which well-extended 2D stanene islands with an apparent height about 1.8 Å covering about two-thirds of Cu(111) surface can be observed. The enlargement from the white square frame in Fig. 1a has been shown in Fig. 1b, many small and bright protrusions resulting from deposited Fe atoms have been recognized on the surface of stanene/Cu(111). Figure 1c displays the atomically-resolved image of the white square frame from Fig. 1b, honeycomb-structured stanene has a (2 × 2) supercell, leading to a lattice constant of about 5.1 Å with respect to the (1 × 1) primitive unit cell of Cu(111)16,32. Interestingly, deposited Fe atoms have spontaneously arranged into three types of atomic structures, including monomer, dimer and trimer, as circled by green, yellow and blue colors, respectively, in Fig. 1c. Note that such self-assembled Fe atomic dopants on stanene/Cu(111) bear a close resemblance to the Co counterpart on stanene/Cu(111) as reported recently (see Supplementary Figs. 1–5 for details)32. A perspective view of schematic atomic model summarizing the Fe monomer, dimer and trimer on stanene/Cu(111) has been illustrated in Fig. 1d.

Fig. 1: Fabrication of Fe atom assembly on monolayer stanene/Cu(111).
figure 1

a STM topographic overview of a dilute amount of Fe atoms deposited onto stanene/Cu(111) held at 80 K. The surface coverage of stanene is about 0.38 ML and a step height about 1.8 Å can be extracted on Cu(111) surface. b Zoom-in area marked by a white square frame in (a). The deposited Fe atoms can be seen as small bright protrusions atop the stanene. c Atomically resolved STM image acquired from the white square frame in (b), where the self-assembled Fe monomer, dimer and trimer are marked by green, yellow, and blue circles, respectively. d A perspective view of schematic atomic model summarizing the Fe monomer, dimer and trimer structures on stanene/Cu(111). (scanning parameters are Vs = − 1 V and It = 400 pA for all topographic images).

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Monomer

Figure 2a is atomic-scale zoom-in of Fe monomer on stanene/Cu(111), where Fe atom appears as a bright protrusion in the center and surrounding Sn atoms in honeycomb lattice have a uniform atomic corrugation indicating the flatness of stanene monolayer. The Fe atom exhibits an apparent height about 12 pm in average, i.e., much lower than adsorbing single Fe atom directly on top of stanene, suggesting a possible substitution of one Sn atom by the highly energetic Fe atom from thermal e-beam evaporation. Figure 2b represents the resultant atomic structure obtained from self-consistent lattice relaxations in DFT, where Fe monomer is analogous to the surface doping of single Fe atom onto stanene32. Note that STM simulations have been further performed to crosscheck bias-dependent topographic images and line profiles with experimental observations (see Supplementary Fig. 2 for details).

Fig. 2: Field-dependent Kondo resonance and Kondo temperature measured from Fe monomer.
figure 2

a Atomically resolved STM topography of Fe monomer made from the substitution of one Sn atom on the honeycomb lattice of stanene/Cu(111). b Atomic Fe monomer structure fully relaxed from DFT calculations. c Magnetic field-dependent dI/dU spectra measured on the Fe monomer from 0 to 9 T fitted by the superposition of Frota functions. All spectra are vertically shifted for clear visibility. d Zeeman splittings of Kondo resonance fitted by effective spin Hamiltonian with magnetic anisotropy terms involved. e Temperature-dependent evolution of Kondo resonance from 0.3 to 10 K and corresponding Frota fits. f HWHM of Kondo resonance extracted from the Frota fits of (e) and TK = 7.70 K can be determined by using Eq. (2). Error bars are defined by the standard deviation of fitting analyses. (stabilization parameters: Ub = +10 mV, It = 1.0 nA for all dI/dU curves).

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A pronounced peak feature at EF that could be attributed to the Kondo effect of Fe monomer has been resolved in the zero-magnetic-field dI/dU spectrum of the Fig. 2c (top curve), which is absent from the dI/dU spectra acquired on stanene/Cu(111) and pristine Cu(111) substrate, respectively. The dI/dU spectra evolving with external magnetic fields applied normal to sample surface from 0 T to 9 T have been displayed in Fig. 2c, a clear splitting of the Kondo peak has been further revealed. Note that spin degeneracy of Kondo singlet state is lifted when an external magnetic field has been applied, resulting in the splitting of Kondo peak proportional to the strength of Zeeman energy8,10,11,12,13,14,15. It is also worth noting that zero-bias Kondo resonance was not observed even down to 0.3 K in the Co/stanene/Cu(111) due to a small magnetic moment of Co atom32. In addition to Cu(111) substrate, there are also energy bands around the Fermi level of single-atomic-layer stanene, which should contribute to the conduction electron bath that screens out the Fe atomic spin.

To advance our understandings on the magnetic-field-dependent splitting of Kondo peak, experimental dI/dU spectra have been fitted to the Frota function7,33,34. We denote that both Frota and Fano fitting results are in line with each other (see Supplementary Fig. 6 for details), although Fano function may give a higher Kondo temperature9. In Fig. 2c, superposition of three Frota functions has been used to simulate the experimental data, Frota-1 and Frota-2 curves (blue and red lines) are referred to the Kondo resonances below and above the EF, and Frota-3 curves (green dashed lines) are used to compensate the broad background. The consequent Zeeman energy (Δ(B)), i.e., the half of Kondo peak splitting, has been extracted as black dots in Fig. 2d8,10,11,12,13,14,15,35. A linear fitting using the relation Δ(B) = gμBB provides an unrealistically large value of g = 3.64 ± 0.10, which excludes the simple assumption of Seff = 1/2 case. To identify the spin state of Fe monomer, our experimental results have been further analyzed by effective spin Hamiltonian12,36,37,38,39:

$${\hat{{{{\mathcal{H}}}}}}_{{{{\rm{spin}}}}}=g{\mu }_{{{{\rm{B}}}}}{\hat{\overrightarrow{{{{\bf{S}}}}}}}_{{{{\rm{eff}}}}}\cdot \overrightarrow{{{{\bf{B}}}}}+D{\hat{S}}_{{{{{\rm{eff}}}}}_{z}}^{2}+E({\hat{S}}_{{{{{\rm{eff}}}}}_{x}}^{2}-{\hat{S}}_{{{{{\rm{eff}}}}}_{y}}^{2})$$
(1)

where g is the Landé factor, μB is the Bohr magneton, \({\hat{\overrightarrow{{{{\bf{S}}}}}}}_{{{{\rm{eff}}}}}\), \({\hat{S}}_{{{{{\rm{eff}}}}}_{x}}\), \({\hat{S}}_{{{{{\rm{eff}}}}}_{y}}\), and \({\hat{S}}_{{{{{\rm{eff}}}}}_{z}}\) are the total effective spin operator and its projections on x, y, and z directions, D and E are the longitudinal and transverse magnetic anisotropies (see Supplementary Note). Since a single Kondo peak at zero bias in zero magnetic field can only be established by the first-order spin excitation in a degenerate ground state, where energy levels are connected by Δm = ± 1, therefore neglecting the possibility of the integer Seff. Moreover, for a negative longitudinal anisotropy (D < 0), energy levels with the highest ∣m∣ of half-integer spins give rise to the ground state that no longer connects with Δm = ± 1, restricting our analyses to D > 0, i.e., an easy-plane magnetic anisotropy. As the red solid line shown in Fig. 2d, our experimental data points (black dots) are fairly described by the Eq. (1) after diagonalizing Hamiltonian matrix, selecting correct spin sector and solving energy ground state, yielding a total effective spin Seff = 3/2 with g = 1.98 ± 0.02, D = 3.13 ± 0.24 meV, and E = 0.17 ± 0.02 meV. Because of atomic-scale orbital hybridization, bonding reconfiguration and charge density redistribution occurring in the substitution of Sn atoms8,15,32, the Fe monomer has Seff = 3/2 smaller than Siso = 4/2 for the isolated Fe atom. Note that magnetic moment values ranged from 2.2 to 2.5 μB have been calculated for the Fe atom assembly/stanene/Cu(111) by DFT (see Supplementary Fig. 7 for details), supporting Seff = 3/2 as extracted from Zeeman-split Kondo resonance experimentally. We would also like to denote that not only a reasonable g value and an easy-plane anisotropy (D > 0) with z-axis parallel to surface have been obtained, but also a very small E ( ≈ 0.05D), i.e., almost magnetically isotropic along transverse direction, has been characterized. On the other hand, the opposite outcome of z-axis normal to surface (blue dashed line) produces essentially the same behavior with the isotropic case of Seff = 1/2 (D = E = 0) as the green solid line plotted in Fig. 2d37,38,39.

Apart from magnetic-field-dependent splitting, temperature-dependent evolution of Kondo resonance from 0.3 K to 10 K has been shown in Fig. 2e. A Kondo temperature (TK) of 7.70 ± 0.23 K has been determined by fitting the half-width at half-maximum (ΓHWHM) as a function of sample temperature (black dots in Fig. 2f) with the following expression14,15,37:

$${{{\Gamma }}}_{{{{\rm{HWHM}}}}}(T)=3.7\sqrt{{(\alpha {k}_{{{{\rm{B}}}}}T)}^{2}+{({k}_{{{{\rm{B}}}}}{T}_{{{{\rm{K}}}}})}^{2}}$$
(2)

where ΓHWHM are extracted from the Frota fits of Fig. 2e (black curves) by using ΓHWHM = 2.542 × ΓFrota, kB is the Boltzmann constant, T is the sample temperature, TK is Kondo temperature and α is a constant parameter. Note that the numerical prefactor of 3.7 is used to get the correct zero-temperature limit of ΓHWHM(0) = 3.7kBTK, as described in the Wilson’s definition of Kondo temperature40 and NRG calculations41.

Dimer

Unlike the Fe monomer, Fe dimer structure forms when two energetic Fe atoms jointly substitute two Sn atoms from stanene honeycomb lattice. As atomically resolved STM image shown in Fig. 3a, Fe dimer appears as a dumbbell-like protrusion in the center and exhibits an apparent height about 25 pm (see Supplementary Fig. 3 for details). The atomic structure model of Fe dimer fully relaxed from DFT has been shown in Fig. 3b, where neighboring Sn atoms are rearranged as a result of the local incorporation of heterogeneous Fe atoms. We would like to denote that several different structural models have been proposed for the Fe dimer, like previous studies on the Co dimer/stanene/Cu(111)32, but they are not consistent with either experimental observations or theoretical calculations.

Fig. 3: Field-dependent Kondo resonance and Kondo temperature measured from Fe dimer.
figure 3

a Atomically resolved STM topography of Fe dimer on the honeycomb-structured stanene/Cu(111). b Atomic Fe dimer structure fully relaxed from DFT calculations. c Magnetic field-dependent dI/dU spectra measured on the Fe dimer from 0 to 9 T fitted by the superposition of Frota functions. All spectra are vertically shifted for clear visibility. d Zeeman splittings of Kondo resonance fitted by effective spin Hamiltonian with magnetic anisotropy terms involved. e Temperature-dependent evolution of Kondo resonance from 0.3 to 10 K and corresponding Frota fits. f HWHM of Kondo resonance extracted from the Frota fits of (e) and TK = 7.10 K can be determined by using Eq. (2). Error bars are defined by the standard deviation of fitting analyses. (stabilization parameters: Ub = +10 mV, It = 1.0 nA for all dI/dU curves).

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From the dI/dU spectrum measured on Fe dimer/stanene/Cu(111), zero-bias Kondo peak has also been observed (topmost, 0 T curve in Fig. 3c). The Δ(B) obtained from the fittings of Fig. 3c have been arranged into Fig. 3d. The best fit of experimental Δ(B) (black dots) has been successfully formulated by effective spin Hamiltonian in Eq. (1), providing Seff = 3/2, g = 2.01 ± 0.02, D = 3.06 ± 0.15 meV, E = 0.23 ± 0.01 meV, and z-axis parallel to the surface as the red line plotted in Fig. 3d. Temperature-dependent Kondo resonance has been shown in Fig. 3(e), where the Frota fitting analyses (black curves) reproduce the broadening of zero-bias peak in the dI/dU spectra (grey squares). Figure 3f summarizes the ΓHWHM(T), and the TK value of 7.10 ± 0.28 K by fitting the Eq. (2) derived for the Fe dimer/stanene/Cu(111).

Trimer

Enlightened by the Fe dimer formation, thermally energetic Fe atoms would likely nucleate first on top of stanene, and then substitute the Sn atoms to develop the Fe trimer structure32. An upside-down triangle composed of three Fe atoms, i.e., bright dot-like protrusions, for the Fe trimer on stanene/Cu(111) has been atomically resolved in the Fig. 4a. Inferring from bias-dependent atomic resolution images as well as apparent height about 28 pm in average (see Supplementary Fig. 4 for details), the Fe trimer structure deduced from the DFT structural relaxations has been shown in Fig. 4b.

Fig. 4: Field-dependent Kondo resonance and Kondo temperature measured from Fe trimer.
figure 4

a Atomically resolved STM topography of Fe trimer on the honeycomb-structured stanene/Cu(111). b Atomic Fe trimer structure fully relaxed from DFT calculations. c Magnetic field-dependent dI/dU spectra measured on the Fe trimer from 0 to 9 T fitted by the superposition of Frota functions. All spectra are vertically shifted for clear visibility. d Zeeman splittings of Kondo resonance fitted by effective spin Hamiltonian with magnetic anisotropy terms involved. e Temperature-dependent evolution of Kondo resonance from 0.3 to 10 K and corresponding Frota fits. f HWHM of Kondo resonance extracted from the Frota fits of (e) and TK = 5.88 K can be determined by using Eq. (2). Error bars are defined by the standard deviation of fitting analyses. (stabilization parameters: Ub = +10 mV, It = 1.0 nA for all dI/dU curves).

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The zero-bias Kondo peak in the dI/dU spectrum of Fe trimer has also been resolved (topmost, 0 T curve in Fig. 4c). The magnetic-field-dependent Kondo peak splitting of Fe trimer is presented in Fig. 4c, where the comparative analyses of Δ(B) following the equivalent methodology for monomer and dimer have been arranged in Fig. 4d. The Δ(B) has been best fitted by the red line in Fig. 4d, when Seff = 3/2, g = 2.01 ± 0.05, D = 3.06 ± 0.48 meV, E = 0.22 ± 0.05 meV and surface parallel z-axis are extrapolated from the Eq. (1). Figure 4e summarizes the temperature-dependent dI/dU spectra (grey squares), where the thermal broadening of Kondo peak has been captured by using the Frota fits (black curves). The ΓHWHM(T) has been plotted in Fig. 4f, and the TK value about 5.88 ± 0.48 K has been obtained by fitting to the Eq. (2). Comparing with Fe monomer and dimer, both zero-field Kondo resonance amplitude and TK decrease all the way to the Fe trimer on stanene/Cu(111). Note that projected density of states (PDOS) of Fe atom assembly/stanene/Cu(111) have been calculated (see Supplementary Fig. 8 for details) and there is no clear trend in PDOS(EF) related to the decrease of TK from Fe monomer to trimer.

Evolution of Kondo temperature and Kondo resonance

Inspecting from the consistently decreased TK values and the successive attenuation of zero-field Kondo resonance summarized in Fig. 5a, one would expect the mutual interplay between Kondo effect and magnetic interaction, especially when individual Fe dopants are self-assembled in a close proximity. The magnitude of Fe-Fe coupling in terms of dipolar, exchange and RKKY interactions42,43,44 has been estimated (see Supplementary Note for details). Since dipole-dipole interaction decreases as the inverse cube of Fe-Fe interatomic distance and exchange interaction vanishes from a lack of direct 3d orbital hybridization between Sn-separated Fe atoms (see Supplementary Fig. 9 for details), the RKKY interaction turns out to be a relatively favorable mechanism.

Fig. 5: NRG analyses of RKKY-mediated Kondo resonance on Fe atom assembly.
figure 5

a Zero-field conductance spectra measured over the Fe impurities in monomer, dimer, and trimer structures. b The RKKY interaction between two magnetic impurities as a function of 2kFR, where, kF is the Fermi wave vector for stanene/Cu(111) and R is the distance between impurities showing an oscillating behavior between FM and AFM for different dimensionality. The grey shaded area shows the range covering R = 5.1 Å and kF from 0.05 to 0.14 Å−1 for the Fe atomic dopants on stanene/Cu(111). c The configuration of the three-impurity Kondo model mapped on a semi-infinite chain. d The intensity of spectral density of the zeroth impurity in the three-impurity Kondo model as a function of energy ω normalized by the bandwidth of the conduction band D (see Experimental and Theoretical Methods).

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Hence, we have modeled the host conduction electrons as free electron gas with Fermi wave vector kF and the Fe atoms as magnetic impurities with quantum wells separated by a distance R. The RKKY interaction in one, two, and three-dimensional systems can be formulated by the following Hamiltonian45:

$${H}_{{{{\rm{RKKY}}}}}=-\mathop{\sum}\limits_{{{{\bf{R}}}},{{{{\bf{R}}}}}^{{\prime} }}{J}_{d}^{{{{\rm{RKKY}}}}}(| {{{\bf{R}}}}-{{{{\bf{R}}}}}^{{\prime} }| )\overrightarrow{{{{\bf{S}}}}}({{{\bf{R}}}})\cdot \overrightarrow{{{{\bf{S}}}}}({{{{\bf{R}}}}}^{{\prime} }),$$
(3)
$${J}_{d}^{{{{\rm{RKKY}}}}}(R)=\pi {E}_{{{{\rm{F}}}}}{N}_{{{{\rm{F}}}}}^{2}{J}_{{{{\rm{K}}}}}^{2}{F}_{d}(2{k}_{{{{\rm{F}}}}}R)$$
(4)

where a subscript d is a spatial dimension, EF is a Fermi energy, NF is a density of states at the fermi energy and JK is a Kondo coupling constant. The functions Fd(2kFR) for d = 1, 2, 3 are given by

$${F}_{1}(2{k}_{{{{\rm{F}}}}}R)=\frac{\pi }{2}-\int\nolimits_{0}^{2{k}_{{{{\rm{F}}}}}R}{{{\rm{d}}}}y\frac{\sin y}{y},$$
(5a)
$${F}_{2}(2{k}_{{{{\rm{F}}}}}R)=-{J}_{0}({k}_{{{{\rm{F}}}}}R){N}_{0}({k}_{{{{\rm{F}}}}}R)-{J}_{1}({k}_{{{{\rm{F}}}}}R){N}_{1}({k}_{{{{\rm{F}}}}}R),$$
(5b)
$${F}_{3}(2{k}_{{{{\rm{F}}}}}R)=\left[-2{k}_{{{{\rm{F}}}}}R\cos (2{k}_{{{{\rm{F}}}}}R)+\sin (2{k}_{{{{\rm{F}}}}}R)\right]/(4{(2{k}_{{{{\rm{F}}}}}R)}^{4}).$$
(5c)

Jn(x) and Nn(x) in the F2(2kFR) are the Bessel functions of the first and second kinds. Using the distance between Fe atoms (R ≈ 5.1 Å) determined experimentally and the Fermi wave vector (kF ≈ 0.05 Å−1) derived from the DFT calculation (see Supplementary Fig. 10 for details), we are able to plot the \({J}_{d}^{{{{\rm{RKKY}}}}}/(\pi {E}_{{{{\rm{F}}}}}{N}_{{{{\rm{F}}}}}^{2}{J}_{{{{\rm{K}}}}}^{2})\) as a function of 2kFR for all considered dimensions and have found that the sign of JRKKY remains positive for one, two and three-dimensional systems as grey shaded region in Fig. 5b, highlighting the ferromagnetic (FM) RKKY coupling. Therefore, the RKKY interaction between the Sn-separated Fe atoms becomes substantial not only in the competition with Kondo physics, but also in the appearance of magnetic phase transition. Furthermore, we would like to denote that the theoretically deduced Kondo temperatures of dimer and trimer with respect to monomer are quantitatively comparable to experimental values, underlining the importance of RKKY-mediated Kondo effect46,47,48 (see Supplementary Note).

The amplitude of Kondo peak declining from Fe monomer to trimer has also been investigated by considering the RKKY interaction in numerical renormalization group (NRG) method. The Kondo impurity model1,40,49 in principle involves three Hamiltonian terms, i.e., H = Himp + Hcon + Hhyb, where Himp is the Kondo impurity, Hcon is the conduction electrons, and Hhyb describes the interaction between the Kondo impurity and the conduction electrons. Following the standard NRG formalism40,50,51,52 and consider magnetic impurities that couple to a semi-infinite chain as the schematic drawing shown in Fig. 5c, the Hamiltonian becomes:

$$\begin{array}{ll}H={H}_{{{{\rm{imp}}}}}+\mathop{\sum }\limits_{n=0}^{\infty }\mathop{\sum}\limits_{\sigma =\uparrow ,\downarrow }({\epsilon }_{n}{c}_{n\sigma }^{{\dagger} }{c}_{n\sigma }+{t}_{n}({c}_{n\sigma }^{{\dagger} }{c}_{n+1\sigma }+{c}_{n+1\sigma }^{{\dagger} }{c}_{n\sigma }))\\\qquad+\mathop{\sum}\limits_{\alpha ,k,\sigma }{V}_{\alpha k}({f}_{\alpha \sigma }^{{\dagger} }{c}_{k\sigma }+{{{\rm{h}}.{\rm{c}}.}}),\end{array}$$
(6)

with

$${H}_{{{{\rm{imp}}}}}=\mathop{\sum}\limits_{\sigma =\uparrow ,\downarrow }\mathop{\sum}\limits_{\alpha =0,1,2}{\epsilon }_{{f}_{\alpha }}{f}_{\alpha \sigma }^{{\dagger} }{f}_{\alpha \sigma }+U{f}_{\alpha \uparrow }^{{\dagger} }{f}_{\alpha \uparrow }{f}_{\alpha \downarrow }^{{\dagger} }{f}_{\alpha \downarrow }+\frac{1}{2}\mathop{\sum}\limits_{\alpha ,\beta =0,1,2}{J}_{\alpha \beta }({\overrightarrow{{{{\bf{S}}}}}}_{\alpha }\cdot {\overrightarrow{{{{\bf{S}}}}}}_{\beta }).$$
(7)

Here \({c}_{n\sigma }^{({\dagger} )}\) is the fermionic operator of conduction electrons with energy ϵn, while \({f}_{\alpha \sigma }^{({\dagger} )}\) is the fermionic operator of fασ electrons with onsite energy ϵf at the α’s impurity. U is the Hubbard interaction of the Kondo impurities, Vαk is the hybridization strength between the fασ electrons of the α’s impurity and the conduction electrons, the exchange couplings between impurities are Jαβ = Jβα, and \({[{\overrightarrow{{{{\bf{S}}}}}}_{\alpha }]}_{ij}=\frac{1}{2}{f}_{\alpha i}^{{\dagger} }{[\overrightarrow{\sigma }]}_{ij}{f}_{\alpha j}^{{\dagger} }\). The hybridizations Vαk can induce the RKKY interaction between impurities on top of exchange couplings Jαβ, and non-zero hybridization V0k as well as the Jαβ are considered. For simplicity, we assume the constant DOS of conduction electrons within the interval [ − 1, 1], the constant hybridization V, ϵn = 0, and

$${t}_{n}=\frac{(1+{{{\Lambda }}}^{-1})(1-{{{\Lambda }}}^{-n-1})}{2\sqrt{1-{{{\Lambda }}}^{-2n-1}}\sqrt{1-{{{\Lambda }}}^{-2n-3}}}{{{\Lambda }}}^{-n/2}.$$
(8)

The crucial benefit of adopting the semi-infinite chain scheme is that Hamiltonian can be solved numerically via the iteration calculation. By using this NRG method, the Kondo resonance peak can therefore be computed from the impurity spectral function, which is defined for T = 0 K as

$$\begin{array}{ll}A(\omega )=\frac{1}{{Z}_{0}}\mathop{\sum}\limits_{\sigma ,\alpha }| \langle \alpha \sigma | {f_{\sigma }^{{\dagger} }}| 0\sigma \rangle {| }^{2}\delta (\omega +({E}_{\alpha }-{E}_{0}))\\\qquad\qquad+\,| \langle 0\sigma | {f}_{\sigma }| \alpha \sigma \rangle {| }^{2}\delta (\omega -({E}_{\alpha }-{E}_{0})),\end{array}$$
(9)

where Z0 is the partition function at T = 0 K, and \(\left\vert \alpha \sigma \right\rangle\) is the corresponding state with energy Eα in the Kondo impurity model mapped on the semi-infinite chain. As the spectra plotted in Fig. 5d, the intensity of Kondo resonance successively attenuates from Fe monomer to trimer, which not only reflects the essential role of RKKY interaction, but also illustrates the consistency with the experimental results in Fig. 5a. We would like to denote that our results are in agreement with the physical picture of Doniach theory, where the Kondo singlet formation can be suppressed by the RKKY interaction in multi-impurity Kondo systems48,53,54. Despite the Kondo effect from RKKY-coupled magnetic atoms already found in distinct systems, such as Co dimers/Cu(100)55, Co atomic chains/Ag(111)56, Co dimers/Cu2N/Cu(100)57, Mn dimers/MoS2/Au(111)58 and FePc molecules/Au(111)59 etc. it remains unprecedented on the magnetic Fe atom assembly doped in the monolayer stanene/Cu(111), where the atomic spin state, magnetic anisotropy, evolution of Kondo temperature and Kondo resonance have been analyzed in detail. Additionally, for an interesting comparison, the topological band structures of stanene monolayer with/without magnetic Fe dopants have been provided in the Fig. 11 of Supplementary, where the spin-split band features might trigger feasible and associated experiments, for examples, photoemission spectroscopy, quantum transport and magnetometry measurements etc.

In summary, we have combined experiment and theory to systematically study the Kondo effect from magnetic Fe atoms self-assembled on stanene/Cu(111). According to atomic resolution STM images and self-consistent DFT structural relaxations, Fe monomer, dimer and trimer from substituting Sn atoms have been successfully synthesized on the monolayer stanene/Cu(111) at 80 K. By employing tunneling spectroscopy with high spatial and energy resolution, not only zero-bias Kondo resonance, but also magnetic-field-dependent Zeeman splitting has been resolved, where an effective spin Seff = 3/2 with D > 0 and E ≈ 0 for an easy-plane magnetic anisotropy has been deduced from the effective spin Hamiltonian. In addition, a continuous reduction of Kondo temperature, i.e., from \({T}_{{{{\rm{K}}}}}^{{{{\rm{monomer}}}}}\) = 7.70 ± 0.23 K to \({T}_{{{{\rm{K}}}}}^{{{{\rm{trimer}}}}}\) = 5.88 ± 0.48 K, has been revealed from the temperature-dependent evolution of Kondo resonance, which can be explained by the indirect RKKY interaction between Sn-separated Fe atoms. Furthermore, the successive attenuation of Kondo peak amplitude from Fe monomer to trimer has been numerically simulated by using the NRG method with dominant RKKY coupling in the Kondo impurity model. From the atomic-scale magnetic doping realized by unraveling the RKKY-mediated Kondo physics, our results open a pathway toward tuning nontrivial properties of topological bands and stabilizing RKKY-coupled magnetic moments for emergent magnetic ordering in the atomically Fe-doped stanene with only one-atomic-layer thickness.

Methods

Sample preparation

The experiment was performed in the ultra-high vacuum (UHV) environment of p ≤ 2.0 × 10−10 mbar. Clean Cu(111) surface was prepared by several cycles of Ar+ ion sputtering with an ion energy of 0.5 keV at room temperature, and followed by the thermal annealing to 1000 K afterward. To grow the flat stanene on Cu(111), the substrate was first cooled down to 80 K, and then high-purity granular Sn (99.999%, Goodfellow) was evaporated from a pyrolytic boron nitride (PBN) crucible heated in an e-beam evaporator. Subsequently, Fe atoms were deposited onto the stanene by heating a high-purity Fe-rod (99.999%, Goodfellow) in an e-beam heater while keeping the substrate at the same temperature of 80 K. Afterward, the sample was in-situ transferred to STM immediately for the measurements.

STM/STS measurement

A custom-designed low-temperature STM (Unisoku Co. Ltd.) with the base temperature of 300 mK equipped with an out-of-plane superconducting magnet of 9 T was employed to investigate the sample. All STM images were scanned in the constant-current mode with bias voltage U applied to the sample. For scanning tunneling spectroscopy (STS) measurements, a small bias voltage modulation was added to U (frequency ν = 2671 Hz), such that tunneling differential conductance dI/dU spectra can be acquired by detecting the first harmonic signal by means of a lock-in amplifier.

Density function theory (DFT)

First-principles calculations based on density functional theory (DFT) were performed using the Vienna Ab Initio Simulation Package (VASP)60,61,62. The generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE)63 form was used for the exchange-correlation potentials with the projector augmented wave (PAW) pseudopotential64. The crystal structure models of Fe monomer-, dimer- and trimer-substituted stanene on Cu(111) were referred from the previous studies16,32 with the Co atoms replaced by Fe. The lattice structures were geometrically optimized until the total energy and residual atomic forces were converged to 10−4 eV and − 0.005 eV ⋅ Å−1, respectively. A vacuum layer with thickness of 25 Å well-separating slabs and plane wave basis with cutoff-energy of 400 eV were chosen in the slab model calculations. The spin-orbit coupling (SOC) was included in the self-consistent-field calculations for the relaxed Fe/Sn/Cu(111) lattice models using 6 × 6 × 1 Monkhorst-Pack k-grid mesh. As for bare stanene on Cu(111), higher density k-grid mesh of 24 × 24 × 1 was used for the smaller unit cell. The hoping constants were calculated using the WANNIER90 code65 based on the DFT results. The 2D Fermi surface was calculated using WannierTools66 software package based on Wannier Hamiltonian.

Numerical renormalization group calculation

The NRG method requires three steps. The first step is to discretize the density of states (DOS) of conduction electrons using a parameter Λ, which defines the energy intervals [Λ−n, Λn+1] of conduction electrons coupled to the Kondo impurity. The second step is to approximate the conduction electrons in each energy interval by a single state. The last step is to map the above systems to a semi-infinite chain, in which the Kondo impurities couple to the first conduction electron by hybridization strength Vα=0,1,2, and the rest of conduction electrons can be described by the tight-binding model with onsite energy ϵn and hopping parameter tn. We have obtained \(\left({J}_{01},\,{J}_{02},\,{{\Lambda }}\right)=(0,\,0,\,3)\) for the monomer case, \(\left({J}_{01},\,{J}_{02},\,{{\Lambda }}\right)=(-0.5* {10}^{-3},\,0,\,2.4)\) and \(\left({J}_{01},\,{J}_{02},\,{{\Lambda }}\right)=(-0.25* {10}^{-3},\,-0.25* {10}^{-3},\,2.09)\) for the dimer and trimer cases, respectively. And the rest parameter U = 10−3, \({{\epsilon}_{f}}=-{0.5}\,*\,{10}^{-3}\), V0 = 0.004, and V1 = V2 = J12 = 0 for the spectra plotted in Fig. 5d.