Atomic-scale magnetic doping of monolayer stanene by revealing kondo effect from self-assembled fe spin entities

ABSTRACT Atomic-scale spin entity in a two-dimensional topological insulator lays the foundation to manufacture magnetic topological materials with single atomic thickness. Here, we have

successfully fabricated Fe monomer, dimer and trimer doped in the monolayer stanene/Cu(111) through a low-temperature growth and systematically investigated Kondo effect by combining

scanning tunneling microscopy/spectroscopy (STM/STS) with density functional theory (DFT) and numerical renormalization group (NRG) method. Given high spatial and energy resolution,

tunneling conductance (d_I_/d_U_) spectra have resolved zero-bias Kondo resonance and resultant magnetic-field-dependent Zeeman splitting, yielding an effective spin _S_eff = 3/2 with an

easy-plane magnetic anisotropy on the self-assembled Fe atomic dopants. Reduced Kondo temperature along with attenuated Kondo intensity from Fe monomer to trimer have been further identified

as a manifestation of Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between Sn-separated Fe atoms. Such magnetic Fe atom assembly in turn constitutes important cornerstones for tailoring

topological band structures and developing magnetic phase transition in the single-atom-layer stanene. SIMILAR CONTENT BEING VIEWED BY OTHERS ANTIFERROMAGNETISM-DRIVEN TWO-DIMENSIONAL

TOPOLOGICAL NODAL-POINT SUPERCONDUCTIVITY Article Open access 04 February 2023 UNRAVELING THE ELECTRONIC STRUCTURE AND MAGNETIC TRANSITION EVOLUTION ACROSS MONOLAYER, BILAYER, AND MULTILAYER

FERROMAGNETIC FE3GETE2 Article Open access 30 September 2024 CHARGE-DENSITY WAVE MEDIATED QUASI-ONE-DIMENSIONAL KONDO LATTICE IN STRIPE-PHASE MONOLAYER 1T-NBSE2 Article Open access 03

February 2024 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 (_T_K), 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 (_E_F)6. Besides an increase of electrical resistivity, a pronounced electronic resonance at _E_F

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 (d_I_/d_U_) 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 d_I_/d_U_ spectra as well as corresponding Zeeman splitting under external magnetic field have been resolved, where

an effective spin _S_eff = 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. 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). A pronounced peak feature at _E_F that could be attributed to the Kondo effect of Fe monomer has been resolved in the zero-magnetic-field d_I_/d_U_ spectrum of the Fig.

2c (top curve), which is absent from the d_I_/d_U_ spectra acquired on stanene/Cu(111) and pristine Cu(111) substrate, respectively. The d_I_/d_U_ 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 d_I_/d_U_ 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 _E_F, 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_) = _g__μ_B_B_ provides an unrealistically large value of _g_ = 3.64 ± 0.10, which excludes the simple assumption of _S_eff

 = 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, _μ_B 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 Δ_m_ = ± 1, therefore neglecting the possibility of the integer _S_eff. 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 Δ_m_ = ± 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 _S_eff = 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 _S_eff = 3/2 smaller than _S_iso = 4/2

for the isolated Fe atom. Note that magnetic moment values ranged from 2.2 to 2.5 _μ_B have been calculated for the Fe atom assembly/stanene/Cu(111) by DFT (see Supplementary Fig. 7 for

details), supporting _S_eff = 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_ ( ≈ 0.05_D_), 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 _S_eff = 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 (_T_K) 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, _k_B is the Boltzmann constant, _T_ is the sample temperature, _T_K 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) = 3.7_k_B_T_K, 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. From the d_I_/d_U_ 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 _S_eff = 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 d_I_/d_U_ spectra (grey squares). Figure 3f summarizes the ΓHWHM(_T_), and the _T_K 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. The zero-bias Kondo peak in the d_I_/d_U_ 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 _S_eff = 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 d_I_/d_U_ 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 _T_K 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 _T_K 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(_E_F) related to the decrease of _T_K from Fe monomer to trimer.

EVOLUTION OF KONDO TEMPERATURE AND KONDO RESONANCE Inspecting from the consistently decreased _T_K 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 3_d_ orbital hybridization between Sn-separated Fe atoms (see Supplementary Fig. 9 for details), the RKKY

interaction turns out to be a relatively favorable mechanism. Hence, we have modeled the host conduction electrons as free electron gas with Fermi wave vector _k_F 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, _E_F is a Fermi energy, _N_F is a density of

states at the fermi energy and _J_K is a Kondo coupling constant. The functions _F__d_(2_k_F_R_) 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) _J__n_(_x_) and _N__n_(_x_)

in the _F_2(2_k_F_R_) 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 (_k_F ≈ 

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 2_k_F_R_ for all considered dimensions and have found that the sign of _J_RKKY 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_ = _H_imp + _H_con + _H_hyb, where _H_imp is the Kondo impurity, _H_con is the

conduction electrons, and _H_hyb 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 _ϵ__n_, while \({f}_{\alpha \sigma }^{({\dagger} )}\) is the fermionic operator of _f__α__σ_

electrons with onsite energy _ϵ__f_ at the _α_’s impurity. _U_ is the Hubbard interaction of the Kondo impurities, _V__α__k_ is the hybridization strength between the _f__α__σ_ electrons of

the _α_’s impurity and the conduction electrons, the exchange couplings between impurities are _J__α__β_ = _J__β__α_, and \({[{\overrightarrow{{{{\bf{S}}}}}}_{\alpha

}]}_{ij}=\frac{1}{2}{f}_{\alpha i}^{{\dagger} }{[\overrightarrow{\sigma }]}_{ij}{f}_{\alpha j}^{{\dagger} }\). The hybridizations _V__α__k_ can induce the RKKY interaction between impurities

on top of exchange couplings _J__α__β_, and non-zero hybridization _V_0_k_ as well as the _J__α__β_ are considered. For simplicity, we assume the constant DOS of conduction electrons within

the interval [ − 1, 1], the constant hybridization _V_, _ϵ__n_ = 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 _Z_0 is the partition function

at _T_ = 0 K, and \(\left\vert \alpha \sigma \right\rangle\) is the corresponding state with energy _E__α_ 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 _S_eff = 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 d_I_/d_U_ 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_, Λ_n_+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 _V__α_=0,1,2, and the rest of conduction electrons can be described by the tight-binding model with onsite energy _ϵ__n_ and hopping

parameter _t__n_. 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, \({{\epsilon}_{f}}=-{0.5}\,*\,{10}^{-3}\), _V_0 = 0.004, and _V_1 = _V_2 = _J_12 = 0 for the spectra plotted in Fig. 5d. DATA AVAILABILITY The datasets generated during

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224, 405–416 (2018). Article  ADS  Google Scholar  Download references ACKNOWLEDGEMENTS P.J.H. acknowledges support from C.L.H. for helium liquefier system operation in the instrumentation

center of National Tsing Hua University under Grants No. MOST-110-2731-M-007-396-001 and MOST-111-2731-M-007-001, National Science and Technology Council of Taiwan under Grant Nos.

NSTC-112-2636-M-007-006 and NSTC-112-2112-M-007-037, Ministry of Science and Technology of Taiwan under Grants No. MOST-111-2636-M-007-007 and MOST-110-2636-M-007-006, and center for quantum

technology from the featured areas research center program within the framework of the higher education sprout project by the Ministry of Education (MOE) in Taiwan. P.-Y.C. acknowledges

support from National Science and Technology Council of Taiwan under Grant No. NSTC-112-2636-M-007-007. H.-T.J. acknowledges support from National Science and Technology Council of Taiwan

under Grant No. NSTC 112-2112-M-007 -034 -MY3, and also from the NCTS, NCHC, CINC-NTU and AS-iMATE-113-12 in Taiwan. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Department of Physics,

National Tsing Hua University, Hsinchu, 300044, Taiwan Nitin Kumar, Ye-Shun Lan, Iksu Jang, Yen-Hui Lin, Chia-Ju Chen, Tzu-Hsuan Lin, Horng-Tay Jeng, Po-Yao Chang & Pin-Jui Hsu *

Institut für Theorie der Kondensierten Materie, Karlsruher Institut für Technologie, Karlsruhe, 76049, Germany Iksu Jang * Center for Quantum Technology, National Tsing Hua University,

Hsinchu, 300044, Taiwan Horng-Tay Jeng & Pin-Jui Hsu * Physics Division, National Center for Theoretical Sciences, Taipei, 10617, Taiwan Horng-Tay Jeng * Institute of Physics, Academia

Sinica, Taipei, 11529, Taiwan Horng-Tay Jeng Authors * Nitin Kumar View author publications You can also search for this author inPubMed Google Scholar * Ye-Shun Lan View author publications

You can also search for this author inPubMed Google Scholar * Iksu Jang View author publications You can also search for this author inPubMed Google Scholar * Yen-Hui Lin View author

publications You can also search for this author inPubMed Google Scholar * Chia-Ju Chen View author publications You can also search for this author inPubMed Google Scholar * Tzu-Hsuan Lin

View author publications You can also search for this author inPubMed Google Scholar * Horng-Tay Jeng View author publications You can also search for this author inPubMed Google Scholar *

Po-Yao Chang View author publications You can also search for this author inPubMed Google Scholar * Pin-Jui Hsu View author publications You can also search for this author inPubMed Google

Scholar CONTRIBUTIONS N.K., Y.H.L. and I.J. contributed equally to this work. N.K., Y.H.L., C.J.C., T.H.L. and P.J.H. carried out the STM/STS experiments and analyzed the data. Y.S.L. and

H.T.J. performed the DFT calculations. I. J. and P.Y.C. performed the NRG simulations. H.T.J., P.Y.C. and P.J.H. coordinated and supervised the project. All authors discussed the results and

contributed to the paper. CORRESPONDING AUTHORS Correspondence to Horng-Tay Jeng, Po-Yao Chang or Pin-Jui Hsu. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing

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CITE THIS ARTICLE Kumar, N., Lan, YS., Jang, I. _et al._ Atomic-scale magnetic doping of monolayer stanene by revealing Kondo effect from self-assembled Fe spin entities. _npj Quantum

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