This is the accepted version of the article: Choi, Deung-Jang et al. “Structural and magnetic properties of FeMn• chains (x=1-6) supported on Cu₂ N/Cu (100)” in Physical Review B, Vol. 94, Issue 8 (August 2016), art. 85406. Available at: 10.1103/PhysRevB.94.085406. This article version is “all rights reserved” http://www.europeana.eu/rights/rr-f/ Structural and magnetic properties of FeMnx (x =1...6) chains supported on Cu2 N / Cu (100) Deung-Jang Choi,1 Roberto Robles,2 Jean-Pierre Gauyacq,3 Markus Ternes,4 Sebastian Loth,5, 4 and Nicolás Lorente6, 7 1 CIC nanoGUNE, Tolosa Hiribidea 78, 20018 Donostia-San Sebastian, Spain 2 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science and Technology, Campus UAB, Bellaterra, 08193 Barcelona, Spain 3 Institut des Sciences Moléculaires d’Orsay, ISMO, CNRS, Université Paris-Sud, Université Paris-Saclay, Bâtiment 351, 91405 Orsay Cedex, France 4 Max Planck Institute for Solid State Research, 70569 Stuttgart, Germany 5 Max Planck Institute for the Structure and Dynamics of Matter, 22761 Hamburg, Germany 6 Centro de Fisica de Materiales, CFM/MPC (CSIC-UPV/EHU), Paseo Manuel de Lardizabal 5, 20018 Donostia-San Sebastian, Spain 7 Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain (Dated: July 11, 2016) Heterogeneous atomic magnetic chains are built by atom manipulation on a Cu2 N/Cu (100) sub- strate. Their magnetic properties are studied and rationalized by a combined scanning tunneling microscopy (STM) and density functional theory (DFT) work completed by model Hamiltonian studies. The chains are built using Fe and Mn atoms ontop of the Cu atoms along the N rows of the Cu2 N surface. Here, we present results for FeMnx (x=1...6) chains emphasizing the evolution of the geometrical, electronic, and magnetic properties with chain size. By fitting our results to a Heisen- berg Hamiltonian we have studied the exchange-coupling matrix elements J for different chains. For the shorter chains, x ≤ 2, we have included spin-orbit effects in the DFT calculations, extracting the magnetic anisotropy energy. Our results are also fitted to a simple anisotropic spin Hamiltonian and we have extracted values for the longitudinal-anisotropy D and transversal-anisotropy E con- stants. These parameters together with the values for J allow us to compute the magnetic excitation energies of the system and to compare them with the experimental data. PACS numbers: 68.37.Ef, 71.15.Nc, 73.23.-b, 72.25.-b I. INTRODUCTION atomic spins maintain their nominal values on the sur- face (SF e = 2, SM n =5/2 and SCo = 3/2), showing the interest of using Cu2 N to preserve much of the magnetic Bottom-up approaches give access to systems of very identity of transition-metal (TM) atoms. Studies of Fe reduced dimensionality with unique physical proper- chains17 and of Mn12 reveals that close-packed TM chains ties. Among these systems, chains of a single-atom on Cu2 N/Cu (100) couple antiferromagnetically due to a cross section are of great interest.1 When the chains are N-mediated superexchange mechanism, largely explain- formed by magnetic atoms, spin-spin correlations can ing experimental findings.5,22 come into place leading to new phenomena and applica- In the present work, we report on a different type of tions.2 Building structures from their atomic constituents magnetic atomic chain. These chains are heterogeneous, can be achieved by the atomic-manipulation capabilities including two types of magnetic atoms, Fe and Mn, on of the scanning tunneling microscope (STM). Magnetic a Cu2 N/Cu (100) substrate. An initial account of the atoms have been positioned one by one at different dis- results has been given in a separate publication.23 The tances and with different arrangements on a variety of experimental study is based on STM and IETS results substrates.3–7 Spin-polarized STM8 or inelastic electron of FeMnx (x = 1, · · · , 6) and confirms that, as in the tunneling spectra (IETS)9,10 have granted us a detailed previous cases, the magnetic ordering along the chain is vision of the magnetic mechanisms at play on the atomic antiferromagnetic. The different anisotropy of Fe and scale. Mn on Cu2 N/Cu (100) leads to two possible orientations Recent developments in density functional theory of the magnetic moments (along the chain and out-of- (DFT) have also permitted us to attain a deep un- plane). Contrary to the case of homogeneous chains, this derstanding of the phenomena revealed by the above may indicate a non-collinear arrangement. When x is experiments. Due to their experimental interest, Mn an odd number, a simple-minded evaluation of the total chains are amongst the most studied ones.11–16 Fe spin yields 1/2, which is compatible with the appear- chains are also well known both experimentally and ance of a Kondo feature at zero bias.23 However, when x theoretically.4,5,15,17–21 . The work by Lin and Jones13,22 is even, the expected spin is 2 and, correspondingly, no on Fe, Co and Mn atoms on Cu2 N/Cu (100) reveals that Kondo peak is observed. In the present work, we per- 2 form extensive calculations and compare them with the experiments. The comparison permits us to conclude on the spin arrangement (it is collinear along the chain) and on the behavior of the exchange coupling between the two different atomic species of the chain in good agree- ment with the experimental observations. The antiferro- magnetic coupling is confirmed and traced back to the N-mediated superexchange mechanism as in the homo- geneous chains. Finally, the obtention of model Hamilto- nians to study the magnetic structure of these chains is discussed within the framework of DFT calculations.18,24 II. EXPERIMENTAL METHOD Experiments were performed in an ultrahigh vacuum low temperature STM at a base temperature of 0.5 K as FIG. 1. Constant current STM image of a FeMn3 chain as- sembled on a Cu2 N layer and atomic scheme (on the right of has been partially reported in Ref. [23]. Differential con- the figure) that can be inferred from the position of the upper- ductance was used as a local spectroscopic tool that gives right Fe atom and manipulation with the STM tip. The STM information via inelastic electron tunneling (IETS).9,10 image was scanned at 10 mV and 100 pA. The scanned area The differential conductance was directly measured us- is 6 × 6 nm2 . The transition metal atoms were always placed ing lock-in detection with 72-µV rms modulation at 691 ontop of a Cu atom between two nitrogen atoms. The topog- Hz to the sample bias voltage V. raphy was processed with WSXM25 . The Cu (100) surface was cleaned by Ar sputtering and then annealed up to 850 K. After having big terraces of Cu(100) crystal, a monolayer of Cu2 N was formed the bulk lattice parameter such that the length along the as a decoupling layer by nitrogen sputtering and post- [010] direction increases as (x+3), where x is the number annealing at 600 K. Single Fe and Mn atoms were de- of Mn atoms. In this way, we keep the distance between posited onto the pre-cooled surface. Pt-Ir tips were pre- chain images constant for all sizes, being of 3 lattice con- pared by sputter-anneal cycles and coated with copper stants in the unrelaxed configuration. The unit cell for in vacuo by soft indentations into the Cu bridges. The FeMn2 is shown in Fig. 2 as an example. The bottom Cu tip status was monitored through STM images and con- layer was kept fixed and the remaining atoms were al- trolled to manipulate the atoms. All atomic chains were lowed to relax until forces were smaller than 0.01 eV/Å. built using vertical atomic manipulation. After identi- The k-point sample was varied accordingly to the unit fying a given adatom by its spectroscopic features, we cell, and tests were performed to assure its convergence. picked it up by voltage pulsing, dropped it off on a nitro- The charge and magnetic moments have been calculated gen site and hop it to a copper site. We build the chain using the Bader analysis.30 atom by atom in a close-packed configuration to ensure A critical aspect of the calculations is the use of a static an AF coupled spin chain. By doing so, all different kinds Coulomb charging energy U . Lin and Jones13 perform a of FeMnx (x=1... 10) chains were constructed. constrained DFT calculation to evaluate U . For Mn they find U = 4.9 eV when Mn is sitting ontop a surface Cu atom, while it is reduced to U = 3.9 eV ontop a N site. III. THEORETICAL METHOD This is in agreement with their previous result22 , where they found U = 5.0 eV for Mn ontop of Cu, and U = 2.0 Ab initio calculations are performed within the eV for Fe. To the best of our knowledge, these are the density-functional theory (DFT) framework as imple- only actual computations of U for Mn on the Cu2 N/Cu mented in the VASP code.26 We expand the wave func- (100) substrate. Rudenko et al.12 take an effective value tions using a plane-wave basis set with a cutoff energy of U − J=5 eV where J is the intra-atomic exchange cou- 300 eV. Core electrons are treated within the projector pling. Nicklas et al.17 do not take any value of U for Fe. augmented wave method.27,28 The PBE form of the gen- As Lin and Jones13 show, the actual value of U greatly af- eralized gradient approximation is used for the exchange fects the computed exchange couplings, indeed they find and correlation functional.29 a difference of a factor of 2 between their calculations The system consists of a Cu(100) surface covered by a with U = 4.9 eV and without U , finding DFT+U results Cu2 N layer. To model the surface we use a slab geome- in better agreement with the experiment. try with four Cu layers plus the Cu2 N layer. Following In this work, we have used the GGA+U method of Du- the above experimental procedure, Fe and Mn atoms are darev31 with a Ueff = U − J of 1 eV for Fe and 4 eV for deposited on top Cu atoms, forming a chain in the [010] Mn. The chosen values correspond to roughly subtract- direction. We use a unit cell [3 × (x+3)] in units of ing J ≈ 1 to U = 4.9 eV (U = 2 eV) as computed by Lin 3 and Jones13 for Mn (Fe) ontop a Cu atom. As we will see, the values of Ueff become critical for the determi- nation of the exchange couplings. The effect of different values for Ueff is discussed below. To rationalize our results we have fitted them to a Heisenberg Hamiltonian in the form: X H=− Jij Si · Sj , (1) i,j>i where Jij are the exchange couplings between spins Si and Sj . Evaluating different magnetic configurations we have been able to extract the Jij values by fitting the DFT energies to the Heisenberg Hamiltonian, Eq. (1). For the shorter chains (FeMn and FeMn2 ) we have included spin-orbit coupling (SOC)32,33 into our scalar- relativistic Hamiltonian. We have considered a simple anisotropic spin Hamiltonian, in the form H = DSz2 + E(Sx2 − Sy2 ). (2) Here x, y and z are such that z is the computed easy axis, while x and y are two orthogonal directions in the plane perpendicular to z (hard-plane) in principle, differ- FIG. 2. Top (upper panel) and side (down panel) views of the ent from the surface directions. The energy of the sys- relaxed geometry of FeMn2 . We show Fe atoms in red; Mn tem has been self-consistently calculated including SOC atoms in green; N atoms in blue; Cu atoms in the first layer for the different orientations x, y, and z of the magneti- in brown; and the rest of Cu atoms in pink. Blue lines delimit zation axis and, from these energies, values for D and E the unit cell used in the calculations. The vacuum side of the in Eq. (2) have been fitted using the evaluated magnetic unit cell and the three bottom Cu layers have been removed moments. for pictorial reasons. IV. RESULTS It is interesting to study the evolution of the geome- try of the chains as their size increases. In Table I we A. Geometries and energetics show the distance between TM atoms within the chain for all considered sizes. We observe that these distances Constant current images were obtained between atom are bigger than the unrelaxed value (3.65 Å correspond- manipulation sequences. This permitted us to have pre- ing to the PBE lattice parameter of Cu due to the ar- cise knowledge of the atomic arrangement. Unfortu- rangement of the chain on the surface). For example, nately, the structure of the tip that is optimal for atom d[Fe-Mn1 ] for FeMn is 3.81 Å, 4% bigger than the un- manipulation is not necessarily good for image produc- relaxed value. This distance tends to decrease as the tion and the obtained images do not contain much infor- chain size is increased, reaching 3.73 Å for FeMn6 . The mation, as can be seen in Fig. 1. In the inset, we show the same behavior can be observed for other positions in the configuration as inferred from the atomic manipulation chain. Another interesting distance to track involves the procedure that corresponds to a FeMn3 chain. Cu atoms just underneath Fe and Mn atoms. As can be The calculation yields precise insight on the actual ge- seen in the side view of Fig. 2, these Cu atoms (CuFe , ometries. In Fig. 2 we show the FeMn2 geometry as an Cu1 , Cu2 ) are pushed downwards by around 0.4 Å. For example. As we can see in the side view, after depo- FeMn, the Fe-CuFe distance is very close to the value sition of the TM atoms the final chain includes the N for the isolated atom (2.37 Å vs. 2.38 Å). However, it atoms in between them. These atoms are lifted from the increases for longer chains, with a value of 2.40 Å for surface. N atoms at the edge of the chain, like N1 , are FeMn6 . The same behavior is observed for the Mn atoms, moved upwards by ≈0.7 Å, while N atoms in between again close to the single adsorbate for the case of FeMn TM atoms, like N2 , are lifted by ≈1.6 Å with respect (2.50 Å vs. 2.47 Å), and it increases for longer chains to their unrelaxed positions in the bare CuN2 surface. reaching 2.56 Å for FeMn6 . Hence, as the number n Therefore, we can assume that the final chains have the of Mn atoms increases, we find that the intra-chain dis- form FeMnx Nx+2 . Of course, this is just a conventional tances diminish, see Table I, while the distance between choice to identify the final chain, since other atoms in the chain atom and the top Cu ones increases. Both be- the surface are also significantly disturbed and could be haviors can be rationalized as stress built up as the chain considered as parts of the chain, like CuS1 and CuS2 . increases its size. Indeed, the elongation of these dis- 4 d[Fe-Mn1 ] d[Mn1 -Mn2 ] d[Mn2 -Mn3 ] d[Mn3 -Mn4 ] d[Mn4 -Mn5 ] d[Mn5 -Mn6 ] FeMn 3.81 FeMn2 3.77 3.84 FeMn3 3.75 3.80 3.82 FeMn4 3.74 3.77 3.78 3.79 FeMn5 3.73 3.75 3.76 3.77 3.77 FeMn6 3.73 3.74 3.75 3.74 3.77 3.74 TABLE I. Distances between the different TM atoms for the 6 considered chains, (in Å). Note that the unrelaxed value would be 3.65 Å in agreement with the DFT lattice parameter of the Cu substrate. tances can be easily identify with some destabilization of the chains as length is increased. To check for the stability of the chains we have studied the chain’s energetics. First, we start by analyzing the atomization energy per TM atom, Eat [F eM nx ], defined from Eat [F eM nx ] = (3) 1 − [E[F eM nx ] − E[F e] − xE[M n] − Ex [Cu2 N ]], x+1 where E[FeMnn ] is the total energy of the system; E[Fe] (E[Mn]) is the energy of a gas-phase Fe (Mn) atom; and Ex [Cu2 N ] is the energy of the [3 × (x+3)] unit cell of the bare surface. The atomization energy per TM atom FIG. 3. Atomization energy, Eat from Eq. (3), (black x’s) and is a measure of how much average energy per atom one the gain in energy by adding one more Mn atom the chain, needs to give to the adsorbed chain to separate it in its ∆x EF from Eq. (4), (red stars). As we can see, atomization of constituent Fe and Mn gas-phase atoms and the pristine the chain becomes less difficult as the chain becomes longer, Cu2 N/Cu(100) substrate. Figure 3 shows the atomiza- and correspondingly, adding one more atom does not reduce tion energy per TM atom as a function of the number the total energy as much for longer chains. These values reveal of Mn atoms, x. The atomization energy per TM atom an increasing destabilization of the chain as it size increases. tends to decrease as the chain size increases, implying The dotted lines are a guide for the eye. that long chains become energetically less favorable. We can extract more information by looking at another quantity that reflects better the experimental procedure an Fe atom, which mimics the experimental procedure to construct the chain. Let us remember that the chains to build the system. The calculated values of ∆x EF are are constructed by STM manipulation. First, Fe and Mn always positive indicating that building the chains is an atoms are deposited on the surface, and then they are energetically favorable process. The energy gain is mod- moved with the STM tip to form the desired structure. est (≈ 0.4 eV ), and decreases by 37% from FeMn to To account for this procedure, we can define a formation FeMn6 (Fig. 3). Experimentally, the longest chains con- energy where the reference of energy is the one of the tained 9 Mn atoms.23 Our results seem to suggest that deposited TM atom, and then to study the gain in energy longer chains will be difficult to form due to the accumu- by adding one more atom to the chain. The definition lated stress imposed by the surface lattice constant and that we used for the formation energy is given by the Fe–Mn distance. EF [F eM nx ] = −[Eat [F eM nx ] − Eat [F e] − xEat [M n]], (4) B. Electronic and magnetic properties and charge where we use the atomization energies defined in equa- transfer tion 3. From EF we can define the gain in energy by adding one more TM atom, ∆x EF 34 , as The magnetic properties are one of the main motiva- ∆x EF = EF [F eM nx ] − EF [F eM nx−1 ] = tions for the study of these systems. They largely stem −[Eat [F eM nx ] − Eat [F eM nx−1 ] − Eat [M n]]. (5) from the electronic structure of the atomic chains on the surface. An analysis of the electronic structure by pro- This quantity, ∆x EF , defines the energy gained by jection onto atomic orbitals have been shown in Ref. [23]. adding a Mn atom to an existing chain starting from The magnetic properties were shown to be given by the 5 µT µ[Fe] µ[Mn1 ] µ[Mn2 ] µ[Mn3 ] µ[Mn4 ] µ[Mn5 ] µ[Mn6 ] Fe 3.82 3.35 Mn 4.97 4.80 FeMn -1.30 3.16 -4.65 FeMn2 3.68 3.17 -4.55 4.71 FeMn3 -1.43 3.19 -4.56 4.56 -4.73 FeMn4 3.78 3.22 -4.56 4.57 -4.57 4.75 FeMn5 -1.36 3.23 -4.56 4.58 -4.58 4.58 -4.77 FeMn6 3.86 3.23 -4.57 4.58 -4.58 4.59 -4.58 4.79 TABLE II. Magnetic moments (in µB ) of the TM atoms in the chains. Values for isolated Fe and Mn atoms on the surface are included for comparison. The values have been obtained by performing a Bader analysis.30 The total magnetic moment µT of the calculation cell also includes contributions from nearby copper atoms that spin polarize, hence µT slighlty differs from the sum of the magnetic moments of Fe and Mn. population of Fe and Mn d orbitals. In the case of Fe the d2z orbital was basically filled (where z is the direc- tion perpendicular to the surface in the present case), not contributing to the atomic magnetic moment. The Mn atoms show a larger spin polarization due to the complete polarization of its d-shell after inclusion in the FeMnx chain. The exchange splitting of d-bands is the smallest for Fe, but still it is roughly 5 eV, showing how robust FIG. 4. Spin density of the FeMn3 chain. Red (yellow) in- magnetism is in these chains. The electronic structure dicates majority (minority) spin areas. The chosen isovalue near the Fermi energy is largely due to the Cu and N is 0.01 eV/Å3 . The color code of the atoms is the same of states and with negligible magnetic polarization. Fig. 2. It is interesting to inspect the total magnetic moment of the chains shown in Table II. We can see how the magnetic couplings between Fe-Mn and Mn-Mn atoms the atom, but net spin close to zero. Indeed, the bigger are antiferromagnetic (AF) in the most stable configura- values for the induced magnetic moments of N atoms are tion. In addition, the magnetic moment of atomic Fe on very small, for example 0.11µB for N1 and 0.08µB for the surface is 3.35µB (formally S=2), while the value for N2 . Superexchange also leads to a change in the atomic Mn is 4.80µB (formally S= 52 ). Therefore, formally when angles: the Fe-N-Mn and Mn-N-Mn angles tend to ap- adding Mn atoms to a Fe one we would get systems with proach 180◦ maximizing the AF interactions.12,17 S= 12 for an odd number of Mn atoms, x, and S=2 for an The magnetic moments tend to increase with increas- even number. In the computed results for the total mag- ing chain size, which is consistent with the previously netic moment in the cell (µT ) we observe the expected mentioned progressive separation of the chain from the even–odd behavior, with values of 1.30–1.43 Bohr mag- surface. The magnetic moments of the Mn atoms within netons, µB , for x odd, and 3.68–3.86µB for the x even. a given chain are quite constant, with the exception of the Values of 1.30–1.43µB correspond well with the simple final atom, which is ≈4% bigger. The reason is the lower result S= 21 . The difference from 1µB has two different coordination of the edge atom, reflected in the slightly sources that are present in these calculations. On the one longer distance with the Cu atom below and the longer hand, the TM atoms present fractional occupancies that distance with the adjacent N atoms. lead to non-integer multiples of µB . On the other hand, The analysis of charge transfer in the system yields these calculations are mean-field approximations to the that each TM atom in the chain loses around one s elec- difficult solutions of correlated AF ground states. As a tron to form the bond with N atoms in the chain and consequence, the mean-field solution averages over the the Cu atom underneath the TM atom. The d charge is possible atomic magnetic moments found in the multiple similar to the atomic case, i.e., there are 6 electrons in spin configurations of the correct AF ground state. the d manifold for Fe, and 5 for Mn, in agreement with The AF coupling between the TM atoms is mediated the computed magnetic moments. by the N atoms via a superexchange mechanism as pre- viously shown for Mn12 and Fe chains.17 To illustrate the superexchange interaction, we show the spin-difference- C. Exchange coupling constants density of the FeMn3 chain in Fig. 4. The spin polariza- tion of the intercalated N atoms adopt the expected form We have fitted our results to the Heisenberg Hamilto- for superexchange, with induced spin polarization within nian shown in Eq. (1). In order to do so we have calcu- 6 1st neighbors 2nd neighbors 3rd neighbors 4th J12 J23 J34 J45 J13 J24 J35 J14 J25 J15 FeMn -10.25 FeMn2 -6.51 -2.09 0.97 FeMn3 -7.30 -1.97 -1.97 1.14 0.32 0.10 FeMn4 -7.96 -2.69 -2.10 -2.16 0.73 0.05 0.27 -0.36 -0.01 -0.02 TABLE III. Values for the exchange coupling constants (in meV) obtained by fitting Eq. (1) to the GGA+U total energies. lated different spin solutions corresponding to different magnetic moments. It is probably due to the sudden magnetic ordering among the TM atoms in the chain. In appearance of an extra neighbor that symmetrizes the all calculations, we have used the geometry of the most interactions on the central atom of the chain. stable configuration, which is always the AF case shown in Table II. For the atomic values of S, in Eq. (1), we had the choice of either using the formal value (S=2 for Fe, S= 52 for Mn), or taking the computed value from the DFT calculation since for a single atom Sz is a good quantum number. We have opted for the later option, which gives values for the exchange coupling constants J’s which are between 15% and 20% higher. We can justify this choice because the DFT values reflect the lo- cal magnetic moment that interacts via other atoms (su- perexchange) with the local magnetic moment of the next neighbor in the calculation. The results are shown in Ta- ble. III. As expected, J’s involving first neighbors are negative, which indicate an AF coupling. Second neigh- bor J’s are positive, indicating a ferromagnetic coupling which further stabilizes the global AF solution. Further order J’s are smaller, and its omission in the fit only im- plies a small error. The convergence with the number of coupled atoms FIG. 5. Evolution of J12 for FeMn with respect to the vari- considered in the Heisenberg chain, Eq. (1), is very fast. ation of Ueff . We have changed one of the values of U while We have done a systematic study for FeMn4 where differ- keeping constant the other one. ent number of neighbors are included in solving Eq. (1). Considering just first neighbors introduces a maximum As Rudenko et al.12 and Lin and Jones13 note, there is error of 0.2 meV. Considering first and second neighbors a dependence of the values of the interatomic exchange the error is less than 0.1 meV. These calculations show coupling on the chosen value of Ueff = U − J for the that the effective interactions in these spin chains are GGA+U approximation. Figure 5 shows the dependence very short range and first-neighbors truncation is indeed of J12 for FeMn on Ueff . We find that as the value of Ueff an accurate approximation. decreases, |J12 | increases. For the values of the present Analyzing the evaluated exchange couplings, J, we ob- study13 (Ueff [Fe] = 1 eV and Ueff [Mn] = 4 eV) J12 serve that the biggest value is obtained for the first- = -10.25 meV. If Ueff [Fe] is reduced by 1 eV this value neighbors interaction between Fe and Mn1 . Thus the changes to -11.60 meV. When Ueff [Mn] is reduced by AF coupling between Fe and Mn is stronger than the 1 eV the effect is more pronounced, obtaining a value one between Mn-Mn. FeMn is a special case since it of -16.44 meV (60% increase in absolute value). These involves two edge atoms, the resulting AF coupling is values give us an interval of exchange couplings that will the strongest one. Curiously, the exchange coupling be- be compared with our experimental results in the follow- tween Fe and Mn1 (J12 ) presents a sharp drop for FeMn2 ing sections. to start increasing again for FeMn3 and FeMn4 . The coupling between Mn1 and Mn2 (J23 ) presents the same behavior, it shows a minimum for FeMn3 . Similar behav- D. Magnetic anisotropy ior is obtained for pure Mn chains by Rudenko et al..12 These minima in the exchange couplings seem to be inde- We have studied the magnetic anisotropy energy pendent from the atomic geometry, where the behavior (MAE) of the system by including SOC in our calcu- with chain length is monotonous, and with the atomic lations. We have performed self-consistent calculations 7 MAE (meV) dir Ez − Ey Ex − Ey in the calculation, the convergence of non-collinear solu- tions is very challenging. We have been able to stabilize a Fe y 0.74 1.88 non-collinear configuration for FeMn where the magnetic Mn z -0.15 -0.02 moment of Fe is along the chain, while the moment for FeMn y 0.75 1.51 Mn forms an angle of 27◦ with the y axis. However, this FeMn2 y 0.48 1.46 solution is 3.79 meV higher in energy than the collinear solution. For longer chains we have not been able to TABLE IV. Magnetic anisotropy energies (MAE) (in meV). stabilize any non-collinear solution. Therefore, we have We also show the easy axis directions, where x is the direction just considered collinear configurations for the rest of our in plane and perpendicular to the chain, y along the chain, calculations. and z perpendicular to the surface. The larger MAE of Fe forces the full-chain magnetic axis to align with the Fe easy axis. This is indeed seen in Table IV, where the value for MAE decreases when in three directions: along the chain (y), perpendicular to increasing the chain length. This reduction in MAE can the chain in the plane (x), and perpendicular to the plane be understood in terms of the addition of Mn atoms that (z). We have also fitted the anisotropic spin Hamiltonian tend to align their magnetic easy axis perpendicular to of Eq. (2) to the singly adsorbed atoms using their com- the Fe one. These results qualitatively agree with the puted magnetic moments to obtain values for D and E results of anisotropic spin Hamiltonians where the to- using the equations: tal magnetic anisotropy is approximated by adding the individual contributions of D and E. This approxima- 2Ei − (Ej + Ek ) D= tion does not capture the changes in geometry of the S(2S − 1) chain with increasing lentgth. Nevertheless, adding up Ej − Ek the MAE contributions of each atom leads to overestima- E= , (6) S(2S − 1) tions of MAE that are below the accuracy of our DFT calculations. where Ei , Ej , Ej are the magnetic anisotropy energies The study of the magnetic anisotropy in these chains when the spin, S, is aligned along the easy or hard axis reveals their composite structure. All FeMnx for odd- i and the two transversal axes j and k, see Table IV. x show a quasi -spin of 1/2, which should present zero Fe shows an easy axis along the N-direction, i = y, with anisotropy if these chains were macrospins of spin 1/2, a MAE of 1.88 meV. For Mn we get an out-of-plane easy see. Eq. (2). Our DFT calculations show sizeable axis, with a smaller MAE of 0.15 meV. This results qual- anisotropies that underscore the complexity of the mag- itatively agree with the STM experimental data of Hir- netic states of these antiferromagnetic structures. jibehedin et al,22 . However, using the conventional choice to select D and E our values of MAE lead to a situation where the y − z plane is an easy plane, with D=0.77 meV and E=0.19 meV. Our results have been tested again E. Magnetic excitation energies convergence with unit cell size35 and present a negligible dependence. Moreover, our values are also robust against The magnetic structure of the chains can be ob- the values of U . Indeed, these results seem to be largely tained by studying the inelastic electron tunneling spec- independent on the exchange-and-correlation functional. tra (IETS)9 obtained with the STM. Figure 6 shows the Our results are within a few µeV off the LDA MAE ob- differential conductance obtained for the different FeMnx tained by Shick and co-workers36 . The values of D and chains. The features that appear in these spectra are due E differ from their calculated ones probably due to the to magnetic excitations, very similarly to the ones of Mnx different way of obtaining them from the evaluated MAE. chains shown in Ref. [22]. Nevertheless, there are notice- Our calculations compared better with their values if the able differences regarding both the peak at zero bias for nominal magnetic moments are used (D=-0.44 meV and the odd-x chains and the detailed structure of the steps. E=0.19 meV using S = 2 for Fe. For Mn the values do Indeed, odd-x chains are singlets in the case of Mn chains, not change because the DFT magnetic moment agrees here however, the ground states of odd-x chains present with the nominal one). Using GGA+U, Barral et al37 a doublet (S ≈ 12 ) magnetic structure. obtained similar results to Shick et al.36 . Recent calcu- We solve Eq. (1) and study the different magnetic lations by Panda et al.38 yield similar values for Fe with states for each chain using the methodology of Ref. [39]. LDA, and values closer to the experiment using dynam- The study of the ground state is particularly important ical mean field theory (DMFT). to understand the behavior of the Kondo physics appear- Taking into account that Mn and Fe have perpendic- ing in Fig. 6.23 Indeed, all odd-x chains display a Kondo ular easy axes, one may wonder if SOC might induce a zero-bias peak, while this peak is absent from the chains non-collinear alignment of the magnetic moments in the with even x. Due to the Heisenberg correlations, the ex- chains. We have tested that possibility for the shorter perimental ground state is multi-configurational. Solving chains by performing an explicit non-collinear calcula- Eq. (1) with the anisotropy terms, Eq. (2) shows that tion including SOC.33 Due to the small energies involved many of these configurations are S = 21 states, with a 8 the Kondo scattering.39 The effect of the potential scat- tering term is to remove the electron-hole symmetry of the excitation spectra of Eq. (1). In order to fit the ex- perimental spectra we have used a JKondo ρ = −0.04 120 FeMn6 (where JKondo is the Kondo exchange coupling with elec- trons from the substrate and ρ is the density of states FeMn5 at the Fermi energy) and a potential scattering term UKondo /JKondo = −0.5. The results indicate that larger dI/dV (nS) values should be used to reproduce the experimental data, implying the need to go beyond third-order per- FeMn4 turbation theory to treat Kondo scattering in FeMn3 . 60 50 FeMn3 (a) FeMn2 40 dI/dV (nS) FeMn -30 -20 -10 0 10 20 30 30 Bias (mV) 20 FIG. 6. Inelastic electron tunneling spectra (IETS) of dif- 6 (b) ferent FeMnx chains obtained by measuring the differential dI/dV (a.u.) conductance with the STM tip placed on a central Mn atom. weight larger than 20% in the total state. Hence, a spin- flip process is possible at zero-energy cost, which explains 3 the appearance of Kondo peaks. Likewise, even-x chains do not have degenerate ground states and Kondo physics -30 -20 -10 0 10 20 30 is absent. The absence of degenerate ground states for even-x is due to a total spin S ≈ 2 where the large lon- Sample bias (mV) gitudinal MAE of the Fe atom lifts the degeneracy of the ground state. The presence of anisotropy prevents S FIG. 7. Inelastic electron tunneling spectra (IETS) of from being a good quantum number.23 FeMn3 , (a) experimental, (b) two computed solution using The solutions of Eq. (1) permit us to compare the com- our calculated spin-chain parameters for (U − J)Mn = 4 eV puted data for the chains with the experimental data. and (U −J)Fe = 1 eV (blue) or scaling the computed exchange Figure 7, (a) shows the IETS for the FeMn3 chain mea- coupling by a 1.6 factor as found for FeMn with (U −J)Mn = 3 sured of the edge Mn atom, and (b) shows two calcu- eV and (U − J)Fe = 1 eV (green), section IV C. The STM tip lations. The first one (blue) performed with the ex- is placed on the edge Mn atom. change couplings computed from our GGA+U calcula- tions with an effective U value for Mn of 4 eV and for Fe The magnetic behavior of the inelastic thresholds is of 1 eV, see section IV C, and adding up the experimen- correctly reproduced by a Zeeman shift. This permits tal atomic magnetic anisotropies. Unfortunately, DFT us to extract the value of the gyromagnetic ratio g, seems to underestimate the anisotropy of Fe on Cu2 N, Fig. 8. For the present case we find that the atomic g’s see section IV D. The second one (green) scales the ex- (gF e = 2.1 and gM n = 1.9 from Ref. [22]) are good ap- change couplings by 1.6, using the scaling found for FeMn proximations to obtain the correct behavior of the mag- when the effective U value for Mn was reduced to 3 eV, netic global states with external B, Fig. 8. To a large and keeping the corresponding value of Fe constant. We extend, the atomic spin preserves its character, although see that the energy thresholds are in good agreement in very entangled due to the sizable Heisenberg exchange the second case, and the solution of Eq. (1) with the interactions. third-order perturbation method of Ref. [39] largely re- produces the dynamical phenomenology of the magnetic chain including higher-energy excitations like the one at V. DISCUSSION AND CONCLUDING ±10 meV. REMARKS The asymmetry of the main inelastic thresholds found in Fig. 7(a) is treated within the third-order perturba- This work is a detailed experimental and theoretical tion method by including a potential scattering term in account of the electronic and magnetic properties of a 9 10 (a) (b) 8 8 Energy (meV) 9T dI/dV (a.u.) 6 7T 5T 6 3T 2T 4 4 1.5 T 1T 0.5 T 2 2 B=0T 0 -30 -20 -10 0 10 20 30 0 1 2 3 4 5 6 7 8 9 Sample bias (mV) B (T) FIG. 8. (a) Inelastic electron tunneling spectra (IETS) for FeMn3 with a magnetic field applied perpendicular to the surface increasing in 1-T steps until 9 T. (b) The behavior of the different magnetic states (black dots) obtained from the experimental figure follows the Zeeman trend expected for a Zeeman term with the atomic g-factors (gF e = 2.1 and gM n = 1.9 from Ref. [22]). heterogeneous type of magnetic atomic chain (FeMnx bors is a sufficient approximation to obtain exchange cou- with x = 1, 6) adsorbed on Cu2 N/Cu (100). The chains plings with an error of 0.2 meV. In the present choice of are assembled by atom manipulation with an STM tip, DFT+U calculations, the computed magnetic exchanges and stable configurations are found when the TM atoms lead to excitation energies smaller than the experimen- (TM = Fe and Mn) sit ontop of Cu atoms. Experimen- tal ones. Unfortunately, there is not a unique way of tally, it is difficult to assemble chains with x > 9 (we determining the value of the Hubbard U for the calcu- rarely succeeded going beyond x = 10 creating a straight lations. Our systematic study of the values show that a FeMnx chain) and theoretically we see that stress builds change of 1 eV in the value of the Mn U leads to a ∼ up as the chain increases size due to the imposed TM-TM 60% change in the value of the evaluated couplings yield- distance by the underlying Cu2 N/Cu (100) substrate. As ing good agreement with the experiment. Despite this the chain increases its size, the TM atoms increase their dependence of the interatomic exchange energies on the mutual distance and also their distance to the chain, en- value of U , the conclusions of this work do not change for ergetically this is translated into a systematic lowering of the range of used values. Moreover, the geometrical, sta- both the atomization energy and the energy gained by bility and magnetic anisotropy results show a negligble the chain every time a new Mn atom is added. dependende with the above values of U . Upon adsorption, Fe and Mn lose one of their s elec- The TM atoms are subjected to magnetic anisotropies trons in the interaction with the substrate, mainly to on this surface. Our calculations show that the MAE form the bond with the neighboring N atoms. There of the full chain is not just the sum of the MAE’s of is considerable distortion and hybridization of the d- each TM atom. Nevertheless, the smallness of Mn MAE electron structure but their occupations remain the free- renders this approximation acceptable. Despite their dif- atom ones, leading to magnetic moment values close to ferent easy axis, the very large MAE of the Fe dominates the gas phase.13,22 and the Mn spins orient along the chain following the The experimental data involve the IETS of different Fe easy axis. This leads to a collinear solution of the chains with detailed information on the excitation ener- initially non-collinear problem. We have not found any gies of the chains. The lower-energy spectral features are spin canting or frustration although we cannot rule it out due to magnetic excitations of the system as was tested for longer chains. Moreover, the study of the magnetic by their magnetic field dependence. In parallel, the val- anisotropy in these chains reveals their composite struc- ues of the Heisenberg Hamiltonian can be obtained from ture. All FeMnx for odd-x show a quasi -spin of 1/2. If DFT by evaluating the energy of different spin arrange- these chains were spins 1/2, their anisotropy would be ments of the chains. This approach gives us a systematic stricly zero. However, our calculations show that they insight on how the different TM atoms relate to each have sizeable anisotropies in agreement with the com- other in the chain, which are mainly driven by antiferro- plexity of the magnetic states of these antiferromagnetic magnetic superexchange mediated by the non-magnetic structures. N atoms of the surface. We find that the Fe-Mn couplings This combined experimental and theoretical work gives are systematically larger than the Mn-Mn ones, and that us direct insight into the different electronic, geometric beyond second neighbors neglection of the magnetic cou- and magnetic properties of these heterogeneous chains. pling is a very good approximation. 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