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1 doi: /nature12759 FORBIDDEN TRANSITIONS IN HO ON PT(111) For a single Ho atom on a Pt(111) surface, the adsorption site s symmetry is given by C 3v. To describe the effect of the crystal field on the lowest multiplet, which is spanned by the eigenstates of J 2 and J z with fixed J 8, J, M M, Stevens method of operator equivalents can be used. The crystal field Hamiltonian has the following form in this case [1]: H B2O B4O B4O B6O B6O B6O 6 6 6, (S1) where the Bn m are crystal field parameters and On m are the Stevens operators [1]: O 0 2 3J 2 z J(J + 1), O J 4 z 30J(J + 1)J 2 z + 25J 2 z 6J(J +1)+3J 2 (J + 1) 2, (S2a) (S2b) O4 3 1 [ ] Jz (J+ 3 + J 3 4 )+(J+ 3 + J )J 3 z, (S2c) O Jz 6 315J(J + 1)Jz Jz J 2 (J + 1) 2 Jz 2 525J(J + 1)Jz Jz 2 5J 3 (J + 1) J 2 (J + 1) 2 60J(J + 1), (S2d) O6 3 1 [ (11J 3 4 z 3J(J + 1)J z 59J z )(J+ 3 + J ) 3 +(J J 3 )(11J 3 z 3J(J + 1)J z 59J z ) ], (S2e) O6 6 1 [ ] J J 6. (S2f) The z-axis is oriented perpendicular to the surface. Since all matrix elements of H between the states M are real, the eigenstates can be chosen to have real coefficients when expanded in the basis M. As the Hamiltonian only contains powers of J z, J+ 3 and J, 3 the eigenstates form three distinctive groups, Ψ 0, Ψ + and Ψ, each coupling M states with M 3. Ψ + i Ψ i k0 c + i,k 8 3k, (S3) c i,k 8+3k, k0 Ψi 0 c 0 i,k 3k k 2 (S4) (S5) The crystal field, being electrostatic in nature, is invariant under time-reversal symmetry, [H, T ] 0. The antiunitary time-reversal operator T anticommutes with the total angular 1

2 momentum operator, JT T J, which implies J 2 T T J 2 and J ± T T J. This leads to the following equations: J 2 {T M} J(J + 1){T M}, J z {T M} M{T M}, J ± {T M} J(J + 1) M(M 1){T M 1}, (S6a) (S6b) (S6c) which have the solution T M ( 1) M M. As J is integer, T 2 1. Applying T to the eigenfunction Ψ +/0/ produces an eigenfunction with the same energy. T Ψ + i T Ψ i (c + i,k ) T 8 3k (S7a) k0 (c + i,k ) ( 1) 8 3k 8+3k (S7b) k0 ( 1) k (c + i,k ) 8+3k k0 (S7c) (c i,k ) T 8+3k (S7d) T Ψ 0 i k0 (c i,k ) ( 1) 8+3k 8 3k (S7e) k0 ( 1) k (c i,k ) 8 3k k0 (S7f) (c 0 i,k) T 3k (S7g) k 2 (c 0 i,k) ( 1) 3k 3k (S7h) k 2 ( 1) k (c 0 k) 3k k 2 (S7i) The Ψ + and Ψ states form doubly degenerate pairs connected by T, i.e. T Ψ ± i Ψ i, while the Ψ 0 states are non-degenerate. This is in agreement with the fact that the C 3v point group contains one- and two-dimensional irreducible representations. In the case that the lowest order term B2O dominates with B2 0 < 0 (out-of-plane easy axis), as observed in similar systems (Fe, Co and Gd atoms on Pt(111) [2 4]), the ground 2

3 state is doubly degenerate. A single electron could flip the magnetic moment of the atom, if the matrix element of the spin-flip operator J σ calculated between the two ground states is non-zero. We shall write the two states as Ψ + G and T Ψ + G. Ψ + G c k 8 3k k0 Ψ G T Ψ + G ( 1) k c k 8+3k k0 (S8a) (S8b) We reorder the indices according to k 5 k. Ψ G ( 1) k c 5 k 7 3k k 0 (S8c) In order to calculate the necessary matrix element, we consider Ψ + G J z Ψ G, Ψ + G J Ψ G and Ψ + G J + Ψ G. Ψ + G J z Ψ G ( 1) k c 5 k c k 8 3k J z 7 3k 0 (S9a) Ψ + G J Ψ G Ψ + G J + Ψ G k0 k 0 k0 k 0 k0 k 0 ( 1) k c 5 k c k 8 3k J 7 3k 0 ( 1) k c 5 k c k 8 3k J + 7 3k k0 k 0 (S9b) (S9c) ( 1) k c 5 k c kδ k,k (1+3k)(16 3k) (S9d) ( 1) k c 5 kck (1+3k)(16 3k) (S9e) k0 ( 1) k c 5 kck (1+3k)(16 3k) k0 ( 1) k c 5 kck (1+3k)(16 3k) (S9f) k3 ( 1) k c 5 kck (1+3k)(16 3k) k0 ( 1) 5 k c k c 5 k (16 3k )(1 + 3k ) k 0 (S9g) 3

4 ( 1) k c 5 kck (1+3k)(16 3k) k0 + ( 1) k c 5 kck (16 3k)(1 + 3k) (S9h) k0 ( 1) [ k c 5 kc k c 5 kck] (1+3k)(16 3k) 0 (S9i) k0 All three matrix elements vanish. Thus direct transitions between two ground states (or any two states with the same energy from the (+) and ( ) subsets) by scattering of electrons are forbidden. The situation changes in the presence of an external magnetic field, since then the time-reversal symmetry is broken. We note here, that the forbidden transitions between the two ground states are not a unique property of the system Ho/Pt(111). Any 4f atom with J 3n + 1 or 3n + 2, for any integer n 0, should also show long lifetimes on a substrate with C 3v symmetry. This, however, excludes most of the rare earths the only promising atoms are praseodymium and promethium (J 4), the latter having no stable isotopes. SWITCHING RATES We model the transitions following [5]. The interaction of the Ho atom with electrons in the substrate and the tunnelling tip leads to transitions between the states Ψ i and Ψ f (i f). The transition rate Γ if can be split into three parts Γ if Γ S S if + Γ T S if + Γ S T if (S10) for processes where the scattered electron remains in the substrate, Γif S S, or where it tunnels between the tip and the substrate, Γ T S if and Γ S T if. The interaction between the Ho atom (total angular momentum J) and electrons (spin σ) depends on the operator [6 8] V J σ 1 2 (J +σ + J σ + +2J z σ z ). (S11) The z-axis is chosen perpendicular to the substrate surface. The initial and final states of the scattering process are the product states Ψ i,σ and Ψ f,σ, where σ, σ ± 1 2 denotes the z-component of the scattered electron s spin. 4

5 The rates can be evaluated in Fermi s Golden Rule approximation, Γ S S if Γ S T if Γ T S if c SS M S S if ζ(e i E f ), (S12a) c TS M S T if ζ(e i E f ev ), (S12b) c TS M T S if ζ(e i E f + ev ). (S12c) Here V accounts for a voltage applied between tip and substrate. The prefactor c SS accounts for the scattering strength of the electrons as well as the substrate density of states, while c TS also depends on tunnelling matrix element and the tip s density of states. The matrix elements are M S S if σ,σ Ψ f,σ V Ψ i,σ 2, (S13a) M S T if σ,σ Ψ f,σ V Ψ i,σ 2 ( ησ ), (S13b) M T S if σ,σ Ψ f,σ V Ψ i,σ 2 ( ησ). (S13c) The spin-polarization of the tip is taken into account by the coefficient η [ 1, 1] which describes that the tip s densities of states are proportional to 1±η for electrons with σ ± The densities of states of the tip and the substrate as well as the tunnelling matrix element are assumed to be constants in the narrow energy regime around the Fermi energy. The function ζ arises as an integral over Fermi distributions F (E) (1+e E/k BT ) 1 and the requirement that the scattered electron s initial state is occupied and the final state empty. It is given by [9] ζ(e) F (E ) [1 F (E + E)] de The dynamics of the system follows from the master equation E 1 e E/k BT. (S14) dp i dt j (Γ ji P j Γ ij P i ), (S15) where P i is the occupation probability of state Ψ i. This system of ordinary differential equations is solved numerically for the initial condition I +8 {P 8 1,P i 0(i 8)} or I 8 {P 8 1,P i 0(i 8)} and all possible 17 states of the system. After the voltage pulse V with variable amplitude is applied for 200 µs, the system is given sufficient time (of order 1 s) to relax to one of the two ground states at the stabilizing voltage of 5

6 5 mv. For small voltage pulses only the lowest states are populated during the pulse, but for stronger pulses, more and more states get important. The pulse drives the system out of its equilibrium state. From this we obtain the switching probability for one run i.e., the conditional probabilities P I +8 8 or P I 8 8. The experiment was done in a sequence of runs, so that the final state of the last run is the new initial state of the next pulse sequence. Hence, the probability P +8 (n) to be in the state Ψ +8 in the n th run is given by P +8 (n) P +8 (n 1) P I (1 P +8 (n 1)) P I (S16) Here we used P 8 (n 1)1 P +8 (n 1). The solution of this iteration is ( ) n ( ( ) ) P I P I P I 8 +8 P +8 (0) P +8 (0) P I P I 8 +8 P I 8 +8 P +8 (n) ( ). (S17) 1 P I P I 8 +8 In the experiment the initial state is not known. We might therefore assume that both states are equally likely P +8 (0) 0.5. But this is not important since after sufficient number of runs the dependence on the initial condition decays anyhow. In fact the experiment cannot distinguish the two states, only the switching between both is observed. The switching probability S(n) from state Ψ +8 to Ψ 8 or vice versa in the n th run is given by S(n) P +8 (n 1) P I P 8 (n 1) P I 8 8. (S18) From this we obtain the total switching probability for a given voltage pulse and number of runs N S tot (N) 1 N N S(n). The symmetric and antisymmetric states shown in Fig. 1c in the main paper n1 Ψ a/s ±6 (S19) have an energy splitting of only E a/s ±6 0.3 µev, when using the anisotropy constants given by first-principles calculations. Compared to the expected tunnelling splitting caused by the transition rates this is very small, i.e. E a/s ±6 Γ a/s ±6f. Therefore the coherence of the states is destroyed by the baths and thus we used for the numerics the states Ψ ±6 ( 1 2 Ψ±6 s ± Ψ±6) a. The prefactors cts, c SS and η were used as fit parameters for the computation. The resulting graph is shown in Fig. 2e in the main paper. The errors shown in the Figure were estimated by approximating the distribution for the number of switching 6

7 a f.c.c. h.c.p. c Height (pm) b d 2 I/dV 2 (arbitrary units) f.c.c. h.c.p Length (nm) f.c.c. h.c.p Voltage (mv) di/dv (arbitrary units) f.c.c. h.c.p Voltage (mv) FIG. S1. Identification of the adsorption sites. a, 3D-STM image of Ho atoms on f.c.c. and h.c.p. adsorption sites of the Pt(111) surface (upper panel) and their corresponding height profiles (lower panel). The size of the STM image is 11.0 nm 19.4 nm. b, Inelastic tunnelling (d 2 I/dV 2 ) spectra and c, di/dv spectra of Ho atoms adsorbed on the f.c.c. and h.c.p. sites of the Pt(111) surface. events by a binomial distribution and calculating the 68 % confidence intervals [10]. The fit only weakly depends on the prefactors, which are estimated as c TS mev 1 s 1 c SS mev 1 s 1, and sensitively depends on η which is ± ADSORPTION SITES OF HO ON PT(111) The adsorption sites (f.c.c. or h.c.p. sites) of Ho atoms on Pt(111) can be identified from STM images. As shown in Fig. S1a, the atoms can be classified into two classes of apparent heights with a difference of 20 pm when imaged at a sample bias V 1.0V. Superimposing the atomically resolved Pt(111) lattice on images with the two kinds of atoms [11] shows that the observed height difference can be attributed to their different adsorption sites. Statistical 7

8 investigations of the atom s apparent heights reveal that there are considerably more atoms with lesser height than those with greater height, indicating a significant difference in the adsorption energy between two adsorption sites. To determine which site is the f.c.c. and which the h.c.p. position, we recorded an atomically resolved STM image of two adjacent Pt(111) terraces separated by a mono atomic Pt step with Ho atoms on the lower terrace (not shown). Geometrical analysis [12] of the position of the atoms with respect to the extended grid of atomic positions of the upper Pt terrace leads to the identification of the adsorption site of the atoms with lesser height as an f.c.c. site and the other as an h.c.p. site. Note that the symmetry of both adsorption sites is C 3v. The observed difference in the heights of atoms sitting on f.c.c. and h.c.p. sites originates from the different occupation of atoms in the Pt layer one atomic layer below the Pt surface (see the inset of Fig. 1c in the main text), which affects the magnetic and electronic properties of the adsorbed atoms. Indeed, inelastic tunnelling spectroscopy (ITS) reveals that the inelastic spin-flip excitation energies of the atoms adsorbed on f.c.c. and h.c.p. sites differ considerably (see Fig. S1b). A higher spin-flip excitation energy is observed for f.c.c. Ho atoms leading to a higher magnetic anisotropy energy of the atoms compared to h.c.p. Ho atoms [3], making the magnetic states of the atoms sitting on the f.c.c. site more stable. The observation of a greater population of f.c.c. atoms, even after deposition at low temperatures, indicates that atoms preferentially adsorb in f.c.c. sites. Moreover, by laterally manipulating single atoms by the STM tip, atoms can be moved from h.c.p. sites to f.c.c. sites but not the opposite way. The adsorption site can also be seen in the electronic structure. Figure S1c shows differential conductance (di/dv ) spectra of the atoms on f.c.c. and h.c.p. sites. Differences are recognizable near the Fermi energy and more clearly at a voltage of about 1 V. While the local density of states (LDOS) of the atoms on the f.c.c. site increases with the positive voltage, a drop of the LDOS is observed for the atoms on the h.c.p. sites. In conclusion, from the heights of the atoms, spin-excitation spectra and di/dv spectra, the adsorption sites of the Ho atoms on Pt(111) can be fully identified. LIFETIMES OF HO ATOMS IN GROUPS Figure S2 shows the decay of magnetic states of Ho atoms (indicated by the dot in the corresponding topographic images) after giving short voltage pulses. The population of 8

9 ± 11 s 2 nm Counts ± 24 s 2 nm ± 15 s 1 nm ± 9 s Time (s) 2 nm FIG. S2. Life times measured on different Ho atoms in f.c.c. positions with different local environment. Exponential decay of the spin population measured on the indicated atoms following 200µs voltage pulses. τ is independent of the local environment within the error. The topographic STM image shows the Ho atoms, on which the experiment was carried out, indicated by a colored dot and its surrounding. states decays exponentially and the lifetime τ is independent of the local environment of the Ho atom within the statistical error. This is due to the fact that spin-flip transitions of single Ho atoms by conduction electrons of the substrate are forbidden and as consequence, also spin-flips via RKKY coupling between Ho atoms are forbidden to first order. AB-INITIO CALCULATIONS To describe structural, electronic and magnetic properties of Ho atoms on the Pt(111) surface, we use a multi-code approach, which combines determination of atomic positions using the Vienna Ab-initio Simulation Package (VASP) [13, 14] and electronic structure calculations with a self-consistent fully relativistic Green s function method specially designed for semi-infinite systems and embedded real space clusters [15, 16]. Since the conventional local density approximation (LDA) fails to describe correctly the strongly localized 4f states of Ho, we applied to them an LDA+U method [17] as well as a self-interaction correction 9

10 method [18] as it is implemented within the multiple scattering theory [19]. The occupancy of the 4f states indicate an orbital moment of almost 6 µ B, in line with Hund s rule. For the LDA+U calculations, we used the value of U eff 5.0 ev. For the calculation of the crystal field parameters, the 4f states were included in the core states and treated within the self-interaction correction method and Slater transition state approach [18, 20, 21]. To determine the atomic positions with the VASP code, the Pt(111) surface was modeled by a 7-monolayers-thick slab. Every slab was separated along [111] by a 1.5 nm thick vacuum layer and each monolayer in the unit cell contained 16 Pt atoms to guarantee a large lateral distance between Ho atoms in the supercell approach. The experimental out-of-plane lattice parameter was used between these monolayers while the two top Pt monolayers and position of Ho were optimized. The relaxation was performed, within the spin-polarized mode using the Monkhorst-Pack k-mesh, until the forces were less than ev Å 1. Thereby, the electron-ion interactions were described by the projector-augmented wave pseudopotential, and the electronic wave functions were represented by plane waves with a cutoff energy of 450 ev. Regarding the Ho adatom, its f.c.c. hollow position is found to be favorable in energy compared to the h.c.p. position. After relaxation, the z-coordinate separation between adatom and the nearest (next-nearest) topmost Pt atoms is 2.30 Å and 2.32 Å in the f.c.c. and h.c.p. hollow positions, respectively. Using the VASP code and the harmonic approximation, we calculated the atomic vibrations for a Ho atom on the Pt(111) surface, which yield 4.9 mev (5.1 mev) in the f.c.c. (h.c.p.) hollow position. These energies are lower then the measured excitation energies. The structural information obtained from our VASP simulations was used as input for further electronic and magnetic structure calculations, using our fully relativistic Green s function method for real space clusters embedded into a surface within the full charge density approach [15, 16]. According to this approach, we computed the Green s function of semiinfinite Pt(111) surface. Then, the Green s function of a cluster around the Ho atom was obtained via a Dyson equation in a 2D periodic structure with the corresponding Fourier transformation into the real space representation. The size of the cluster was varied to reach a certain convergence of the total energy and magnetic spin and orbital moments. A cluster size of 201 atoms was found to be optimal for our calculations. Our fully relativistic calculations yield 4.1 µ B (3.9 µ B ) and 5.6 µ B (5.45 µ B ) for the total 10

11 FIG. S3. Density of states (DOS) of a Ho atom in f.c.c. position. Calculated DOS of a Ho atom (red) and the Pt(111) substrate (blue) for majority (spin up) and minority (spin down) electrons as function of the energy with respect to the Fermi energy E F. spin and orbital moments for a Ho atom in the f.c.c. (h.c.p.) hollow position. The corresponding density of states was calculated for the f.c.c. case with 7 majority and 3 minority 4f electrons below the Fermi level (see Fig. S3). It indicates a total angular momentum of J 8 which is in good agreement with Hund s rules. The magnetic anisotropy energy within the LDA+U approximation (U eff 5 ev) was found to be 45 mev and 32 mev for the f.c.c. and h.c.p. hollow positions, respectively. AB-INITIO DESCRIPTION OF CRYSTAL FIELD PARAMETERS In general, the parameter Bn m of the crystal field Hamiltonian (2) can be calculated from first-principles starting from the interaction energy of the 4f electrons with all other charges [22]: E 4f drdr ϱ(r)ϱ 4f(r) R r. (S20) In order to obtain a Hamiltonian, we apply the first order perturbation theory. An unperturbed 4f state from the ground state multiplet of a free rare earth metal atom R 4f ; JM with n 4f unfilled f shells is disturbed by a small electrostatic potential V (r) and leads to 11

12 the expectation value of the interaction energy [23] n 4f E 4f (M) R 4f ; JM V asph (r i ) R 4f ; JM. i (S21) The non-spherical potential V asph includes all charges in the crystal (nucleus, core electrons and valence electrons) without the non-spherical 4f charge of the considered rare earth atom. This potential is expanded into real spherical harmonics Y L, centered at the the rare earth atoms ( l>0 and L (l, m)) V (r) L V L (r)y L (r). (S22) When we assume every unfilled electron shell has the same radial wave function R 4f, we can separate the radial and spherical contributions E 4f (M) L n 4f R 4f V L (r) R 4f JM Y L (r i ) JM. In the second expectation value, we can use the Stevens operator equivalents [24, 25] i (S23) n 4f Y L (x i,y i,z i ) C L θ l (J)Ol m (J x,j y,j z ), i (S24) where the factors θ l (J) for the rare earth metal atoms R 3+ are tabulated in [24, 25]. The numerical prefactors C L of the real spherical harmonics appear explicitly because the Stevens operator equivalents were formerly written only for the x, y and z depend polynomials of Y L (r). Applied to Eq. (S23), this leads to the interaction energy E 4f (M) drr 2 R4f(r)V 2 L (r) C L θ l (J) JM Ol m (J x,j y,j z ) JM. (S25) L In this formulation, the radial part of the 4f charge density (the square of the wave function corresponds to the charge density) is only a parameter for a Hamiltonian applied on the JM states [22, 25]: B m l θ l (J)A m l r l θ l (J)C L drr 2 ϱ 4f (r)v L (r). (S26) For the evaluation, we consider the potential of the charge density ϱ(r ) of all other charges V (r) dr ϱ(r ) and expand the denominator which leads to a radial potential r r V L (r) 4π dr ϱ(r )Y L ( r ) rl <, (S27) 2l +1 r> l

13 with r < min(r, r ) and r > max(r, r ). Thereby, the crystal field potential will take the form [ A m l r l ] 4π 2l +1 C L dr ρ(r )Y L ( r ) drr 2 ϱ 4f (r) rl <. (S28) r> l+1 The resulting anisotropy constants of Ho atoms in f.c.c. position are given in Table I. From the anisotropy constants the eigenstates of H can be calculated. Their energies and values of J z are given in Table II. Anisotropy constant value energy splitting B µev 45.9 mev B nev 3.3 mev B nev B nev 0.8 mev B 3 6 B nev nev TABLE I. Calculated anisotropy constants, and corresponding maximal energy splitting in the J-multiplet for diagonal Stevens operators. Methods The Pt(111) sample was cleaned by Ar + sputtering followed by annealing to 850 K. The sample surfaces were checked for impurities by STM at 4.2 K prior to deposition of Ho. Ho atoms were deposited onto the clean surfaces at 4.2 K directly in the STM. Unpolarized STM tips were prepared from W wires and were cleaned in situ by flashing to above 2500 K. Spinpolarized tips were prepared by depositing 20 monolayers (ML) of Mn or Cr in situ onto the tip apex followed by gentle annealing. Inelastic d 2 I/dV 2 spectra were recorded using a modulation frequency of 16 khz and a root mean square (rms) of 2.4 mv and detecting the second harmonics with a lock-in technique. Spin-polarized STM measurements were performed recording the di/dv signal with coated tips using a modulation of 720 Hz and 1 mv rms. Voltage pulses were given with the feedback loop open to avoid destabilizing the tip position. Tip changes due to voltage pulses were excluded by checking between every measurement on Ho atoms also the time trace of pulses given on bare Pt(111). For tests of the out-of-plane sensitivity of the spin-polarized tips, one monolayer high Co islands were 13

14 Eigenstate Energy (mev) J z Ψ Ψ Ψ Ψ Ψ±6 s Ψ±6 a Ψ Ψ Ψ Ψ Ψ±3 a Ψ±3 s Ψ Ψ Ψ Ψ Ψ TABLE II. Energies and expectation values of J z for the calculated eigenstates. grown on Pt(111) at 360 K prior to inserting the sample into the STM. All experiments were performed using a home-built cryogenic Joule-Thomson cooled STM [26]. First-principles calculations were carried out within the density functional theory. The crystalline structure was determined with the Ab-initio Simulation Package (VASP) method, well-known for providing precise total energy and forces [13, 14]. The obtained structural information was further used for calculations of the electronic and magnetic properties of single Ho atoms on the Pt(111) surface using a self-consistent relativistic fullcharge Green function method, which is specially designed for semi-infinite systems and embedded real-space clusters [15, 16]. A self-interaction correction method and an LDA+U approximation were applied to provide an adequate description of strongly localized Ho f electrons [17, 18]. 14

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