Magnetic character of holmium atom adsorbed on platinum surface

We address a recent controversy concerning the magnetic state of holmium adatom on platinum surface. Within a combination of the density functional theory (DFT) with the exact diagonalization (ED) of Anderson impurity model, the 〈J z〉 = 0 paramagnetic ground state |J = 8, J z = ±8〉 is found. In an external magnetic field, this state is transformed to a spin-polarized state with 〈J z〉 ≈ 6.7. We emphasize the role of 5d–4f interorbital exchange polarization in modification of the 4f shell energy spectrum.


CHOICE OF THE DOUBLE-COUNTING
No unique choice of the double counting exists in DFT+U and DFT+ED calculations, and "physical" arguments prevail in the choice of W dc . In previously reported LDA+Hubbard I calculations (HIA) for the bulk Ho [1], the W dc was fixed to adjust the position of the first multiplet peak below E F , which was measured by experimental valence-band photoelectron spectroscopy (PES) [2].
Since no PES results exist for Ho@Pt, we assumed the transferability of W dc , and performed the DFT+ED calculations for the bulk Ho in the paramagnetic state (∆ ex =0 in Eq. (1) of the main text) making use of the experimental lattice parameter value a and the c/a ratio . The resulting densities of state (DOS) obtained in DFT+ED with "fully localized" limit (FLL) [3] and "around-mean-field" (AMF) [4] flavours for W dc are shown in Fig. S2. The FLL results are in striking disagreement with the experimental data [2]. DFT+ED-AMF gives the positions and the spectral shape of the occupied 4f -states in a reasonable agreement with the experiment (first multiplet peak at 3.3 eV below E F ), and a bit worse for the empty 4f -states (first multiplet peak at 3.8 eV above E F ). Comparison with experimental valence-band photoelectron spectroscopy (PES) is often taken as important criterion of truthfulness of electronic structure calculations. Thus, we conclude that AMF flavour is a better approximation, and places the 4f -level f in Eq. (1) correctly. Therefore, this form of W dc is used in the further calculations.
In Fig. S2C we show the total DOS per Ho atom for comparison with the results of [5]. There is a good agreement between these two calculations. The difference in the energy positions of the f -peaks is caused by a different choice of W dc . In [5], it was adjusted so that the position of the first multiplet peak below E F matched the experimental PES.  Table S2, in comparison with previously reported DFT+U results of [6,7]. Note that DFT+U results were obtained with the same Korringa-Kohn-Rostoker (KKR) method [8] and differ for a reason which is not completely clear. It is seen that the DFT+U results exceed substantially the experimental values for both M S and M L obtained in the XMCD experiments [9] shown in Table I of the main text. The use of DFT+HIA brings the results closer to the XMCD [9] data for both the spin M S and the orbital M L components of the total moment M J . No noticeable differences are found for Ho@Pt in f cc and hcp positions.
In the above equation τ α are the Pauli matrices, J α are the total moment operators (α = x, y, z), and g is the Lande factor. The t 1 tunnelling matrix element is proportional to the square of hybridization parameters V k shown in Table S1. The differential conductance G = ∂I ∂V is, [10] where, ∆ M M = M − M , P M is the statistical weight of the eigenstate |M , and G S = (g−1) 2 t 2 1 ρ sur ( F )ρ tip ( F ) proportional to the hybridization, and to the surface/tip ρ sur/tip DOS at the Fermi edge. V is an external voltage. The IETS spectrum for Ho adatom on Pt(111) in spin-polarized state (see Table S3) calculated using the model Eq. (S3) is shown in Fig. S3. It is seen that the calculation gives a shallow step at the voltage ±10 meV. Other step(s) occur at the higher energy range (over 40 meV).
The G S in Eq. (S3) is proportional to the square of hybridization strength t 1 multiplied by a prefactor (g −1) 2 where g is the Lande factor. Assuming that the most important contribution to the tunnelling comes from the hybridization occurring in the vicinity of E F , and taking into account that the t 1 ∼ |V k | 2 shown in Table  S1, we obtain the hybridization strength in the range of 12-23 meV. The prefactor G S is reduced further by (g − 1) 2 = 0.05 where the Lande factor g=1.23 for J = 8 many body state.

SCENARIO FOR MAGNETIC INSTABILITY OF HO@PT(111)
After the magnetic field is removed, the magnetization starts to evolve. We keep in mind that the 4f shell and 5d states can react differently, and that the latter is demagnetized faster than the former. This assumption is qualitatively consistent with recent experimental observation of a difference, by one order of magnitude, in the ultrafast magnetic moment dynamics of the localized 4f and itinerant 5d states in the RE metals [11].
We model this magnetic moment evolution in a simplified picture: we assume ∆ ex = 0 while keeping all other parameters in Eq. (1) of the main text the same as in the spin-polarized case. The energy splitting of the seventeen lowest many-body eigenvalues of Eq. (1) for this intermediate state are shown in Fig. S4. Importantly, the lowest eigenstate, |J = 8, J z = 0 , has zero moment and is almost degenerate (within 1 meV energy difference) with a doublet |J = 8, J z = ±6.98 . The zero-field splitting becomes negligibly small, meaning that the system can go in and out of the magnetic state at no energy cost, thus it becomes magnetically unstable. Since the transition to this intermediate state is connected with the "fast" demagnetization of the 5d shell, the Ho adatom becomes magnetically unstable before the f -shell "slowly" transforms to the final fully relaxed paramagnetic state.