Quantum engineering of spin and anisotropy in magnetic molecular junctions

Single molecule magnets and single spin centres can be individually addressed when coupled to contacts forming an electrical junction. To control and engineer the magnetism of quantum devices, it is necessary to quantify how the structural and chemical environment of the junction affects the spin centre. Metrics such as coordination number or symmetry provide a simple method to quantify the local environment, but neglect the many-body interactions of an impurity spin coupled to contacts. Here, we utilize a highly corrugated hexagonal boron nitride monolayer to mediate the coupling between a cobalt spin in CoHx (x=1,2) complexes and the metal contact. While hydrogen controls the total effective spin, the corrugation smoothly tunes the Kondo exchange interaction between the spin and the underlying metal. Using scanning tunnelling microscopy and spectroscopy together with numerical simulations, we quantitatively demonstrate how the Kondo exchange interaction mimics chemical tailoring and changes the magnetic anisotropy.

. Error estimation of the fit: A spin-1 CoH with the best Jρ 0 fit and two fits with Jρ 0 ± 10% of the best-fit value. To help quantify the error, we compute the mean square error of the best fit and imperfect fits. The best fit (red) has a mean square error value of 2.8 %, the fits with ± 10 % Jρ 0 have a mean square error of 5.2 % (green) and 11.6 % (blue).   Hubbard U and J values were taken from self-consistent calculations to be U −J = 3 eV. [16][17][18][19] In general the influence of U was found, as expected, to enhance local magnetism, however not changing the validity of any of the conclusions discussed in the present work (especially the attribution of CoH x complexes to either S = 1 or S = 1/2).
In terms of geometry most structural parameters were obtained self-consistently by min-  We describe the spin system with a phenomenological Hamiltonian, which is sufficient to fully explain the spectroscopic features observed in our scanning tunneling spectroscopy In this equation g is the gyromagnetic factor, µ B Bohr's magneton, D determines the axial anisotropy, and E the transverse anisotropy. B is the external applied magnetic field and S = (Ŝ x ,Ŝ y ,Ŝ z ) T the total spin operator with the components ( = 1): In the absence of a magnetic field the three eigenvectors |Ψ i and eigenenergies i of supplementary equation S1 are calculated in the m z basis to To calculate the tunneling spectrum we use a model based on the perturbative approach established by Appelbaum, Anderson, and Kondo [25][26][27][28] in which spin-flip scattering processes up to the 2nd order Born approximation are accounted for and which has been previously successfully used on quantum spin systems 4,29 . In this model the transition probability W i→f for an electron to tunnel between tip and sample and concomitantly changing the spin state of the CoH complex from its initial (i) to its final (f ) state is Here, M i→j are the matrix elements given by the Kondo-like interaction of the scattering electron |ϕ with the localized spin of the CoH complex |Ψ In this equation |ϕ i , Ψ i is the combined state vector of the localized spin and the interaction electron andσ = (σ x ,σ y ,σ z ) T is the total spin operator for the spin-1/2 electrons, withσ x,y,z as the standard Pauli matrices.
The first term in the supplementary equation S3 is responsible for the conductance steps observed in our spectra. While we assume zero field (B = 0) and no spin-polarization in the two electron reservoirs of tip and sample, the matrix elements are easily calculated to |M i→j | 2 = 0.5 for i = j and |M i→i | 2 = 0 otherwise. This leads at low temperature, i. e. k B T ε 2 , when only the ground state |Ψ 1 is occupied, to two increasing steps in the differential conductance dI/dV with identical amplitude at the energies ±ε 2 and ±ε 3 (dashed line in supplementary figure 2(b)): At the bias voltage where this process changes from being virtual to real, the denominator approaches zero which leads to a temperature broadened logarithmic divergence in the spectrum: The conductance σ 2 changes in a very particular fashion the observed spectra which is the sum of σ 1 and σ 2 : Additional peak-like structures arise at the energy 3 which allow us to determine Jρ 0 very precisely from fits of the supplementary equations S5 and S7 to the spectra measured at zero field.

GIES
Treating the quantum mechanical systemĤ (supplementary equation S1) not as a separated system but coupled to the dissipative bath of the substrate electrons we employ a Bloch-Redfield approach to account for the decay of excited states and coherences in the density matrix 32 . Interestingly, this approach leads for the off-diagonal elements of the reduced density matrix ofĤ not only to a fast decoherence but additionally to an energy shift of the eigenstates due to the interaction betweenĤ and the reservoir. We will restrict ourselves to the Kondo-like scattering between the substrate electrons and the localized spin, as higher excited state at ε 3 is more affected as the low lying state at ε 2 .
For the magnetic anisotropy parameters D and E of the CoH system the shift can be approximated to: with the coefficients α and β given by the integrals of supplementary equation S9.