Quantum Engineering of Spin and Anisotropy in Magnetic Molecular Junctions

Quantum Engineering of Spin and Anisotropy in Magnetic Molecular Junctions

Peter Jacobson p.jacobson@fkf.mpg.de Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany    Tobias Herden Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany    Matthias Muenks Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany    Gennadii Laskin Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany    Oleg Brovko Max Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle, Germany    Valeri Stepanyuk Max Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle, Germany    Markus Ternes m.ternes@fkf.mpg.de Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany    Klaus Kern Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany Institute de Physique de la Matière Condensée, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
Abstract

Single molecule magnets and single spin centers can be individually addressed when coupled to contacts forming an electrical junction. In order 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 center Bogani and Wernsdorfer (2008); Gatteschi and Sessoli (2003); Rau et al. (2014); Miyamachi et al. (2013); Wegner et al. (2009); Heinrich et al. (2013). 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 when coupled to contacts Oberg et al. (2014). Here, we utilize a highly corrugated hexagonal boron nitride () monolayer Laskowski et al. (2007); Herden et al. (2014) to mediate the coupling between a cobalt spin in () complexes and the metal contact. While the hydrogen atoms control the total effective spin, the corrugation is found to smoothly tune the Kondo exchange interaction between the spin and the underlying metal. Using scanning tunneling microscopy and spectroscopy together with numerical simulations, we quantitatively demonstrate how the Kondo exchange interaction mimics chemical tailoring and changes the magnetic anisotropy.

P. J. and T. H. contributed equally.

preprint: CoH/BN/Rh(111) v1.0

Magnetic anisotropy defines the stability of a spin in a preferred direction Gatteschi and Sessoli (2003). For adatoms on surfaces, the low coordination number and changes in hybridization can lead to dramatic enhancement of magnetic anisotropy Rau et al. (2014); Gambardella et al. (2003). Surface adsorption site and the presence of hydrogen has been shown to alter the magnetic anisotropy of adatoms on bare and graphene covered Khajetoorians et al. (2013); Dubout et al. (2015); Donati et al. (2013). Furthermore, the exchange interaction and strain has been invoked for adatoms on islands where the adatom position on the island affects the observed magnetic anisotropy Oberg et al. (2014); Bryant et al. (2013). Studies on single molecule magnets (SMMs) containing or spin centers have revealed that chemical changes to the ligands surrounding the spin affect the magnetic anisotropy Jurca et al. (2011). However, the most important factor for maintaining magnetic anisotropy in SMMs is a low coordination number and a high axial symmetry Miyamachi et al. (2013); Zadrozny et al. (2013); Ungur et al. (2014).

Magnetic anisotropy is not guaranteed in SMMs or single spin centers upon coupling to contacts. The spin interacts with the electron bath through the exchange interaction leading to a finite state lifetime and the decay of quantum coherence Cohen-Tannoudji et al. (2008); Delgado et al. (2014). Additionally, the scattering of the spin with the electron bath results in an energy renormalization of the spin’s eigenstate energy levels, similar to the case of a damped harmonic oscillator Cohen-Tannoudji et al. (2008). In practice, this leads to a net reduction of the magnetic anisotropy, pushing the system closer to a Kondo state. At the heart of the Kondo effect are spin–flip scattering processes between localized states at the impurity spin and delocalized states in the bulk conduction band, resulting in the formation of a correlated quantum state Hewson (1993). The Kondo regime is reached when the magnetic moment of the impurity spin is screened by the electron bath, with the exchange interaction defining the relevant energy scale, the Kondo temperature (). High spin systems with a total spin have the potential for both magnetic anisotropy and the Kondo effect Misiorny et al. (2012); Žitko et al. (2008). Thus, the Kondo exchange interaction with the electron bath can force the impurity spin into a competing Kondo state, where antiferromagnetic coupling with the reservoir reduces or even quenches the magnetic moment. The outcome of this competition can be determined in local transport measurements, but few quantitative measures of this competition exist.

Here, we study complexes coupled to a spatially varying template, the moiré, to observe and model how the environment influences magnetic anisotropy. The monolayer, a wide band gap two dimensional material, decouples and mediates the interactions between and the underlying metal while lattice mismatch leads to a spatial corrugation resulting in an enlarged unit cell with 3.2 nm periodicity corresponding to 13 units on top of 12 atoms Laskowski et al. (2007). The local adsorption configuration of on the is conserved across the moiré unit cell, with the large number of inequivalent adsorption sites allowing us to explore how hybridization affects magnetic anisotropy. To complement our experimental observations, we model transport through the complexes using Hamiltonians that incorporate magnetic anisotropy as well as coupling to the environment. This is accomplished by parameterizing the environment through use of a dimensionless coupling constant , describing the strength of the Kondo exchange interaction, , between the localized spin and the electron density of the substrate near the Fermi level (see SI).

Figure 1: CoH adsorbed on a h–BN/Rh(111) surface. (a) Constant current STM topography with three CoH complexes (protrusions) adsorbed on different sites ( image size, , , ). High symmetry points of the moiré are marked by the white overlay. (b) Sketch of the atom positions for the adsorption of CoH. The -BN registry with Rh(111) shifts across the moiré unit cell with three high symmetry sites: at the valley site (v) the Rh is directly underneath the N, whereas for the two unequal rim sites (r and r) changes in the registry and distance to the surface are observed. (c) Line profiles along the dashed lines indicated in (a) show two CoH systems with adsorption sites r and v (red line) and a -BN reference cut (blue line, offset by ). (d) Differential conductance curves versus bias voltage of three different CoH systems (stabilization setpoint: , , , curves vertically offset for clarity). The upper curve (grey) shows a spin– Kondo resonance centered at zero bias. The two lower curves (red and blue) show step-like conductance increases symmetric around zero bias indicating a spin–1 system. Solid black lines are least-square fits using a perturbative transport model.

Figure 1a shows a representative scanning tunneling microscopy (STM) topograph of the moiré with isolated () complexes, line profiles across the indicate can adsorb at multiple positions within the moiré (Figure 1b) Natterer et al. (2012). On these complexes we measure the differential conductance, , against the applied bias voltage between tip and sample at low-temperature () and zero magnetic field (, details see Methods). The spectra can be divided into two broad classes: a sharp peak centered at zero bias or two symmetric steps of increasing conductance at well-defined threshold energies (Figure 1c). The peak at zero bias is consistent with a spin– Kondo resonance while the steps correspond to the onset of inelastic excitations from the magnetic ground state to excited states. The observation of two steps hints at a spin–1 system with zero field splitting. The two lower spectra (Figure 1c; red, blue curves) are measured on CoH at different parts of the moiré and share the same characteristics but the step positions vary.

Figure 2: CoH and CoH density of states. (a) Ball and stick model of the adsorption of CoH on -BN. The linear adsorption geometry of CoH on the N atom is emphasized and marks the main (axial) magnetic anisotropy () along the -axis. Additional transverse anisotropy () in the plane further breaks the symmetry. (b) Schematic linear crystal field splitting diagram for the shell of Co highlighting the origin of the axial () and transverse () magnetic anisotropy. The magnetic ground state is an antisymmetric superposition of and states ( is the magnetic moment in units of the reduced Planck constant in -direction), the first excited state is the symmetric superposition, and the second excited state is . (c) Plots of the majority and minority spin projected density of states (PDOS) for CoH and CoH. The difference in majority and minority spin spectral weights indicate that CoH has a total spin and CoH has . (d) Plot of the asymmetry between majority and minority PDOS for CoH adsorbed on N at the r (left) and v (right) high symmetry points.

We employ density functional theory (DFT) to correlate the magnetic properties of with the local adsorption configuration. Our calculations (see Methods) show that adsorption in the BN hexagon, i.e. hollow site, is preferable for bare Co. The addition of hydrogen shifts the preferred adsorption site to N, with the hollow site adsorption energy consistently higher. For CoH complexes the preferred hydrogen position was found to be either exactly on top of Co or tilted towards the nearest atom (Figure 2a). An important consequence of the N adsorption site is the linear crystal field acting on the cobalt (i.e. ) removing the 5-fold degeneracy of the -levels (Figure 2b).

In Figure 2c the spin–resolved, symmetry decomposed local density of states of CoH and CoH adsorbed in the valley is plotted. The atomic -levels are split roughly by the intrinsic Stoner exchange giving a bare Co adatom a magnetic moment of 2.2 Bohr magnetons (). Formation of CoH leads to hybridization of the H orbitals and the Co orbitals, slightly reducing the magnetic moment to , equivalent to a configuration (Figure 2b). The second hydrogen changes the picture significantly, with the hybridization sufficient to bring the Co -levels closer together, reducing the magnetic moment to resulting in a configuration. Therefore, from our spectroscopic observations and DFT calculations we identify CoH as an effective spin–1 and CoH as spin– system.

Figure 2d shows the spin density distribution for CoH in a N adsorption configuration at two high symmetry points of the moiré. The strong vertical bond between Co and N leads to an effective spin–polarization along this axis and can be expected to provide the system with out-of-plane magnetic anisotropy. Tilting of the hydrogen and the underlying lattice mismatch reduces the C symmetry and introduces small shifts in the , levels producing a non-negligible in-plane component of the anisotropy.

To model the experimentally observed tunneling spectra and to determine the magnetic anisotropy we use a phenomenological spin Hamiltonian including the Zeeman energy and magnetic anisotropy:

(1)

with as the gyromagnetic factor, the magnetic field, the total spin operator, and and as the axial and transverse magnetic anisotropy Oberg et al. (2014); Bryant et al. (2013); Otte et al. (2008); Hirjibehedin et al. (2007); Lorente and Gauyacq (2009). Transport through the junction is calculated using a Kondo-like interaction between the tunneling electrons and the localized spin system, with as the standard Pauli matrices. We account for scattering up to order in the matrix elements by considering additional exchange processes between the localized spin and substrate electrons of the form Zhang et al. (2013) (see SI):

(2)
Figure 3: Magnetic field behavior of CoH and CoH. (a) Left: Zeeman splitting of the spin– states of a CoH complex in magnetic field. Dots mark the energy differences as determined by least-square fits of the perturbation model to the experimental data in (b). The regression line corresponds to a gyromagnetic factor . Right: Sketch of the CoH complex adsorbed on a N site. (b) Evolution of the differential conductance of a CoH complex in an external magnetic field normal to the surface ( and ; ). (c) Simulated spectra using a order perturbation model and a constant coupling to the substrate of and . (d) Left: Sketch of the spin–1 CoH complex adsorbed on a N site. Right: State energy evolution in magnetic field along the out-of-plane anisotropy axis. Dots mark the experimentally determined step positions, full lines are the calculated eigenstate energies of the model Hamiltonian (see text) using magnetic anisotropy parameters of , , and . (e) Evolution of the differential conductance of a CoH system in an external magnetic field normal to the surface ( and ; ). (f) Simulated spectra using the parameter from (d) and . The spectrum is shown together with a order perturbation theory model, i.e. (dashed line), to highlight the necessity of order contributions. Curves in (b, c) and (e, f) are shifted vertically for better visibility.

To confirm the magnetic origin of the spectroscopic features in CoH and CoH, we measure the differential conductance at magnetic fields up to normal to the surface. Figure 3b shows experimental spectra taken over one CoH complex and Figure 3c the model calculations for the Kondo resonance. Applying an external magnetic field introduces Zeeman splitting to the spin– system (Figure 3a). At low magnetic fields, , the peak broadens and the differential conductance of the resonance is reduced. Increasing the field to , a clear splitting of the Kondo resonance is observed. For the highest fields, the degeneracy of the spin– state is effectively lifted, resulting in a strong reduction of the Kondo resonance and the appearance of an inelastic excitation gap. We can reproduce the peak and its splitting by our perturbative model (Figure 3c) even though at high fields the peak-like conductance is weaker in the experimental data than expected from the model calculation. This indicates that the Kondo temperature of the system lies close to the base temperature of our experiment.

Increasing the external magnetic field has two effects on the spin–1 CoH; Zeeman splitting separates the steps and the ratio between inner and outer conductance step height decreases (Figure 3e). At zero field, the ground and first excited states are a superposition of and states, applying a magnetic field reduces the spin mixing and leads towards a ground and excited state. This accounts for the reduction of the inner step with increased magnetic field, as the transition between ground and first excited state becomes less probable because it would require a change in of two. Reverting to a purely 2nd order simulation, large deviations are observed at both steps, evidence that coupling of the spin to the substrate conduction electron bath must be considered (Figure 3f, dashed line). The experimental data fits excellently when including 3rd order terms, i.e. assuming a finite , an out-of-plane anisotropy axis, and .

Figure 4: Influence of environmental coupling on spectra. (a) Experimentally determined and (red and blue dots) parameters plotted versus the coupling strengths . Full lines show the expected renormalization of and due to virtual coherences calculated with a Bloch-Redfield approach taking exchange scattering with the dissipative substrate electron bath into account. Shaded region shows the experimental uncertainty. (b) Computed differential conductance for different coupling strengths between the localized spin and the electrons of the substrate ranging from to . At stronger couplings () an increase of the outer step’s shoulder is expected concomitant with a reduction of the energy position of the outer step. This is equivalent to a reduced anisotropy energy . (c) Schematic diagram showing the effect of exchange. When the exchange coupling, , between the local spin and the conduction electron bath is weak, a large magnetic anisotropy, , is observed (top). As exchange coupling to the substrate strengthens, the magnetic anisotropy is reduced driving the system closer to a Kondo state (bottom).

Evaluation of more than 30 CoH shows no sharp distribution of the anisotropy parameters and . A transition of the main anisotropy axis into the surface plane occurs when , therefore we have only considered complexes with a clear out-of-plane anisotropy determined by the criterion ; a representative spectrum with in-plane anisotropy is shown in the SI. By considering the values of from our fits, we observe a correlation between the magnetic anisotropy and coupling with the substrate, . The red branch in Figure 4a shows that as the substrate coupling increases, the axial magnetic anisotropy decreases. These results are in line with predictions that increased coupling shifts energy levels. The solid red line shows the best fit to our data and follows the trend , where is a constant describing the bandwidth of the Kondo exchange interaction. The shaded red region accounts for the possible range of by considering an effective bandwidth of (see SI). For the variation in magnetic anisotropy fits exceptionally well, but for small values of , some spread in the axial anisotropy is observed. These fluctuations are not accounted for in our model and indicate that for small additional factors such as strain or defects may contribute to the magnetic anisotropy. While the axial anisotropy shows clear dispersion, the transverse anisotropy is essentially constant (Figure 4a, blue).

Figure 4b shows the influence of on the tunneling spectra calculated using a Bloch-Redfield approach to incorporate virtual correlations between the ground and excited states due to the coupling with the dissipative spin bath in the substrate assuming a flat density of states and an effective bandwidth of (see SI) Oberg et al. (2014); Cohen-Tannoudji et al. (2008); Delgado et al. (2014). As is increased, virtual correlations lead to renormalization and reduce the level splitting. This is observed experimentally as a reduction of the axial magnetic anisotropy. Furthermore, higher order scattering processes in the tunneling influence the conductance leading to an enhanced shoulder at the outer energy step that changes the contours of the spectrum (see SI). The symmetric peaks shift towards zero bias as increases indicating that correlations drive the anisotropic spin–1 system closer to the Kondo state. Figure 4c schematically depicts the observed trend, when the spin is weakly coupled to the conduction electrons the magnetic anisotropy is stabilized. Increasing the exchange interaction introduces correlations between the excited spin states and the conduction electrons, leading to a net reduction in the magnetic anisotropy.

In conclusion, our results show that the Kondo exchange interaction modulates the magnetic anisotropy of single spin CoH complexes. The role of exchange was quantitatively determined by exploiting the corrugated moiré structure. In conjunction with 3rd order perturbation theory simulations, we extracted the precise values of the spin coupling to the environment and its influence on the magnetic anisotropy. Kondo exchange must be considered an additional degree of freedom – beyond local symmetry, coordination number, and spin state – for spins connected to contacts. This parameter is non-local and therefore expected to be discernable at surfaces, in junctions, and perhaps in bulk SMM materials.

Acknowledgements

P.J. acknowledges support from the Alexander von Humboldt Foundation. T.H., M.M. and M.T. acknowledge support by the SFB 767. O.B. and V.S. acknowledge support by the SFB 762.

Author contributions

M.T. and K.K. conceived the experiments. T.H., P.J., M.M., and G.L. performed the STM measurements. P.J. and T.H. analyzed the data using a perturbation theory simulation package developed by M.T. O.B. and V.S. performed first principles density functional theory calculations. P.J., M.T., T.H., and O.B. drafted the manuscript; all authors discussed the results and contributed to the manuscript.

Methods

The surface was prepared by multiple cycles of argon ion sputtering and annealing to . On the final annealing cycle borazine () was introduced at a pressure of for 2 minutes resulting in a monolayer film. Cobalt was deposited onto a cold, , surface via an electron beam evaporator.

Scanning tunneling experiments were performed on a home-built STM/AFM in ultra-high vacuum with a base temperature of and magnetic fields up to . All spectroscopic () measurements presented were obtained with an external lock-in amplifier and a modulation voltage of applied to the bias voltage at a frequency of . The tunneling setpoint before the feedback loop was disabled was and . For measurements on the same adatoms in different external magnetic fields the tip was retracted while the field was ramped and allowed to settle for maximum stability.

First principles calculations have been carried out in the framework of the density functional theory (DFT) as implemented in the VASP code Kresse and Hafner (1993); Kresse and Furthmüller (1996). We use the projector augmented-wave technique Blöchl (1994) where the exchange and correlation were treated with the gradient-corrected PBE functional as formalized by Perdew, Burke and Ernzerhof Perdew et al. (1996). Hubbard and values were taken from self-consistent calculations and fitting to experiments to be Steiner et al. (1992); Osterwalder (2001); Wehling et al. (2010). Full details are presented in the Supplementary Information.

References

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