Corresponding author

Markus Ternes

Max Planck Institute for Solid State Research

References

1.
Kondo, J.
Resistance minimum in dilute magnetic alloys
2.
Hewson, A.C.
The Kondo Problem to Heavy Fermions
3.
Appelbaum, J.A.
Exchange model of zero-bias tunneling anomalies

Department "Nanoscale Science"

Temperature and magnetic field dependence of a Kondo system in the weak coupling regime

Authors

Y. Zhang, S. Kahle, T. Herden, U. Schlickum, M. Ternes, P. Wahl, and K. Kern

Departments

Nanoscale Science (Klaus Kern)

The Kondo effect arises when a localized spin interacts with the electrons of the surrounding host. Scanning tunneling spectroscopy on individual magnetic impurities have renewed the interest in Kondo physics, however, a quantitative comparison with theoretical predictions remained challenging. Here we studied the zero-bias anomaly on an organic radical weakly coupled to a Au(111) surface which we where able to describe with astonishing agreement by perturbation theory as originally developed by Kondo 60 years ago. Our results demonstrate that Kondo physics can only be fully conceived by studying both temperature and magnetic field dependence of the resonance.

The Kondo effect, discovered originally in dilute magnetic alloys [1], emerges ubiquitously in seemingly unrelated contexts, such as the zero-bias anomalies observed in quantum dots and nanowires or the dynamical behavior close to a Mott transition [2]. The simplicity of the underlying model Hamiltonian – a single spin coupled by an exchange interaction to a bath of conduction electrons – contrasts the complex physics that only the development of a completely new theoretical understanding clarified [2]. The Kondo effect is usually considered for an antiferromagnetic (AFM) interaction between a localized spin and an itinerant spin bath with a spin-spin exchange coupling 0. This interaction leads in the strong coupling regime, that is, at temperatures below a characteristic Kondo temperature, TK, to a screening of the impurity magnetic moment and results in a stable, non-magnetic singlet ground state.

<p><strong>Fig. 1:</strong> The different regimes of the Kondo effect and the studied organic radical molecule: (a) For ferromagnetic coupling or at temperatures <em>T</em>&raquo;<em>T</em><sub>K</sub> , the system is in the weak coupling regime, which can be treated perturbatively. At temperatures <em>T</em>&lt;<em>T</em><sub>K</sub> and antiferromagnetic coupling the exchange interaction leads to entangled many-body state effectively screening the impurity spin. (b) Chemical structure of the molecule (C<sub>28</sub>H<sub>25</sub>O<sub>2</sub>N<sub>4</sub>). (c) High-resolution topography of one molecule. Contour lines are at height intervals of 50pm.</p> Zoom Image

Fig. 1: The different regimes of the Kondo effect and the studied organic radical molecule: (a) For ferromagnetic coupling or at temperatures T»TK , the system is in the weak coupling regime, which can be treated perturbatively. At temperatures T<TK and antiferromagnetic coupling the exchange interaction leads to entangled many-body state effectively screening the impurity spin. (b) Chemical structure of the molecule (C28H25O2N4). (c) High-resolution topography of one molecule. Contour lines are at height intervals of 50pm.

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Much less attention has been paid to the weak coupling regime, which is relevant for ferromagnetic (FM) interaction (J>0) or at elevated temperatures (T»TK ) (Fig. 1(a)). For AFM interactions and high temperatures, thermal fluctuations destroy the singlet state. For both cases, FM interaction and AFM interaction in the weak coupling regime, the physics can be described by perturbation theory. In past studies of single Kondo adsorbates by scanning tunneling spectroscopy, a detailed quantitative characterization of the Kondo physics was hampered by orbital degeneracies, spin quantum numbers >1/2, and rather high Kondo temperatures.

Here, we study a purely organic molecule, which has a radical nitronyl-nitroxide side group, adsorbed on a Au (111) surface. Instead of being localized on a specific atom, the unpaired electron is spatially delocalized over the O–N–C–N–O part of the side group (Fig. 1(b)) stabilizing it against chemical reaction and charge transfer, which would lead to a spin zero system. The unpaired electron has no further orbital degeneracy. This enables us to study the physics of a pure spin 1/2 interacting with conduction electrons, which has been the subject of intense theoretical research.

<strong>Fig 2:</strong> Temperature dependence of the zero-bias anomaly: (a) Typical differential conductance spectra taken on the radical side group of the molecule (black) and simulated spectra using perturbation theory (red). All spectra are normalized and offset for visual clarity. (b) Two exemplary spectra (black dots) and the conductance obtained from the model (red line) plotted with logarithmic abscissa. (c) Plot of the effective temperature <em>T</em><sub>eff</sub> versus the experimental temperature. Zoom Image
Fig 2: Temperature dependence of the zero-bias anomaly: (a) Typical differential conductance spectra taken on the radical side group of the molecule (black) and simulated spectra using perturbation theory (red). All spectra are normalized and offset for visual clarity. (b) Two exemplary spectra (black dots) and the conductance obtained from the model (red line) plotted with logarithmic abscissa. (c) Plot of the effective temperature Teff versus the experimental temperature. [less]

Constant-current scanning tunneling microscopy (STM) images acquired at low temperatures show the elongated molecular backbone and the radical side group (Fig. 1(c)). We detect a strong resonance at the Fermi level when measuring the differential conductance on the side group of the molecule (Fig. 2(a)), which is found neither on the backbone of the molecule nor on the clean surface. The resonance shows clear logarithmic voltage dependence over almost two orders of magnitude (Fig. 2(b)) and broadens significantly at increased temperatures – much more than expected for a single-particle resonance. The apparent width of the resonance does not converge to a natural width in the low temperature limit within the temperature range accessible to us, as one would expect for a spin 1/2 Kondo system with AFM interaction in the strong coupling limit at T«TK . The spectra rather have a flat top whose width scales with the thermal energy.

We compare our data with the conductance calculated from the Kondo spin-flip scattering Hamiltonian in a perturbative approach accounting for processes up to third order in the exchange interaction J. In this Anderson-Appelbaum model [3], tunneling between two electrodes via a magnetic impurity leads at zero magnetic field to a temperature-broadened logarithmic resonance in the conductance. We perform least-squares fits, taking the temperature as the only relevant parameter. We denote the temperature obtained from the fits Teff. Figure 2(a),(b) show the excellent agreement between the model and our data. The extracted effective temperatures, Teff, agree well with the temperature T of the experiment (Fig. 2(c)).

<strong>Fig. 3: </strong>Splitting of the resonance in magnetic field:(a) Differential conductance measurements (black) and simulated spectra (red) for successively increased magnetic fields on the radical side group of the molecule. All spectra are normalized and offset for clarity. The blue curve at <em>B</em>=14T shows exemplarily the contribution of the second order in <em>J</em> to the differential conductance. (b) Extracted Zeeman splitting as a function of magnetic field. The red line is a linear fit. (c) Magnetic coupling <em>J&rho;</em><sub>0 </sub>obtained from the ratio between second and third order contributions and from the Zeeman splitting. Zoom Image
Fig. 3: Splitting of the resonance in magnetic field:(a) Differential conductance measurements (black) and simulated spectra (red) for successively increased magnetic fields on the radical side group of the molecule. All spectra are normalized and offset for clarity. The blue curve at B=14T shows exemplarily the contribution of the second order in J to the differential conductance. (b) Extracted Zeeman splitting as a function of magnetic field. The red line is a linear fit. (c) Magnetic coupling 0 obtained from the ratio between second and third order contributions and from the Zeeman splitting. [less]

When we apply a magnetic field perpendicular to the Au (111) surface, the resonance splits into two peaks with superimposed symmetric steps as soon as the Zeeman energy exceeds the thermal energy (Fig. 3(a)). The steps in the differential conductance are due to inelastic spin-flip excitations and can be described with excellent agreement using the same perturbative approach. The perturbation theory not only accurately accounts for the steps, but also for the peaks on top of these steps. The energies of the peak positions and the spin-flip excitations as extracted from the fits scale linearly with the applied magnetic field and yield a Landé factor which slightly differs from the expected value for a free electron. This difference provides an estimate for the exchange interaction 0: due to the coupling of the localized spin with the conduction electrons and their polarization by the magnetic field, the effective Landé factor will be modified compared to the value for the free spin. A second independent estimate of 0 can be arrived by comparing the amplitudes of the different scattering orders in the model: The height of the logarithmic peaks is proportional to (0)3, while the amplitude of the steps due to spin-flip excitations is proportional to (0)2. We find with both approaches 0=0.04±0.02 (Fig. 3(c)). Our fits indicate a weak AFM exchange interaction between the localized spin on the molecule and the Au surface.

We conclude that the resonance we observe is due to a single unpaired spin with weak AFM coupling to the conduction band. Almost 60 years after Kondo identified the correct Hamiltonian for a single magnetic impurity in a metallic host and showed the breakdown of perturbation theory on a characteristic energy scale, our results demonstrate its validity for a single spin ½ impurity in the weak coupling regime, which is characterized by an almost universal temperature and magnetic field dependence. Both differ substantially from the behavior predicted for a spin 1/2 AFM Kondo model in the strong coupling limit.

The identification of a true spin 1/2 system allows to perform quantitative tests of theoretical predictions, which can be tested in measurements performed at even lower temperatures. Besides enabling studies of spin 1/2 Kondo physics on a single impurity, experiments on systems of coupled impurities are conceivable. Here, we note that most theoretical studies of Kondo lattices are built on top of spin 1/2 Kondo impurities. Tailoring of the backbone in the synthesis of the molecules facilitates self-assembled growth of ordered one- or two-dimensional lattices of these molecules. This potentially allows for experimental studies of model Hamiltonians for correlated electron materials.


 
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