The Kondo coupling
Considering the experimental results, two important questions arise when we compare the ac-quired CoHS = 1spectra to the transition rates given by first order perturbations: Where do the intensity peaks at the outer excitation steps originate from and why are the magnetic anisotropy energies different for every system? The Kondo resonance for the CoH2S = 1/2introduced the second order contributions to the total transition rate and indeed, the peaks of the CoHS = 1 are related to the exact same perturbation.
The fits to the datasets in the CoH subsection were performed with a perturbation model that also includes second order contributions. The user-friendly Scilab code together with a fit routine that implements both orders with additional parameters such as possible Coulomb interaction was developed by Dr. Markus Ternes. While the familiar first order contributions only take direct transitions between real states into account:
Wf i0 = 2π
¯
h | hψ|fH1|ψii|2 δ(f −i) (3.27) we can also expand to second order contributions that are mediated by an intermediate state,
|ψim:
whereλ∝J ρ0 is later defined as the interaction strength of the spin system with the substrate.
Since in equation 3.28 two probabilities are multiplied, the resulting transition rate will be much smaller compared to the first order one. However, also virtual intermediate states lead to a possible transition from a real initial to a real final state of the spin system. Now, as the tunnel junction is strongly located in all dimensions we specifically consider the interferences between the first and second order terms for the overall transition probabilities, thus leading to additional third order terms when the fourth order terms are neglected:
Wf i ∝ |Mf i|2+ρ0JX where the matrix elementsM include the eigenstates of the S = 1system and the tunneling electrons. Not only the overall transition rate is enhanced but also the energies of the eigenstates are renormalized and can be approximated in terms of the MAE parameters,DandE, as:
D(J ρs)≈D0(1−α(J ρ0)2) E(J ρs)≈E0(1−β(J ρ0)2) (3.30)
whereαandβdepend on the energy bandwidth,ω0, of the Kondo exchange interactions and are derived from the integration over all possible final state energies as shown in [19]. We estimate ω0between 0.4 eV and 1.2 eV.
-15 -10 -5 0 5 10 -10 -5 0 5 10 15
dI/ dV/ arb. units
Bias voltage / meV Bias voltage / meV
0 1
Figure 3.20: Third order contributions for a CoHS = 1system withD =−5meV andE = 1 meV at 1.1 K. Left: Contribution to the differential conductance (grey shaded area) only using second order terms. Right: The matrix elements of the third order calculation (red shaded area) with J ρ0 = −0.1 add up to the second order matrix elements and lead to the characteristic peaks at the second excitation step around -6 meV and +6 meV. Note that the energy positions of the two steps have slightly shifted to smaller energies.
With the magnetic anisotropy being responsible for the eigenstate energies of the spin sys-tem, the third order contributions due to the Kondo exchange with the substrate electrons can fine-tune these energies. An analogy with an harmonic oscillator helps to clarify the energy renormalization [79]. When a free harmonic oscillator with mass, m, and spring constant k, is displaced from its equilibrium position, it will oscillate with its well known eigenfrequency, f0 ∝p
(k/m). If this motion is subject to friction such as oscillating in air, the eigenfrequency shifts to lower energies and the amplitude gets damped. A microscopic picture that takes every air molecule and its interaction with other molecules as well as the rough oscillator’s surface area into account would be far too complicated to compute. Thus, this damping is described with a general dissipative bath that leads to an energy renormalization. In quantum mechanical systems, these energy corrections can be quantified by a broadening and a shift in energy of the system’s eigenstates. In our case, the CoHS = 1 eigenstates that are subject to an interaction with the electron bath of the substrate will experience this broadening and energy shift as shown in figures 3.20, 3.21 and 3.22.
-15 -10 -5 0 5 10 -10 -5 0 5 10 15
dI/ dV/ arb. units
Bias voltage / meV Bias voltage / meV
0 1
Figure 3.21: Simulation of increasing the Kondo coupling fromJ ρs= 0 (lower black curve) to a realistic value ofJ ρs= -0.2 for a CoHS = 1system withD= −5meV andE = 1meV at 2 K. Not only the overall differential conductance increases together with the Kondo coupling strength but also the step positions shift in energy. A derivative of the curves highlights the energy shifts: The original system’s outer energy step position was at -6 meV according to table 3.2 for the example spin system. A Kondo exchange coupling of J ρ0 = -0.2 has shifted this energy position to -5.81 meV. The outer step is shifted to lower energies (in this case by around 3%).
-15 -10 -5 0 5 10 15
dI/ dV/ arb. units
Bias voltage / meV
Figure 3.22: Three experimental spectra and their third order fits. The higher the coupling to the substrate the smaller the magnetic anisotropy energies: From low coupling ofJ ρs= -0.11 andD= -6.58 meV (blue curve), to an intermediate coupling ofJ ρs= -0.15 andD= -4.66 meV (red curve) to a strong couplingJ ρs= -0.27 andD= -4.13 meV (black curve).
- 1 5 - 1 0 - 5 0 5 1 0 1 5
dI / dV / arb. units
B i a s v o l t a g e / m e V
Figure 3.23: Co atoms on h-BN/Rh(111) at 1.3 K with S = 3/2 signatures. Similarly to the results of Oberg et al. presented in figure 3.24, the intensity of the Kondo resonance is correlated with the energy position of the step excitations. The blue spectrum shows rather large excitation energies around|VB|= 10 meV and a low intensity Kondo resonance, whereas the black spectrum shows small excitation energies around|VB|= 4 meV and a high intensity Kondo resonance. Spectra are offset for better visualization.