Many-body effects in magnetic inelastic electron tunneling spectroscopy

Magnetic inelastic electron tunneling spectroscopy (IETS) shows sharp increases in conductance when a new conductance channel associated with a change in magnetic structure is open. Typically, the magnetic moment carried by an adsorbate can be changed by collision with a tunneling electron; in this process the spin of the electron can flip or not. A previous one-electron theory [Phys. Rev. Lett. 103, 176601 (2009)] successfully explained both the conductance thresholds and the magnitude of the conductance variation. The elastic spin flip of conduction electrons by a magnetic impurity leads to the well-known Kondo effect. In the present work, we compare the theoretical predictions for inelastic magnetic tunneling obtained with a one-electron approach and with a many-body theory including Kondo-like phenomena. We apply our theories to a singlet-triplet transition model system that contains most of the characteristics revealed in magnetic IETS. We use two self-consistent treatments (noncrossing approximation and self-consistent ladder approximation). We show that, although the one-electron limit is properly recovered, new intrinsic many-body features appear. In particular, sharp peaks appear close to the inelastic thresholds; these are not localized exactly at thresholds and could influence the determination of magnetic structures from IETS experiments. Analysis of the evolution with temperature reveals that these many-body features involve an energy scale different from that of the usual Kondo peaks. Indeed, the many-body features perdure at temperatures much larger than the one given by the Kondo energy scale of the system.

Figure 1

Projected density of states on the magnetic adsorbate electronic structure or spectral function $A$ as a function of the electron energy $\omega$ with respect to the Fermi energy of the substrate. The vertical dashed lines show the one-electron inelastic thresholds. The figure presents the adsorbate density of states in the absence of inelastic effects (marked as “No inelastic”), the many-body spectral function, and the one-electron conductance with inelastic effects. The system's temperature is 7 K.

Figure 2

Temperature dependence of the spectral function for an exchange coupling of the impurity's spins equal to 10 times the Kondo temperature, $I=10k_B{T}_K^0$, for the Anderson model, Eq. (8a). The energy axis is expressed in $T_K^0$ units, such that the inelastic thresholds appear at $\pm 10$. As the temperature is raised (from 0.2 $T_K^0$ to 2.4 $T_K^0$), the Kondo-like peaks located close to the inelastic thresholds are smeared and finally even the inelastic gap disappears. Dashed lines are the results for the one-electron theory, that accounts for the inelastic changes of the spectral function where all impurities are assumed to be initially in their singlet state.

Figure 3

Imaginary part of the $T$ matrix times the density of states $\rho$ for different temperatures in $T_K^0$ units. (a) SU(4) case for the spin 1/2 problem obtained when the exchange interaction between the two localized spins is set to zero, $I=0$. (b) The singlet-triplet excitation for a large value of the exchange integral, $I=256k_B{T}_K^0$.

Figure 4

Maximum value of the spectral functions of Figs. 2 and 3 as a function of temperature normalized to the value at $T\approx 0$. Black dots: Peak near the Fermi energy for $I=0$ [Fig. 3a]; mauve down triangles: peak near $\omega =-10k_B{T}_K^0$ and cyan diamonds: peak near $\omega =10k_B{T}_K^0$ for $I=10k_B{T}_K^0$ (Fig. 2); red squares: peak near $\omega =-256k_B{T}_K^0$ and green triangles: peak near $\omega =256k_B{T}_K^0$ for $I=256k_B{T}_K^0$ [Fig. 3b]. The dark blue line is an empirical fit to the results for $I=0$ that displays at $T=T_K^0$ a maximum value half of that at $T=0$. The normalization of the down-triangle curve has been changed above $T=6T_K^0$, due to the coalescence of the two inelastic Kondo-like peaks at high $T$ (see text for details).

Figure 5

Normalized spectral function as a function of the electron energy in units of the Kondo temperature $T_K^0$. Case (a) corresponds to an excitation energy $I=4k_B{T}_K^0$ as depicted by the vertical dashed line. Case (b) corresponds to the $I=32k_B{T}_K^0$ case. The curves correspond to orbital energies $\left|\epsilon \right|=110,220,440,800k_B{T}_K^0$ for the curves in increasing spectral function in the positive-energy part of the graphs. The topmost curve has been computed using the equivalent Coqblin-Schrieffer Hamiltonian, and hence corresponds to the $\epsilon \to -\infty$ case. The inset magnifies the negative-energy peaks of the $I=32k_B{T}_K^0$ case. The temperature is $T=0.1T_K^0$.

Figure 6

Shift of the excitation energy $\left|\Delta I\right|$ vs excitation energy $I$ (crosses). All results are in $T_K^0$ units. (Solid line) Fit of the numerical results to the function $\left|\Delta I\right|=\sqrt{a{x}^2+bx+c}$ where $x=I/ln(I/k_B{T}_K^0)$. Hence, for values of $I$ larger than the ones of the present graph, the threshold shift $\left|\Delta I\right|$ follows the asymptotic behavior $\sim I/ln(I/k_B{T}_K^0)$.