Orbital Kondo effect in Cobalt-Benzene sandwich molecules
We study a Co-benzene sandwich molecule bridging the tips of a Cu nanocontact as a realistic model of correlated molecular transport. To this end we employ a recently developed method for calculating the correlated electronic structure and transport properties of nanoscopic conductors. When the molecule is slightly compressed by the tips of the nanocontact the dynamic correlations originating from the strongly interacting Co shell give rise to an orbital Kondo effect while the usual spin Kondo effect is suppressed due to Hund’s rule coupling. This non-trivial Kondo effect produces a sharp and temperature-dependent Abrikosov-Suhl resonance in the spectral function at the Fermi level and a corresponding Fano line shape in the low bias conductance.
pacs:73.63.Rt, 71.27.+a, 72.15.Qm, 31.15.A–
The discovery of ferrocene and bisbenzene chromium Kealy and Pauson (1951); ? over half a century ago was the starting point for experimental and theoretical work both in chemistry and physics concerning the intricate properties of organometallic compounds. The investigation of these and related sandwich complexes is driven by their relevance in various chemical applications (e.g. catalysis) and more recently also because of their prospective nanotechnological applications for example as molecular magnets Gatteschi et al. (2006) or spintronic devices Xiang et al. (2006); ?.
Molecules with a transition metal (TM) center coupled to aromatic groups are also of high interest from a fundamental point of view. The strong electronic correlations in the shell of the TM can modify the ground state and electronic transport properties of such molecules, leading to many-body phenomena like the Kondo effect Hewson (1997) as recently observed in TM-phthalocyanine molecules Zhao et al. (2005); ?. The importance of dynamical correlations in nanoscopic devices in general is further substantiated by the recent observation of the Kondo effect in nanocontacts made from TMs Calvo et al. (2009); Jacob et al. (2009). Also recently a so-called underscreened Kondo effect has been reported for a molecule trapped in a breakjunction Parks et al. (2010). While the Kondo effect is commonly associated with the screening of a local magnetic moment by the conduction electrons, it is also possible that another internal degree of freedom associated with a degeneracy gives rise to a Kondo effect Cox and Zawadowski (1998). One example is the so-called orbital Kondo effect where the pseudo-spin arising from an orbital degeneracy is screened by the conduction electrons Kolesnychenko et al. (2002); ?.
Such complex many-body phenomena call for a theoretical description beyond the standard treatment with Kohn-Sham (KS) density functional theory (DFT) which cannot capture many-body physics beyond the effective mean-field picture. The incorrect behavior of the KS DFT for strongly correlated systems can be remedied by augmenting the DFT with a local Hubbard-like interaction. This DFT++ approach is the de facto standard in the theory of solids Kotliar et al. (2006); ?. Recently this approach has been adapted to the case of nanoscopic conductors Jacob et al. (2009); Jacob and Kotliar (2010); ?; Lucignano et al. (2009); ?; ?; ?.
In this letter we apply this method to investigate the transport properties of cobalt benzene sandwich molecules (CH)Co(CH) (CoBz in the following). We find that dynamical correlations in the Co shell give rise to an orbital Kondo effect in a doubly degenerate level of the Co shell while the usual spin Kondo effect is suppressed due to Hund’s rule coupling.
We perform DFT calculations using the CRYSTAL06 code Dovesi et al. () employing the LDA Kohn and Sham (1965), PW91 Perdew and Wang (1986) and the hybrid functional B3LYP Becke (1993), and using the all electron Gaussian 6-31G basis set. The geometries of the wires were relaxed beforehand and kept fixed during the calculations. The geometry of the molecule in contact with the wires was relaxed employing the B3LYP functional geo ().
In order to capture many-body effects beyond the DFT level, we have applied the DFT+OCA (Density Functional Theory + One-Crossing Approximation) for nanoscopic conductors developed by one of us in earlier work Jacob et al. (2009); Jacob and Kotliar (2010). To this end the system is divided into three parts: The two semi-infinite leads L and R and the device region D containing the molecule and part of the leads. The KS Green’s function (GF) of region D can now be obtained from the DFT electronic structure as where is the KS Hamiltonian of region D and are the so-called lead self-energies which describe the coupling of D to L and R and which are obtained from the DFT electronic structure of the nanowire leads.
Next the mean-field KS Hamiltonian is augmented by a Hubbard-like interaction term that accounts for the strongly interacting electrons of the Co shell. Here we use a simplified interaction which only takes into account the direct Coulomb repulsion and the Hund’s rule coupling . These are different from the bare interactions due to screening processes. Here we assume that the interactions are somewhat increased as compared to their bulk values since the screening should be weaker than in bulk. Hence we use eV and eV; indeed we find that our results are qualitatively stable for a reasonable range of values for and (see below).
The interacting Co shell coupled to the rest of the system (benzene+leads) constitutes a so-called Anderson impurity model (AIM). The AIM is completely defined by the interaction parameters and , the energy levels of the orbitals and the so-called hybridization function . The latter describes the (dynamic) coupling of the Co -shell to the rest of the system and can be obtained from the KS GF as where is the chemical potential, are the KS energy levels of the orbitals and is the KS GF projected onto the subspace. The energy levels are obtained from the static crystal field where as usual in DFT++ approaches a double counting correction (DCC) has to be subtracted to compensate for the overcounting of interaction terms. Here we employ the so-called fully localized or atomic limit DCC Czyżyk and Sawatzky (1994) .
The AIM is solved within the OCA Haule et al. (2001); Kotliar et al. (2006). This yields the electronic self-energy which accounts for the electronic correlations of the -electrons due to strong electron-electron interactions. The correlated GF is then given by . Correspondingly, the correlated GF for D is given by where and only act within the subspace. From we can calculate the transmission function where . For small bias voltages , the transmission yields the conductance: .
We consider a single CoBz molecule in contact with two semi-infinite Cu nanowires as shown in Fig. 1a. The CoBz molecule is the smallest instance of a general class of MBz complexes, where M stands for a metal atom, that have been prepared and investigated Kurikawa et al. (1999); ?; ?; Xiang et al. (2006); ?; ?. The semi-infinite Cu wires exhibit the hexagonal symmetry of the molecule and correspond to the (6,0) wires described in Ref. Tosatti et al., 2001. We investigate the system at three different Cu-tip-Co distances 3.6 Å, 4.0 Å and 4.3 Å (see Fig. 1a,b). As can be seen from Fig. 1b the Bz-Bz distance varies between 3.4 Å and 3.65 Å depending on the distance of the Cu tip to the Co atom in the center of the molecule. At distances Å and Å the molecule is slightly compressed compared to its free height of about Å, whereas it is slightly stretched at Å.
The hexagonal symmetry of the system leads to a lifting of the degeneracy of the five orbitals. The symmetry adapted representations are: the group consisting of the orbital only, the doubly degenerate group consisting of the and orbitals and the group consisting of the and orbitals.
Figs. 1c+d show the imaginary parts of the hybridization functions calculated from the LDA electronic structure. The imaginary part of the hybridization function exhibits a distinct peak close to the Fermi level () in the channel, whose position, width and height depend significantly on the molecular geometry, specifically on the Bz-Co distance. The other channels and show only a negligible hybridization close to . The dominant feature stems, similarly as shown for graphene Wehling et al. (2010); Jacob and Kotliar (2010) from hybridization with the orbital state of the benzene rings. The feature does not depend qualitatively on the DFT functional used, as we have found the same feature within GGA and also in B3LYP calculations. The presence of strong molecular resonances in the hybridization function makes this case different from the case of nanocontacts with magnetic impurities where the hybridization functions are generally much smoother (see Ref. Jacob et al. (2009) for comparison).
Fig. 2a compares the correlated spectral functions of the Co electrons for the three distances considered here at high temperatures on a large energy scale. The spectra vary considerably as the distance changes. Most importantly, for around 3.6 Å when the molecule is slightly compressed, a sharp temperature-dependent peak appears right at , as can be seen from Fig. 2b. The peak is strongly renormalized (i.e. it only carries a small fraction of the spectral weight) due to the strong electron-electron interactions.
The sharp peak in the spectral function at that starts to develop already at temperatures of eV K stems from the channel which is the only channel with appreciable hybridization near , see Figs. 1b+c. Correspondingly, the transmission function (Fig. 2c) shows a Fano-like feature around zero energy. A renormalized, sharp and temperature-dependent resonance in the spectral function at is commonly associated with the Kondo effect. Looking at the orbital occupations, we find that the channel that gives rise to the resonance for Å has an occupation of about while the total occupation of the shell is . The fractional occupation numbers indicate the presence of valence fluctuations where the charges in the individual impurity levels fluctuate in contrast to the pure Kondo regime where these fluctuations are frozen.
Analyzing the atomic states of the Co -shell contributing to the ground state of the system we find that the principal contribution (%) is an atomic state with electrons and a total spin of as shown in Fig. 3a. The total spin 1 stems from holes in the and channels. The charge fluctuations in the channel are mainly due to the contribution (%) of an atomic state. There are considerably weaker contributions (%) from atomic and states. The individual contributions of the remaining atomic states are very small (below 1%) but add up to a total contribution of 34%.
By exclusion of individual atomic states from the calculation of the spectra we can determine which fluctuations are responsible for the different spectral features. We find that the fluctuations between the and the states are primarily responsible for the three spectral features close to including the sharp Kondo-like peak right at , as illustrated in Fig. 3b. Also the broad peak around eV below originates from these fluctuations while the broad peak about eV above arises from fluctuations between the atomic state and the principal atomic state. The two peaks in the spectral function nearest to the Kondo resonance arise from the strong energy dependence of the hybridization function whose real part has poles just below and above roughly at the positions of these two spectral features.
Note that the fluctuations from the to the states that give rise to the Kondo-like peak at actually cannot lead to a spin Kondo effect since the spin of the state is higher than the spin of the principal state. Instead the fluctuations between the and the states which give rise to the Kondo-like resonance at correspond to an orbital Kondo effect in the doubly-degenerate levels of the Co shell as illustrated in the upper panel of Fig. 3c. Here the index labeling the two orbitals with symmetry takes over the role of a pseudo spin. In the principal atomic state the levels are occupied with three electrons and hence have a pseudo spin of . By excitation to the state the electron with minority real spin and with some pseudo spin state is annihilated. By relaxation to the principal electronic state a minority real spin electron can now be created in one of the two pseudo spin states. Those processes that lead to a flip of the pseudo spin then give rise to the orbital Kondo effect and the formation of the Kondo peak at .
The absence of a normal spin Kondo effect where the total spin 1 of the principal atomic state is screened, is easily understood on the following grounds: First, in general the Kondo scale decreases exponentially with increasing spin of the magnetic impurity Nevidomskyy and Coleman (2009). In addition, here the level does not couple at all to the conduction electrons around (no hybridization). Thus the spin associated with it cannot be flipped directly through hopping processes with the conduction electron bath.
On the other hand, an underscreened Kondo effect as reported in Ref. Parks et al. (2010) where only the spin within the shell is screened is also suppressed compared to the orbital Kondo effect due to Hund’s rule coupling: Screening of the spin in the shell can take place by fluctuations to the () state. However, the Hund’s rule coupling favors the high spin () state over the low spin () state as can also be seen from the smaller weight of the latter compared to the former. Hence the Kondo scale is considerably lower for the underscreened Kondo effect than for the orbital Kondo effect found here. At lower temperatures (not accessible with OCA) the two Kondo effects may in fact coexist.
We have also checked the dependence of the LDA+OCA spectra on the DCC as well as on the interaction parameters and (not shown sup ()). In general we find the spectra and also the Kondo peak to be qualitatively robust against shifts of the impurity levels in energy over a range of several electron volts, and changes of between 3 and 7 eV, and of between 0.6 to 1 eV. As expected for the Kondo effect the sharp resonance stays pinned to the Fermi level when shifting the impurity levels in energy, and only height and width somewhat change.
Stretching the molecule by displacing the tips of the Cu nanowires the Kondo resonance and the concomitant Fano line shape in the transmission disappear for distances Å (not shown). This is accompanied by an increase of the occupation of the Co shell. The new regime is characterized by a strong valence mixing between the and atomic state of roughly equal contribution indicating that the system is now in the so-called mixed valence regime (see e.g. Ref. Hewson, 1997, Chap. 5). Hence the orbital Kondo effect and the associated spectral features can be controlled by stretching or compressing the molecule via the tip atoms of the Cu nanocontact. This strong sensitivity on the molecular conformation stems from the sharp features in the hybridization function which change considerably when the molecule is stretched or compressed as can be seen from Fig. 1d. This peculiar behavior is qualitatively different from the case of the nanocontacts containing magnetic impurities where the hybridization functions are much smoother Jacob et al. (2009).
In conclusion, we have shown for a CoBz sandwich molecule coupled to Cu nanowires that the dynamic correlations originating from the strongly interacting Co electrons give rise to an orbital Kondo effect in the doubly degenerate levels while a spin Kondo effect is suppressed by Hund’s rule coupling. Due to the sensitivity of the electronic correlations on the molecular conformation it is possible to control the appearance of the orbital Kondo effect by stretching and compressing the molecule with the tips of the Cu leads. It should be possible to prepare a setup similar to the one considered here in an actual experiment and measure the orbital Kondo effect, e.g. by contacting a CoBz molecule deposited on a Cu(111) surface with a scanning tunneling microscope. Finally, we mention that we have also explored the case of a NiBz sandwich. In this case we do not find a Kondo effect. Instead the system is in the mixed valence regime characterized by a strong peak with weak temperature dependence close to but not at sup ().
We thank K. Haule for providing us with the OCA impurity solver and for helpful discussions. Support from SFB 668, LEXI Hamburg and ETSF are acknowledged. MK gratefully acknowledges the hospitality of the Max-Planck-Institute in Halle (Saale).
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