Inverse indirect magnetic exchange
Magnetic moments strongly coupled to the spins of conduction electrons in a nanostructure can confine the conduction-electron motion due to scattering at almost localized Kondo singlets. We study the resulting local-moment formation in the conduction-electron system and the magnetic exchange coupling mediated by the Kondo singlets. Its distance dependence is oscillatory and induces robust ferro- or antiferromagnetic order in multi-impurity systems.
pacs:71.70.Gm, 75.10.Lp, 75.75.-c, 85.75.-d
The appearance of magnetic order in condensed-matter systems Yosida (1996); Nolting and Ramakanth (2009) requires (i) the existence or the formation of local magnetic moments, (ii) a coupling mechanism favoring a certain alignment of the moments, e.g. ferro- or antiferromagnetically, and (iii) the stability of long-range magnetic order against different types of thermal or quantum fluctuations and against competing ordering phenomena.
Local-moment formation typically results from incompletely filled localized orbitals or from strong local correlations and can be described by Hubbard-, Anderson- or Kondo-type models Anderson (1961); Hubbard (1963); Kondo (1964). Among the different known coupling mechanisms, such as the direct Heisenberg exchange Heisenberg (1928) or the indirect Anderson super exchange Kramers (1934); Anderson (1950), the Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction Ruderman and Kittel (1954); Kasuya (1956); Yosida (1957) provides a mechanism for longer-ranged coupling between magnetic impurities in a -dimensional metallic system which can be either ferro- or antiferromagnetic, depending on the distance. It originates from a local exchange coupling which, for weak , mediates an indirect interaction .
RKKY exchange has gained much interest recently in the context of nanostructures, e.g. in double-dot semiconductor quantum devices Craig et al. (2004) with tunable RKKY-mediated control on spin degrees of freedom. Nanostructures with tailored magnetic properties can be engineered using scanning-tunnelling techniques by positioning magnetic atoms on non-magnetic metallic surfaces at certain distances where is ferro- or antiferromagnetic Zhou et al. (2010); Khajetoorians et al. (2011, 2012). Furthermore, RKKY-mediated magnetism competes Doniach (1977) with the Kondo effect Kondo (1964); Hewson (1993), i.e., the screening of the local magnetic moments due to non-local antiferromagnetic correlations induced by . A subtle Kondo-vs.-RKKY competition takes place at weak in nano-systems with strong electron-confinement effects Thimm et al. (1999); Schwabe et al. (2012).
Here, we study an exchange mechanism where the roles of conduction electrons and impurities are “inverted”. We show that the Kondo effect helps (i) to form local moments, (ii) to couple the moments and (iii) leads to magnetic order in certain nano-structured geometries: For strong , almost local Kondo singlets are formed which act as hard scattering centers for the itinerant conduction electrons and may confine their motion, depending on the impurity positions. In certain geometries, this tends to localize the conduction electrons and leads to the formation of local magnetic moments in the a priori uncorrelated conduction-electron system. These moments are found to couple magnetically via virtual excitations of the Kondo singlets.
We study the resulting “inverse indirect magnetic exchange” (IIME) by means of strong-coupling perturbation theory and different numerical techniques. The IIME shows an oscillatory distance dependence. For extended systems, it triggers long-range magnetic order which is robust against charge fluctuations on the impurities but sensitively depends on the quantum confinement of the conduction electrons, e.g., on the geometry of magnetic adatoms in an experimental setup using scanning-tunnelling techniques.
Furthermore, for certain geometries, the IIME at strong can be understood as evolving by adiabatic connection from the standard RKKY coupling at weak . This is ensured by quantum confinement and by exact results Shen (1996); Tsunetsugu (1997) based on Lieb’s concept of reflection positivity in spin space Lieb (1989) available for Kondo systems on bipartite lattices at half-filling.
From RKKY to inverse exchange.
We consider a system with spins , with spin-quantum numbers , which are coupled locally via an antiferromagnetic exchange to the local spins of a system of itinerant and non-interacting conduction electrons. The conduction electrons hop with amplitude between non-degenerate orbitals on neighboring sites of a -dimensional lattice of sites:
Here, annihilates an electron at site with spin projection , and is the local conduction-electron spin at , where is the vector of Pauli matrices. Impurity spins couple to the local conduction-electron spins at the sites . We investigate the half-filled system with electrons.
To illustrate the crossover from conventional RKKY indirect magnetic exchange at weak to the inverse indirect exchange at strong , we first analyze a simple model with a small number of sites and spins at and using exact diagonalization (see Fig. 1). In the RKKY regime for , the low-energy sector of is exactly described by an effective RKKY two-spin model with . For a “ferromagnetic distance” the two impurity spins form a non-local triplet in the ground state.
As is seen in Fig. 1, the ground state is unique (apart from the spin degeneracy) for any , . The absence of a ground-state level crossing at half-filling and for a bipartite lattice is in fact enforced by analytical results Shen (1996); Tsunetsugu (1997). Consequently, the ground-state symmetry is preserved, and the triplet stays intact in the entire range. However, its character must change. With increasing , the Kondo effect, which for is cut by the finite-size gap , sets in and dominates for with a shrinking and eventually local screening cloud. This results in two completely local and magnetically inert “Kondo” singlets while the ground state must be a triplet. The numerical analysis of spin-correlation functions shows that this triplet is formed by two spins formed in different parts of the conduction-electron system which couple ferromagnetically. Opposed to a coupling of impurity spins mediated by the metal host for weak , i.e. standard indirect RKKY exchange, this type of interaction represents an “inverse indirect magnetic exchange” (IIME) mediated by local Kondo singlets (see insets in Fig. 1).
Before we analyze this exchange mechanism in detail, we demonstrate its usefulness to understand ferromagnetic order for large systems . To this end we applied the density-matrix renormalization group (DMRG) White (1992); Schollwöck (2011) to study one-dimensional models for different and for spins ( odd) coupled to the sites (B sites, see Fig.2, inset), i.e. we study a “diluted” Kondo lattice with impurity spins at “ferromagnetic” distances. At half-filling this model is known Shen (1996) to exhibit a ferromagnetic ground state.
Our implementation (see Ref. Titvinidze et al. (2012) for details) makes use of conservation of the -component of the total spin . The total spin is obtained by computing the ground-state expectation value . For the system shown in Fig. 2 (, ), we in fact find a large in the entire range. This is consistent with the prediction by Shen Shen (1996).
The adiabatic connection between RKKY and IIME is more subtle in this case: For , standard RKKY theory would predict . In the present case, exactly one of the impurity spins, however, is Kondo screened by the single electron occupying the spin-degenerate one-particle energy level at the Fermi wave vector of the non-interacting conduction-band system (see Ref. Schwabe et al. (2012)). This results in . For all (finite but large) systems studied here, the ground state turns out to be a smooth function of . Therefore, must be the same in both limits. For , this large spin must then result from a ferromagnetic coupling of local magnetic moments at the sites (A-sites, see Fig.2, inset) which are formed as a result of the increasing confinement of electrons due to the formation of local Kondo singlets at the B sites. The DMRG calculations indeed yield strong antiferromagnetic local spin correlations , vanishing RKKY correlations , and local-moment formation at A-sites for .
Fig.2 shows the ordered magnetic moments at the central impurity , at the “sub-impurity” B site and a neighboring A-site , as obtained from the ground state with maximum . With increasing , there is a clear crossover from the RKKY regime, with , , to the IIME regime for , where the magnetization of the system results from ordered moments at A-sites. The results are characteristic for the infinite system as is obvious by comparing results for and (see in Fig. 2).
To analyze the mechanism generating a ferromagnetic coupling between magnetic moments at next-nearest neighboring A-sites and , we treat the hopping term in Eq. (1) perturbatively. The starting point is the highly degenerate ground state of the model consisting of local Kondo singlets and an arbitrary electron configuration. A non-trivial effective model capturing the low-energy sector of in the limit is obtained at fourth order in through processes where e.g. an electron hops from via the neighboring B site to and, again via B, back to . Here, the local Kondo singlet at B must be excited at an energy cost first and restored again on the way back. Calculations are lengthy but straightforward and will be published elsewhere Titvinidze et al. (2013). For and keeping terms up to we find:
The effective model is governed by a single energy scale and describes spin and charge degrees of freedom on the A sites only. This is opposed to a strong-coupling variant of the RKKY theory TDH+13 () where the focus is on the effective coupling between impurity spins.
The first term in Eq. (2) represents a Heisenberg-type ferromagnetic spin interaction and indeed explains the ferromagnetic IIME through a local Kondo singlet. Ferromagnetism due to the IIME competes with formation of a charge-density wave or pairing Tsunetsugu et al. (1997) as favored by the second term. This includes the local isospin . Note that the total isospin and the total spin are the generators of the SO(4) symmetry group of the half-filled Kondo model on the bipartite lattice Tsunetsugu et al. (1997) – and of the effective model as well. The effective isospin interaction is “antiferromagnetic”. Analogous to the Mermin-Wagner theorem Mermin and Wagner (1966), and opposed to ferromagnetic spin order, antiferromagnetic (staggered) isospin order would be suppressed by quantum fluctuations of the order parameter for . The necessary formation of local isospin moments in the ground state is suppressed anyway by the repulsive Hubbard term (third term in Eq. (2)). On the contrary, the Hubbard interaction favors formation of local magnetic moments. Finally, there is a correlated hopping term in which, however, is only active between a spin at and an isospin at or vice versa. Exact diagonalization of for systems with a few A-sites is easily done and in fact yields a ferromagnetic ground state with while .
The effective model and thus the IIME concept is also valid for fillings , as long as the local Kondo singlets in the ground state are unbroken, i.e. for fillings . In dimensions essentially the same effective model is obtained. Generalizations to non-bipartite lattice structures are possible.
The IIME mechanism is robust against charge fluctuations on the impurities. This is demonstrated by DMRG calculations where the spin-1/2 Kondo impurities are replaced by Anderson impurities, i.e., correlated sites with Hubbard interaction coupled to the conduction electrons by a local hybridization . For weak with the results for the two models agree (filled and open symbols in Fig. 2), as prescribed by the Schrieffer-Wolff transformation Schrieffer and Wolff (1966). With increasing , however, we again find a crossover from RKKY to IIME, and for strong confinement of A-site electrons is due to the formation of strongly bound states at the sites. In the case of Kondo impurities the crossover takes place between and while in the Anderson case it is located around .
Dynamical mean-field theory.
While non-local correlations due to the RKKY interaction in the metastable paramagnetic state are not accessible to a mean-field description, at least for weak Titvinidze et al. (2012), DMFT is able, however, to describe the symmetry-broken ferromagnetic state. The solid and dashed lines in Fig. 2 show the results of DMFT calculations for Kondo and Anderson impurities, respectively. We employ a standard Lanczos implementation Titvinidze et al. (2012); Caffarel and Krauth (1994). In case of the diluted Kondo-lattice model, Eq. (1), the effective self-consistently determined impurity problem consists of the local spin , the corresponding B site and up to 8 bath sites Momoi and Kubo (2000); Otsuki et al. (2009), while up to 9 bath sites are used in the Anderson case. Almost perfect agreement with the DMRG data is found, see Fig. 2.
DMFT can be employed to describe the magnetic properties of nanostructures of magnetic atoms on a metallic surface layer (isolated from the substrate by a spacer). Motivated by the agreement with DMRG for the symmetry-broken state of the bulk system, we again expect quantitatively reliable predictions. This can be checked to some extent in case of a bipartite structure as the total spin quantum number is fixed by reflection positivity in spin space Shen (1996); Tsunetsugu (1997). More important from a physical point of view, however, is to confine, with the help of the Kondo effect, conduction electrons in certain geometries to avoid a dissipation of the total spin into a large (bulk) layer.
Fig. 3 displays an example for a layer with several magnetic impurities. Calculations are done using the real-space generalization of DMFT Potthoff and Nolting (1997) for the corresponding diluted Kondo lattice model. A symmetry-broken ground state is found: Electrons in the chain of A-sites 1, 2, …, 8 with relative distance are confined. Their local moments in fact order ferromagnetically. The moment at A-site 9 (distance ) couples ferromagnetically while the moments at sites 10 and 11 ( and ) couple antiferromagnetically to the chain. Note that the local moments at 10 and 11 are formed by confinement due to surrounding local Kondo singlets. If such moments are weakly coupled to the rest of the system, a tiny Weiss field produced by the chain is sufficient to result in an almost full polarization. is in fact found to slightly increase with increasing distance from the chain. We conclude that the IIME is oscillatory and decreasing with distance – similar to the RKKY case.
More characteristic features of the IIME can be studied qualitatively (see Fig. 3): Neighboring A-sites with higher effective coordination mutually support magnetic polarization. This explains the slightly enhanced at and around A-site 12. Confinement of an odd number of electrons is important: There is almost no moment at sites 13 and 14 while the structure 15, 16, 17 is polarized. Confinement as such (with respect to all dimensions) is essential: Electrons at 18 or 19 are not confined, no local moments are formed and thus no polarization found. There is a proximity effect, however, as can be seen at 20, 21, or 22. Furthermore, spin-dependent multiple scattering of conduction electrons at the magnetic structures causes an interference pattern, see sites 23, 24, for example. The real-space DMFT is found to give quantitatively reliable results: Summing up the local magnetic moments for the 396 uncorrelated sites and the 57 impurities, we find a ground-state spin moment of which is, within numerical uncertainties, equal to the exact value which can be obtained analytically Shen (1996); Tsunetsugu (1997); Tsunetsugu et al. (1997) by counting the number of impurities on B (filled) and on A sites (open dots).
We have analyzed an indirect magnetic exchange mechanism where confinement of conduction electrons due to scattering at Kondo singlets leads to local-moment formation at a priori uncorrelated sites and to spin and isospin couplings via virtual excitations of the Kondo singlets. This IIME is “inverse” to the conventional RKKY coupling. Its oscillatory distance dependence can be utilized to construct nanostructures with tailored magnetic properties, e.g., by placing magnetic atoms in certain geometries on a metallic layer, similar to RKKY-based artificial structures Zhou et al. (2010); Khajetoorians et al. (2011, 2012). Alternatively, systems of ultracold fermionic atoms trapped in optical lattices Jaksch and Zoller (2005); Bloch et al. (2012) may realize multi-impurity Kondo systems in the strong-coupling regime essential to the IIME. Future theoretical work may explore systems with correlated conduction electrons and preformed local moments. Spin-only, e.g., Kondo necklace models may be considered to get an analytical expression for the effective IIME spin coupling. DMFT and DMRG studies of filling dependencies appear particularly exciting. Temperature dependencies are accessible to quantum Monte-Carlo techniques Assaad (2002) on bipartite lattices at half-filling.
Acknowledgements.We thank D. Gütersloh for instructive discussions. Support of this work by the Deutsche Forschungsgemeinschaft within the SFB 668 (project A14), by the excellence cluster “The Hamburg Centre for Ultrafast Imaging - Structure, Dynamics and Control of Matter at the Atomic Scale” and by the SFB 925 (project B5) is gratefully acknowledged.
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