Magnetic Order Beyond RKKY in the Classical Kondo Lattice

We study the Kondo lattice model of band electrons coupled to classical spins, in three dimensions, using a combination of variational calculation and Monte Carlo. We use the weak coupling ‘RKKY’ window and the strong coupling regime as benchmarks, but focus on the physically relevant intermediate coupling regime. Even for modest electron-spin coupling the phase boundaries move away from the RKKY results, the non interacting Fermi surface no longer dictates magnetic order, and weak coupling ‘spiral’ phases give way to collinear order. We use these results to revisit the classic problem of 4f magnetism and demonstrate how both electronic structure and coupling effects beyond RKKY control the magnetism in these materials. The Kondo lattice model describes local moments on a lattice coupled to an electron band. Such local moments arise from electron correlation and Hunds coupling in the d shells of transition metals or the f shells of rare earths. Although historically the ‘Kondo lattice’ arose as the lattice version [1] of the Kondo impurity problem, and refers to S = 1/2 moments coupled to conduction electrons, there are also systems with local electron-spin coupling where the moment is due to a spin with 2S ≫ 1. In that case the quantum fluctuations of the local moment, and the Kondo effect itself, are not relevant. Such a system can be described by a classical Kondo lattice model (CKLM). This limit is relevant for a wide variety of materials, e.g, the manganites [2], where S = 3/2 moments couple to itinerant electrons via Hunds coupling, or 4f metals [3–6], e.g, Gd with S = 7/2, or the Mn based dilute magnetic semiconductors [7] where S = 5/2. In some of these materials, notably the manganites and the magnetic semiconductors, the coupling scale is known to be large, while in the f metals they have been traditionally treated as being weak. The CKLM involves the ordering of ‘classical’ spins, but the effective interaction between spins is mediated by electron delocalisation and cannot be described by a short range model. In fact the major theoretical difficulty in analysing these systems is the absence of any simple classical spin model. Nevertheless, there are two limits where the CKLM is well understood. (a). When the electron-spin coupling is small, one can perturbatively ‘integrate out’ the electrons and obtain the celebrated Ruderman-KittelKasuya-Yosida (RKKY) model [8]. The effective spin-spin interaction in this limit is oscillatory and long range, controlled by the free electron susceptibility, χ0(q), and the magnetic ground state is generally a spiral. (b). When the electron-spin coupling is very large compared to the kinetic energy, the ‘double exchange’ (DE) limit, the electron spin is ‘slaved’ to the orientation of the core spin and the electronic energy is minimised by a ferromagnetic (FM) background [9]. This leads intuitively to a spin polarised ground state. In many materials the ratio of coupling to hopping scale is ≥ 1, but not quite in the double exchange limit. In that case one has to solve the coupled spin-fermion model from first principles. Doing so, particularly in three dimensions and at finite temperature, has been a challenge. We study this problem using a combination of variational calculation and full spin-fermion Monte Carlo. Our principal results are the following: (i) We are able to map out the magnetic ground state all the way from the RKKY limit to double exchange, revealing the intricate evolution with coupling strength. (ii) We demonstrate that the phase boundaries depend sensitively on electronic hopping parameters. This is not surprising in the RKKY regime, but the dependence at stronger coupling is unknown. (iii) We use our results to revisit the classic 4f magnets, widely modelled as RKKY systems, and suggest that with increasing 4f moment, the effective coupling in these systems pushes them beyond the RKKY regime. We work out the signatures of this ‘physics beyond RKKY’.