The quantum origins of skyrmions and half-skyrmions in CuOSeO
The Skyrme-particle, the skyrmion, was introduced over half a century ago and used to construct field theories for dense nuclear matter.Skyrme (1962); Witten (1983) But with skyrmions being mathematical objects — special types of topological solitons — they can emerge in much broader contexts.Bogdanov and Yablonskii (1989); Bogdanov and Hubert (1994); Rößler et al. (2006) Recently skyrmions were observed in helimagnets,Mühlbauer et al. (2009); Tonomura et al. (2012); Yu et al. (2010); Seki et al. (2012a); Yu et al. (2012) forming nanoscale spin-textures that hold promise as information carriers.jonietz2010 ; fert2013 Extending over length-scales much larger than the inter-atomic spacing, these skyrmions behave as large, classical objects, yet deep inside they are of quantum origin. Penetrating into their microscopic roots requires a multi-scale approach, spanning the full quantum to classical domain. By exploiting a natural separation of exchange energy scales, we achieve this for the first time in the skyrmionic Mott insulator CuOSeO. Atomistic ab initio calculations reveal that its magnetic building blocks are strongly fluctuating Cu tetrahedra. These spawn a continuum theory with a skyrmionic texture that agrees well with reported experiments. It also brings to light a decay of skyrmions into half-skyrmions in a specific temperature and magnetic field range. The theoretical multiscale approach explains the strong renormalization of the local moments and predicts further fingerprints of the quantum origin of magnetic skyrmions that can be observed in CuOSeO, like weakly dispersive high-energy excitations associated with the Cu tetrahedra, a weak antiferromagnetic modulation of the primary ferrimagnetic order, and a fractionalized skyrmion phase.
Skyrmionic spin textures in magnetic materials correspond to magnetic topological solitons as depicted in Fig. 1. They were first observed in the non-centrosymmetric B20 helimagnets MnSi,Mühlbauer et al. (2009); Tonomura et al. (2012), FeGe,Yu et al. (2012) and FeCoSi.Yu et al. (2010) These skyrmionic textures are encountered also in a completely different branch of physics: in the theoretical description of nuclear matter.Skyrme (1962); Witten (1983) In this setting the skyrmions are of course not related to magnetic degrees of freedom, but rather to particles emerging from cold hadron vector fields at densities a few times that of ordinary nuclear matter. This is the density range relevant for compact astronomical objects such as neutron stars.Ouyed and Butler (1999); Luckock and Moss (1986); Lee2010 The perhaps perplexing connection between these two seemingly disparate fields of physics is borne out of the underlying mathematical structures.Bogdanov and Yablonskii (1989); Bogdanov and Hubert (1994); Rößler et al. (2006); Lee2010 ; Ouyed and Butler (1999); Luckock and Moss (1986); Castillejo et al. (1989); Kugler and Shtrikman (1989) The physical phenomena in the two different settings are both governed by an emerging set of differential equations with topological solitonic solutions: the skyrmions found first by Skyrme in the 1960s.Skyrme (1962)
In this context we investigate the formation and microscopic origin of the observed magnetic skyrmions in helimagnets (Fig. 1). These skyrmions are large objects compared to the atomic length-scale: they are about three orders of magnitude larger in size than the inter-atomic lattice spacing. Understanding the origin of these nanometer-scale skyrmions therefore requires a multi-scale approach. In the above mentioned B20 helimagnets such is however not viable because all these materials are metallic. The metallicity causes low-energy, delocalized electronic and magnetic degrees of freedom to mix so that they intrinsically involve multiple energy and spatial scales, which renders a multi-scale approach presently intractable.
This is very different in the recently discovered skyrmionic material CuOSeO, a large band-gap Mott insulator (Fig. 2). The band gap enforces a natural separation between electronic and magnetic energy scales. CuOSeO is actually the first example of an insulating material displaying the chiral helimagnetism that is desired for skyrmion formation while sharing the non-centrosymmetric cubic space group of the metallic B20 phases, but with a unit cell that is much more complex, containing 16 Cu atoms. Due to the presence of a magnetoelectric couplingSeki et al. (2012a); Belesi et al. (2012) its skyrmions can be manipulated by an electric field,White et al. (2012) which is in principle very energy efficient as this avoids losses due to joule heating.
A multi-scale approach to elucidate the quantum origin of the skyrmion textures in CuOSeO has to start from a calculation of magnetic interactions at the atomic level. In a CuOSeO crystal, the magnetic Cu ions make up a 3D network of corner-sharing tetrahedra (Fig. 2, b) with two inequivalent Cu sites, Cu(1) and Cu(2) that are inside Cu(1)O bi-pyramids and distorted Cu(2)O plaquettes, respectively.Bos et al. (2008); Belesi et al. (2010) Each tetrahedron contains Cu(1) and Cu(2) in a ratio of 1:3. The resulting net of magnetic Cu ions in CuOSeO thus has a structure that is rather different from the previously mentioned metallic B20 helimagnets such as MnSi, in which the magnetic Mn ions constitute instead a three dimensional corner-sharing net of triangles, commonly referred to as the trillium lattice. The more complex crystal structure of CuOSeO leads to five inequivalent superexchange coupling constants between neighboring copper spins and and also five different Dzyaloshinskii-Moriya (DM) vectors in the microscopic magnetic Hamiltonian
where denotes the Cu quantum-spin at site . We have determined these coupling constants by means of an extended set of ab initio density functional based electronic structure calculations. The obtained values were cross-checked by calculating the magnetic and the temperature dependence of both the magnetization and magnetic susceptibility by means of large scale Quantum Monte Carlo (QMC) simulations. These simulations agreeing very well with the measurements inspire further confidence in the accuracy of the values calculated from first principles.
Having fixed the microscopic coupling constants, we proceed by establishing a hierarchy of magnetic energy scales. We first observe that the calculated ’s between Cu(1) and Cu(2) are antiferromagnetic (AFM) and between Cu(2) and Cu(2) ferromagnetic (FM). A more detailed examination of the magnetic energy scales reveals a striking difference between two groups of exchange couplings, splitting the system into two kinds of Cu tetrahedra: one with strong ( K) and the other with weak ( K) superexchange couplings. The DM terms are in turn much smaller than the exchange couplings, .
The four spins of a Cu tetrahedron with strong superexchange interactions can couple together to form either a total singlet, triplet or quintet state (with total spin , 1 or 2, respectively). The tetrahedron having 3 AFM and 3 FM exchange couplings, renders the ground state (GS) a total triplet, see Fig. 2 (b-c). The triplet GS is separated from the other spin-multiplets by a large energy gap of 275 K. An important point is that the Cu tetrahedron triplet wavefunction is not the classical (tensor product) state (where the double arrow labels the Cu(1) site in the tetrahedron) but rather a coherent quantum superposition of four classical states
with labeling the three orthogonal triplet states with = , 0, 1 (for brevity only the wavefunction is given above). Although these are not the exact tetrahedron basis states due to the presence of a small triplet-quintet mixing, this representation is qualitatively correct. This effective spin wavefunction is in full agreement with the experimental observation of a locally ferrimagnetic order parameter.Bos et al. (2008); Belesi et al. (2010) The quantum fluctuations ingrained into these triplet wavefunctions, however, give rise to a substantial reduction of the local moments, providing a natural explanation for the origin of the small moments observed experimentally.Bos et al. (2008) As opposed to transversal spin fluctuations arising from spin waves (which are expected to be small in the present case owing to the dimensionality, the ferrimagnetic nature of the order parameter and the absence of frustration), these local quantum fluctuations are longitudinal in character and hence directly affect the effective magnitude of the spin. This picture is confirmed by a lattice QMC simulation for the full model of Cu spins.
This establishes Cu tetrahedra carrying magnetic triplets as building blocks in CuOSeO at the next step of the multi-scale approach. Within this abstraction, each of these tetrahedra can be contracted to a single lattice point. The resulting structure turns out to consist of corner-shared triangles which together constitute a trillium lattice, which is precisely the same lattice that is formed by the Mn atoms in the B20 structure of MnSi and the Fe atoms in FeGe. This establishes a very close analogy between Mott insulating CuOSeO and these well-known metallic helimagnets, despite the fundamental differences in electronic structure. However in CuOSeO, the effective triplet interactions can be derived relying on rigorous microscopic results. At this point both the weaker superexchange couplings and the DM interaction become crucial. A straightforward perturbative calculation reveals that their net effect is a weak FM interaction between nearest-neighbor (NN) and next-nearest-neighbor (NNN) spins, with an effective exchange coupling of about K, which reflects the tendency of the system towards FM ordering. This drastic reduction of the energy scale in the effective model is caused by the renormalization of the local spin lengths and the strong quantum correlations inside the strongly coupled tetrahedra. Not only an exchange interaction, but also a DM coupling between NN and NNN spins emerges. This is crucial because in the GS of a single strong tetrahedron all diagonal matrix elements of the DM couplings within the tetrahedron vanish by symmetry. The twisting mechanism that causes chiral helimagnetism in CuOSeO, therefore originates from the effective DM couplings between the strong Cu tetrahedra: these will be the root cause for skyrmions to emerge.
Having established the effective trillium lattice model of CuOSeO, we now proceed to the long wavelength magnetic continuum theory that governs the skyrmion formation in CuOSeO on the mesoscopic scale. The resulting continuum equations involve two magnetic constants, and , which from a direct calculation are evaluated to be K and K. With the characteristic period of the double-twisted skyrmion structures being , with wavenumber and lattice constant a, the calculated magnetic constants result in a helix period of nm which has the correct order of magnitude compared with the experimentally measured valueSeki et al. (2012a) of 50 nm. Besides this agreement with basic experimental observations, our multi-scale description also provides two essential predictions. Firstly, a very distinct set of weakly dispersive, high-energy intra-tetrahedra excitation modes should appear. Secondly, a specific antiferromagnetic superstructure emerges that is the dual counterpart of the weak ferromagnetism present in chiral acentric bipartite antiferromagnets.Bogdanov et al. (2002) Both these effects can immediately be tested experimentally, for instance by neutron scattering.
Having quantified in detail the microscopic couplings responsible for the chirally twisted spin order in CuOSeO, its magnetic phase diagram can be assessed. In the continuum description of CuOSeO, the Dzyaloshinskii model of chiral cubic ferromagnets, we can merge the ab initio parameters and the exchange stiffness (obtained from QMC calculations) to determine quantitatively the thermodynamic (Landau) potential. The value of the weakest primary coupling, the DM interaction, is fixed by the experimentally observed helix period. With this approach both the magnetic field and temperature scale are fully determined.
In this framework, one can calculate the critical field for the continuous transition from the conical helix state into the field-enforced ferrimagnetic collinear state. We find 80 mT at 50 K, in very good agreement with experimental data. We also determined the temperature window for the precursor region at around the magnetic ordering temperature where meso-phases, that are potentially of skyrmionic character, can be formed and find K . In this temperature interval, the skyrmionic cores are energetically favorable compared to one-dimensional helix solutions.Rößler et al. (2006) The computed range is in agreement with the interval of about 2 K in which the so-called A-phases appear under magnetic field in CuOSeO crystals.Seki et al. (2012b); Adams et al. (2012) From symmetry considerations it is immediately clear that the dominant anisotropy in CuOSeO is cubic with a coefficient , which can stem from the magnetoelectric effect and the dielectric polarizability of CuOSeO. The experimental value for the magnetic field at which the conical helix closes and becomes a field-enforced saturated state fixes this anisotropy at J/m (corresponding to 3.3 eV/Cu atom).
With all this in place, we can fully determine the equilibrium solutions and thereby the phase diagram (Fig. 3 a and b). In the precursor region, we find as equilibrium states two competing skyrmionic phases (Fig. 3 c to f). The first one (Fig. 3 c and e) is the standard field-driven “”-skyrmion phase of Fig. 1(b) with the radial skyrmions ordered in a hexagonal lattice.Bogdanov and Yablonskii (1989); Bogdanov and Hubert (1994) The other one (Fig. 3 d and f) is the “”-skyrmionBogdanov and Hubert (1994); Wilhelm2012 state of Fig. 1(c), which actually is the stable state at zero and low fields, because the fractionalization of skyrmions into half-skyrmions yields a higher packing density of the energetically advantageous skyrmionic cores. The fractionalized skyrmion textures contain defects like hedgehogs, narrow line or wall defects, where the magnetic order parameter passes through zero, leading to a broad distribution of local moments that can be discernible by local probes such as SR and NMR, or neutron diffraction methods. The emergence of the half-skyrmion phase in the vicinity of the -skyrmion lattice of CuOSeO opens a new venue to study properties of textures with split topological units in experiment. Observations of these defect-ridden topological phases, together with the predicted weakly antiferromagnetic indentations of the ferrimagnetic order, the strong fluctuations of the local moments, and the weakly dispersive high-energy magnetic excitations, that are associated with the rigidly coupled spins of tetrahedra, allow to probe the quantum origin of the magnetic skyrmions in experiments on CuOSeO.
Acknowledgements.We acknowledge fruitful discussions with J.-P. Ansermet, A. N. Bogdanov, V. A. Chizhikov, V. E. Dmitrienko, and Y. Onose. IR was supported by the Deutsche Forschungsgemeinschaft (DFG) under the Emmy-Noether program. OJ and AT were partly supported by the Mobilitas program of the ESF, grant numbers MJD447 and MTT77, respectively.
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