Antiferromagnets with anisotropic spin-spin interactions: Stability of the zero-field structure in an external field.

Abstract

The transition from the polarized paramagnetic state to the antiferromagnetic phase in an applied external magnetic field has been investigated theoretically by linear spin-wave theory at T=0. Our analysis applies to spins arranged in a lattice of cubic symmetry. In addition to the Zeeman term, the Hamiltonian consists of bilinear interactions. Antiferromagnetic transition to a state described by an ordering vector Q is discussed in terms of softening of the corresponding spin-wave excitation in the paramagnetic phase. It is shown that the onset of antiferromagnetic order can be calculated by solving an eigenvalue problem. The smallest eigenvalue of the Fourier transformed 2\ifmmode\times\else\texttimes\fi{}2 interaction matrix, which describes the spin-spin interactions in the plane perpendicular to B, determines ${\mathit{B}}_{\mathit{c}}$, Q, and the direction of the antiferromagnetic component. For sufficiently anisotropic spin-spin interactions, these quantities can depend on the direction of B with respect to the crystalline axes.However, when the spin structure shows an easy-plane anisotropy, which is possible for ordering vectors of the type Q=(h,0,0) and Q=(h,h,h), and for some vectors at the Brillouin-zone boundary, the direction of B has no such effect. The general results were first applied to investigate the stability of the easy-axis type-III antiferromagnetism of the fcc lattice, characterized by Q=\ensuremath{\pi}/a(1,1/2,0). It was shown that, if the spin-spin interactions are sufficiently anisotropic, type-III order becomes unstable against type-I order [Q=\ensuremath{\pi}/a(1,0,0)] when a strong enough field B is applied along a [111] crystalline axis. If the anisotropy is comparable to the isotropic next-nearest-neighbor coupling, like in ${\mathrm{K}}_{2}$${\mathrm{IrCl}}_{6}$, a high-field ordering vector, between the type-I and -III vectors, is predicted. As another application, the magnetic phase diagram of nuclear spins in copper was investigated. Antiferromagnetic type-I ordering has been found in this fcc metal below ${\mathit{T}}_{\mathit{N}}$=60 nK.We studied the puzzle presented by the neutron-diffraction measurements of Annila et al., which show that type-I order is absent in the high-field reigon below ${\mathit{B}}_{\mathit{c}}$=0.25 mT when B\ensuremath{\parallel}[111], although this kind of ordering was observed in the same fields when B\ensuremath{\parallel}[100] or [110]. Soft-mode analysis shows that the high-field ordering vector for B\ensuremath{\parallel}[111] is of the general type, Q=(h,k,l), where \ensuremath{\Vert}h\ensuremath{\Vert}, \ensuremath{\Vert}k\ensuremath{\Vert}, and \ensuremath{\Vert}l\ensuremath{\Vert} are all unequal and nonzero, in agreement with our previous suggestion based on the mean-field theory. We predict various (h,k,l) structures in fields B\ensuremath{\lesssim}${\mathit{B}}_{\mathit{c}}$ for several field alignments other than [111]. The magnetic phase diagram of nuclear spins in copper can be explained on the basis of the previously calculated spin-spin interactions at least in fields B\ensuremath{\lesssim}${\mathit{B}}_{\mathit{c}}$ if the calculated parameters are changed only slightly.