Ordered spin structures of β -MnO$_{2}$ (rutile-type) systems with competing exchange interactions: numerical approach using equi-energy contour plot Masatsugu Suzuki ∗ and Itsuko S. Suzuki † Department of Physics, State University of New York at Binghamton, Binghamton, New York 13902-6000 (Dated: October 24, 2018) Using numerical calculations of equi-energy contour plot of the Fourier transform of the spin Hamiltonian, we study the magnetic phase diagram ( J$_{2}$ and J$_{3}$ ) of the rutile type β -MnO$_{2}$, where J$_{1}$ ( < 0) is fixed and is the antiferromagnetic interaction along the diagonal direction, J$_{2}$ is the interaction along the c axis, and J$_{3}$ is the interaction along the a axis. The magnetic phase diagram consists of the multricritical point (the intersection J$_{2}$J$_{3}$ = J 2 1 and J$_{2}$ + J$_{3}$ = 2 J$_{1}$ ), the helical order along the c axis, the ( h = 1 / 2 , k = 0 , l = 1 / 2) phase, the helical order along the a axis, and the phase ( h = 0 , k = 0 , l = 1). The shift of the location of the magnetic Bragg peak in the ( h, 0 , l ) reciprocal lattice plane is examined with the change of J$_{2}$ and J$_{3}$ in the phase diagram. The shift is discontinuous on the first-order phase transition, and is continuous on the second-order phase transition. The detail of our magnetic phase diagram is rather different from that reported by Yoshimori. PACS numbers: 75.10.-b,75.30.Kz It is well known that β -MnO$_{2}$ (with the rutile-type structure) is one of the systems with a helical spin order along the c axis. Erickson 1 found from magnetic neutron scattering on β -MnO$_{2}$ that the spins are helically ordered with a period of 7 c along the c axis. The direction of spins in the c plane turns from one plane to the next plane by an angle of 129 ◦ (5 π/ 7). In 1958, Yoshimori 2 theoretically demonstrated the origin and stability of the helical spin order in β -MnO$_{2}$, based on the experimental results from Erickson. The spin Hamiltonian of β -MnO$_{2}$ is described by a form of the Heisenberg-type, where the combination of nearest neighbor (n.n.) exchange inter- actions, next nearest neighbor (n.n.n.) exchange inter- actions, and so on are included. His success lies in the use of the Fourier transform of the spin Hamiltonian in the reciprocal lattice space. Then the Fourier transform depends only on exchange interactions and wavevectors Q . The location of the magnetic Bragg points in the re- ciprocal lattice space are uniquely determined from the condition that the Fourier transform takes a maximum. In spite of the success in theory established by Yoshimori, 2 Erickson has not published his detailed data of the magnetic neutron scattering of β -MnO$_{2}$ as far as we know. Since 1958, there have been several papers on the magnetic structure of β -MnO$_{2}$ using magnetic neu- tron scattering 3–5 and magnetic x-ray scattering. 6,7 In recent years, Sato et al, have reported that the mag- netic Bragg peaks appear at the wave vector Q = (1, 0, 2 + ϵ ) in the units of a ∗ (= 2 π/a ), b ∗ (= 2 π/b ), and c ∗ (= 2 π/c ) in the reciprocal lattice space, where ϵ = 0.297 at 10 K, increases with increasing temperature, and reaches 0.2992 just below T$_{N}$ (= 92 K). The value of ϵ is rather different from 2/7 derived by Yoshimori. 2 This indicates that the helical spin structure along the c axis is incommensurate with the c - axis lattice constant. Because of the crystal field (distorted octahedron In the present paper, we study the magnetic phase dia- gram of β -MnO$_{2}$ type structure using the molecular field theory developed by Yoshimori, 2 where Mn 4+ spins are localized. The exchange interactions J$_{1}$ along the diago- nal axis of the system, J$_{2}$ along the c axis and J$_{3}$ along the a axis are taken into account. Note that J$_{1}$ is assumed to be fixed and be antiferromagnetic. According to Sato et al., 8 the intra-atomic t$_{2}$$_{g}$ − e$_{g}$ exchange interaction (Hund coupling), the transfer interaction between adjacent e$_{g}$ orbitals, and the occupancy of the e$_{g}$ orbitals are defined by J$_{Hund}$ , t , and c , respectively. Since | J$_{n}$ | ≫ | cJ$_{Hund}$ | and | J$_{n}$ | ≫ | ct | for β -MnO$_{2}$ ( n = 1 , 2 , 3), 8 this means that the spin structure of β -MnO$_{2}$ below T$_{N}$ is not af- fected by the effect of cJ$_{Hund}$ and ct at al. In other words, the spin structure is well described by the localized spin model with J$_{1}$ , J$_{2}$ , and J$_{3}$ . We find that our magnetic phase diagram ( J$_{2}$ vs J$_{3}$ ) consists of four phases including the helical phase along the c axis, the phase with (1/2, 0, 1/2), the helical phase along the a axis, and the phase formed by O 2 − ions) in the vicinity of Mn 4+ ion, the ground orbital state of Mn 4+ ion (3 d $^{3}$, L = 3 and S = 3/2) is split into the t$_{2}$$_{g}$ ( dϵ ) level (lower energy level, triple degenerate) and the e$_{g}$ ( dγ ) level (upper energy, double degenerate). As a result, the ground state is now orbital singlet, indicating that the orbital angular mo- mentum is quenched. The e$_{g}$ electrons are responsible for the metallic conduction, while the localized t$_{2}$$_{g}$ elec- trons are responsible for the magnetism. In recent years, Sato et al. 8 have reported the transport properties of a single crystal β -MnO$_{2}$, such as electrical resistivity, ther- mopower, Hall effect, and magnetoresistance. Their data show an appreciable anomaly near T$_{N}$ . This implies that there is a strong correlation between conduction ϵ$_{g}$ elec- trons and localized t$_{2}$$_{g}$ magnetic moment through the Hund’s rule. The DC magnetic susceptibility shows a sig- nificant deviation from a molecular-field theory based on a localized spin model, which was assumed by Yoshimori 2 on his helical spin order. I. INTRODUCTION arXiv:1001.1539v1 [cond-mat.str-el] 10 Jan 2010