Orbital order and partial electronic delocalization in a triangular magnetic metal ${\text{Ag}}_2{\text{MnO}}_2$

Magnetic and electrical properties of ${\text{Ag}}_2{\text{MnO}}_2$ were examined by elastic and inelastic neutron-scattering measurements and by density-functional calculations. The spins of the triangular antiferromagnet metal ${\text{Ag}}_2{\text{MnO}}_2$ are found to freeze into a gapless short-range collinear state below 50 K because of a ferro-orbital ordering and spin-orbit coupling of the high-spin ${\text{Mn}}^{3+}$ ions. The decrease in the spin-spin correlation lengths of ${\text{Ag}}_2{\text{MnO}}_2$ in the order, ${\xi }_b⪢{\xi }_a⪢{\xi }_c$, is explained by the spin-exchange interactions calculated for the ferro-orbital ordered state. The electronic states around the Fermi level have significant contributions from the spin-polarized $\text{Mn}3d$ and $\text{O}2p$ states, which makes electron-electron scattering dominate over electron-phonon scattering at low temperatures leading to the $\rho \propto T^2$ behavior below 50 K.

Figure 1

(Color online) Neutron powder-diffraction data measured (a) at 580 K and (b) 300 K. Circles are the data and the line represents the calculated intensity based on the lattice parameters listed in Table . The inset of (a) shows a closeup of a narrow range of wave vector $Q$ while the inset of (b) shows the monoclinic splitting of the two peaks around $Q\sim 2.5{\text{Å}}^{-1}$ into three peaks below 540 K. (c) The stacking of the ${\text{MnO}}_2$ and Ag layers along the $c$ axis in the high-temperature trigonal phase. (d) The local Jahn-Teller distortion of the ${\text{MnO}}_6$ octahedra in the ${\text{MnO}}_2$ layer.

Figure 2

(Color online) (a)–(c) Color contour maps of neutron-scattering intensity as a function of wave vector, $Q$, and energy transfer, $\hslash \omega$, measured with $\lambda =2.4\text{Å}$ at (a) 80 K, (b) 40 K, and (c) 1.4 K. (d) $\hslash \omega$ dependence of low-energy spin fluctuations measured with $\lambda =4.9\text{Å}$ at various temperatures spanning the phase transition. (e) $T$ dependence of the elastic magnetic intensity obtained by the neutron intensity over $1<Q<2{\text{Å}}^{-1}$ and $\left|dE\right|<0.1\text{meV}$. $T_f$ was determined from Fig. 4d.

Figure 3

(Color online) (a) Elastic neutron-scattering intensity, $\tilde{I}\left(Q\right)$, at 4 and 100 K. For $4\text{K}<T_f$, nonmagnetic background measured at 100 K was subtracted to get magnetic contributions only. The reflection indices are in the monoclinic notations. The lines are described in the text. (b) $ab$ projection of the ${\text{MnO}}_2$ triangular layer with a magnetic structure that is consistent with the characteristic wave vector, ${\mathit{q}}_m={\left(0.5,0.5,0\right)}_{\text{mono}}$, of $\tilde{I}\left(Q\right)$. The bigger and smaller spheres are Mn and O ions, respectively. The ovals represent the $d_{3r^2-z^2}$ orbitals with the local $z$ axis along the axially elongated O-Mn-O bonds. The thick (thin) blue lines refer to the short (long) bonds between neighboring Mn ions. The dotted lines represent the chemical unit cell of the monoclinic phase.

Figure 4

(Color online) (a)–(c) Energy dependence of the imaginary part of the dynamic susceptibility, ${\chi }^{″}\left(\omega \right)$, obtained by integrating and converting the inelastic neutron-scattering intensity $I\left(Q,\omega \right)$ shown in Fig. 2 over $1<Q<2{\text{Å}}^{-1}$ at (a) 100 K, (b) 40 K, and (c) 1.5 K. Dashed lines (black) and solid lines (blue) represent fits by spectral weight functions of Lorentzian type and arctan which give linewidths $\left({\Gamma }_0\right)$ and the lower limit $\left({\Gamma }_1\right)$ of spin-relaxation rates, respectively. (d) Spin-relaxation rates as a function of temperature. Lines are described in the text.

Figure 5

(Color online) The spin-exchange paths of ${\text{Ag}}_2{\text{MnO}}_2$ (a) within the ${\text{MnO}}_2$ layer and (b) between adjacent ${\text{MnO}}_2$ layers. The circles represent the ${\text{Mn}}^{3+}$ ions, and the numbers 1–6 refer to the spin exchanges $J_1–J_6$, respectively. (c) The spin arrangement in the ${\text{MnO}}_2$ layer showing that the NNN interchain interactions are not frustrated. The arrows represent some ferromagnetic $J_4$ interactions

Figure 6

The ordered spin states, defined in terms of the $\left(2a,4b,2c\right)$ magnetic supercell, used to extract the spin-exchange parameters $J_1–J_6$ by $\text{GGA}+U$ calculations with $U=2.5$ and 4.5 eV. Only the ${\text{Mn}}^{3+}$ ions of two ${\text{MnO}}_2$ layers are shown, and the unshaded and shaded circles represent the ${\text{Mn}}^{3+}$ ions with up and down spins, respectively. The relative energies (in millielectron volt per 32 FUs) of the seven ordered spin states are given in parentheses, where the first and second numbers refer to the relative energies with respect to the FM state obtained with $U=2.5$ and 4.5 eV, respectively.

Figure 7

(Color online) The electronic structure of the ordered spin state [Figs. 1c, 5c] of ${\text{Ag}}_2{\text{MnO}}_2$ obtained from $\text{GGA}+U$ calculations. All the results shown were obtained with $U=2.5\text{eV}$ except the ones with $U=4.5\text{eV}$ shown in the lower panels in (a) and (b). (a) The total DOS (black line) and PDOS of the Ag atoms (gray shading) in number of states/eV/4 formula units as a function of energy in electron volt. The up- and down-spin DOS are represented by plus and minus numbers, respectively. (b) The PDOS of the $\text{Mn}3d$ states calculated for one up-spin Mn site. (c) The PDOS calculated for the $\text{Mn}3d$, $\text{O}2p$, $\text{Ag}5s$, $\text{Ag}5p$, and $\text{Ag}4d$ states around the Fermi level. The horizontal axis covers the energy from $-0.2$ to $+0.2\text{eV}$. The up- and down-spin DOS plots are represented by solid and dashed curves, respectively. (d) The charge-density distribution associated with the Fermi level, obtained by considering the occupied and unoccupied states lying within 0.01 eV from the Fermi level.