Electronic structure and magnetic properties of pyroxenes $(\mathrm{Li},\mathrm{Na})\mathrm{TM}{(\mathrm{Si},\mathrm{Ge})}_2{\mathrm{O}}_6$: Low-dimensional magnets with 90° bonds

The results of the $\mathrm{LSDA}+U$ calculations for pyroxenes with diverse magnetic properties $(\mathrm{Li},\mathrm{Na})\mathrm{TM}{(\mathrm{Si},\mathrm{Ge})}_2{\mathrm{O}}_6$, where TM is the transition metal ion (Ti, V, Cr, Mn, Fe), are presented. We show that the anisotropic orbital ordering results in the spin-gap formation in $\mathrm{Na}\mathrm{Ti}{\mathrm{Si}}_2{\mathrm{O}}_6$. The detailed analysis of different contributions to the intrachain exchange interactions for pyroxenes is performed both analytically using perturbation theory and basing on the results of the band structure calculations. The antiferromagnetic $t_{2g}-t_{2g}$ exchange is found to decrease gradually in going from Ti to Fe. It turns out to be nearly compensated by ferromagnetic interaction between half-filled $t_{2g}$ and empty $e_g$ orbitals in Cr-based pyroxenes. The fine tuning of the interaction parameters by the crystal structure results in the ferromagnetism for $\mathrm{Na}\mathrm{Cr}{\mathrm{Ge}}_2{\mathrm{O}}_6$. Further increase of the total number of electrons and occupation of $e_g$ subshell makes the $t_{2g}-e_g$ contribution and total exchange interaction antiferromagnetic for Mn- and Fe-based pyroxenes. Strong oxygen polarization was found in Fe-based pyroxenes. It is shown that this effect leads to a considerable reduction of antiferromagnetic intrachain exchange. The obtained results may serve as a basis for the analysis of diverse magnetic properties of pyroxenes, including those with recently discovered multiferroic behavior.

Figure 1

(Color online) Crystal structure of pyroxenes. Two chains made of $\mathrm{TM}{\mathrm{O}}_6$ octahedra and its connections via $X{\mathrm{O}}_4$ (where $X$ is usually Si or Ge) tetrahedra are shown. Purple balls represent Li(Na) ions.

Figure 2

(Color online) (a) Triangularlike topology of the TM network in pyroxenes. Solid lines are intrachain links and dashed and double-dashed lines are the interchain connection via one or two $X{\mathrm{O}}_4$ tetrahedra; $J$, $J_1$, and $J_2$ are the corresponding exchange parameters. (b) Sketch of the one of $\mathrm{TM}{\mathrm{O}}_6$ chains. Two different types oxygens are shown. Those depicted as solid circles belongs to two different $\mathrm{TM}{\mathrm{O}}_6$ octahedra and open circle only to one. The later oxygens are called “side oxygens” in the text. The natural choice of the local coordinate system, when $z$ axis is directed along the single 180° O–TM–O bond connecting two neighboring $\mathrm{TM}{\mathrm{O}}_6$ octahedra, is shown.

Figure 3

(Color online) Typical orbital ordering between two occupied $\mathrm{TM}d$ orbitals via the same $\mathrm{O}2p$ orbital, which results to AFM interaction for the case of 90° TM–O–TM bond.

Figure 4

(Color online) Typical orbital ordering between occupied (filled) and unoccupied (unfilled) $d$ orbitals of TM via the same $\mathrm{O}2p$ orbital, which results to FM interaction for the case of 90°TM–O–TM bond.

Figure 5

(Color online) Typical orbital ordering between two occupied $d$ orbitals of TM via different $\mathrm{O}2p$ orbitals, which results in a FM interaction for the case of 90° TM–O–TM bond.

Figure 6

(Color online) Total and projected DOS for $\mathrm{Na}\mathrm{Ti}{\mathrm{Si}}_2{\mathrm{O}}_6$ obtained in $\mathrm{LSDA}+U$ calculation. Total DOS is the sum for both spins. Up (down) panel corresponds to spin majority (minority). Fermi level is zero energy.

Figure 7

(Color online) The orbital ordering obtained in $\mathrm{LSDA}+U$ calculations for $\mathrm{Na}\mathrm{Ti}{\mathrm{Si}}_2{\mathrm{O}}_6$ is shown.

Figure 8

(Color online) Total and projected DOS for $\mathrm{Na}\mathrm{Cr}{\mathrm{Ge}}_2{\mathrm{O}}_6$ obtained in $\mathrm{LDA}+U$ calculation. Fermi level is zero energy.

Figure 9

(Color online) $\mathrm{Cr}3d$ DOS for $\mathrm{Na}\mathrm{Cr}{\mathrm{Si}}_2{\mathrm{O}}_6$ and $\mathrm{Na}\mathrm{Cr}{\mathrm{Ge}}_2{\mathrm{O}}_6$ obtained in LDA. It illustrates the decrease of the $t_{2g}-e_g$ crystal-field splitting with substitution Si by Ge. Fermi level is zero energy.

Figure 10

(Color online) Sketch illustrating mechanism of the spin polarization on oxygen ions in Fe-based pyroxenes.