Orbital and magnetic transitions in geometrically frustrated vanadium spinels: Monte Carlo study of an effective spin-orbital-lattice coupled model

We present our theoretical and numerical results on thermodynamic properties and the microscopic mechanism of two successive transitions in vanadium spinel oxides $A{\mathrm{V}}_2{\mathrm{O}}_4$ ($A=\mathrm{Zn}$, Mg, or Cd) obtained by Monte Carlo calculations of an effective spin-orbital-lattice model in the strong correlation limit. Geometrical frustration in the pyrochlore lattice structure of V cations suppresses development of spin and orbital correlations, however, we find that the model exhibits two transitions at low temperatures. First, a discontinuous transition occurs with an orbital ordering assisted by the tetragonal Jahn-Teller distortion. The orbital order reduces the frustration in spin-exchange interactions and induces antiferromagnetic correlations in one-dimensional chains lying in the perpendicular planes to the tetragonal distortion. Second, at a lower temperature, a three-dimensional antiferromagnetic order sets in continuously, which is stabilized by the third-neighbor interaction among the one-dimensional antiferromagnetic chains. Thermal fluctuations are crucial to stabilize the collinear magnetic state by the order-by-disorder mechanism. The results well reproduce the experimental data, such as transition temperatures, temperature dependence of the magnetic susceptibility, changes of the entropy at the transitions, and the magnetic ordering structure at low temperatures. Quantum fluctuation effect is also examined by the linear spin-wave theory at zero temperature. The staggered moment in the ground state is found to be considerably reduced from saturated value and reasonably agrees with the experimental data.

Figure 1

(Color) Cubic unit cell of the lattice structure of vanadium spinel oxides $A{\mathrm{V}}_2{\mathrm{O}}_4$. (a) The pyrochlore lattice of vanadium cations (red balls). (b) The 3D edge-sharing network of $\mathrm{V}{\mathrm{O}}_6$ octahedra. Oxygen ions are on the corners of octahedra. Tetrahedral $A$ sites are omitted.

Figure 2

(Color) Spin-ordering structure proposed for $\mathrm{Zn}{\mathrm{V}}_2{\mathrm{O}}_4$ on the basis of the neutron scattering results. The ordering pattern consists of staggered AF chains in the $ab$ plane (red solid lines), which stack with a four-times period in the $c$ direction as up-up-down-down-… (blue dashed lines). (b) Projection of (a) from the $z$ direction. Symbols + and − denote the up and down spins, respectively.

Figure 3

(Color) Hopping integrals for the $\sigma$ bonds; $t_{\sigma }^{\mathrm{nn}}$ and $t_{\sigma }^{3\text{rd}}$ are for the nearest-neighbor sites and for the third-neighbor sites, respectively. The overlaps between $d_{xy}$ orbitals within the $xy$ plane (the solid arrows) and those between $d_{yz}$ orbitals within the $yz$ plane (the dashed arrows) are shown. The $d_{zx}$ overlaps are similarly taken into account. The gray arrow shows an example of second-neighbor pair.

Figure 4

(Color) (a) Tetragonal distortion and the level splitting. (b) The orbital-ordering pattern for the model (1) predicted by the mean-field argument in Ref. 33. The ferro-type (antiferro-type) orbital bonds are shown by the blue solid (red dashed) lines.

Figure 5

(a) The internal energy per site [Eq. (17)] and (b) the specific heat per site [Eq. (18)] at $J_3∕J=0.02$. Error bars are smaller than symbol sizes in (a). Typical error bars are shown in (b).

Figure 6

(a) The sublattice orbital moment in Eq. (20) at $J_3∕J=0.02$. The inset shows the three-state clock vector for the orbital state. (b)–(d) Electron density in each orbital for four sublattices 1–4 shown in Fig. 4b for $L=12$ [Eq. (21)]. Error bars are smaller than the symbol sizes.

Figure 7

(Color) Orbital-ordering structure obtained by MC calculations. Dark blue and light yellow octahedra in (a) contain V cations where $(d_{xy},d_{zx})$ and $(d_{xy},d_{yz})$ orbitals are occupied, respectively. They stack alternatively in the $z$ direction. Some of $d_{zx}$ and $d_{yz}$ orbitals are shown by white and black lobes, respectively. (b) is the projection of (a) from the $z$ direction. $d_{xy}$ orbitals are singly occupied at all the sites and not shown in the figures.

Figure 8

The average of the JT distortion defined in Eq. (22) at $J_3∕J=0.02$. Error bars are smaller than the symbol sizes.

Figure 9

(a) The staggered moment defined in Eq. (23). Error bars are smaller than the symbol sizes. (b) The staggered magnetic susceptibility in Eq. (27). (c) The Binder parameter defined in Eq. (28). The lines are guides for the eyes. All the results are calculated at $J_3∕J=0.02$.

Figure 10

(Color) Two sublattices; one consists of the [110] chains (blue dashed lines) and the other consists of the $\left[1\overline{1}0\right]$ chains (red solid lines). In each sublattice, chains are connected by the third-neighbor exchange $J_3$ in the $yz$ and $zx$ directions. Black (white) arrows show spins in the sublattice moment ${\mathbf{M}}_1\left({\mathbf{M}}_2\right)$.

Figure 11

The collinearity defined in Eq. (29) at $J_3∕J=0.02$. The lines are guides for the eyes.

Figure 12

The uniform magnetic susceptibility in Eq. (31) at $J_3∕J=0.02$. Error bars are smaller than the symbol sizes.

Figure 13

The staggered magnetic moment along (a) the $xy$ chains [Eq. (32)] and (b) the $yz$ chains [Eq. (33)] at $J_3∕J=0.02$. Error bars are smaller than the symbol sizes.

Figure 14

System-size extrapolations of the staggered magnetic moment along (a) the $xy$ chains and (b) the $yz$ chains at $J_3∕J=0.02$. Error bars are smaller than the symbol sizes. The gray lines are the fits to a linear function of $1∕\sqrt{L}$. See the text for details.

Figure 15

The spectral weights for the $x$, $y$, and $z$ directions calculated by Eqs. (40, 41, 42) at $J_3∕J=0.02$. Error bars are smaller than the symbol sizes. The solid (dashed) line shows $T_{\mathrm{N}}\left(T_{\mathrm{O}}\right)$.

Figure 16

Temperature dependences of the specific heat at (a) $J_3∕J=0.0$, (b) 0.01, (c) 0.02, (d) 0.03, (e) 0.04, and (f) 0.05, respectively. The dashed lines (the downward arrows) indicate $T_{\mathrm{O}}\left(T_{\mathrm{N}}\right)$. Typical error bars are shown.

Figure 17

Temperature dependences of the uniform magnetic susceptibility at (a) $J_3∕J=0.0$, (b) 0.01, (c) 0.02, (d) 0.03, (e) 0.04, and (f) 0.05, respectively. The dashed lines (the downward arrows) indicate $T_{\mathrm{O}}\left(T_{\mathrm{N}}\right)$. Error bars are smaller than the symbol sizes.

Figure 18

Temperature dependences of the orbital sublattice moment and the staggered magnetic moment at (a) $J_3∕J=0.0$, (b) 0.01, (c) 0.02, (d) 0.03, (e) 0.04, and (f) 0.05, respectively. The dashed lines (the downward arrows) indicate $T_{\mathrm{O}}\left(T_{\mathrm{N}}\right)$. Error bars are smaller than the symbol sizes.

Figure 19

Phase diagram for the spin-orbital-lattice coupled model (1) determined by classical Monte Carlo calculations. The shaded area $T_{\mathrm{N}}<T<T_{\mathrm{O}}$ shows the orbital- and lattice-ordered phase. The hatched area below $T_{\mathrm{N}}$ is the AF-ordered phase concomitant with the orbital and lattice orders. The orbital and lattice transition at $T_{\mathrm{O}}$ (the dashed line) is a first-order transition, while the AF transition at $T_{\mathrm{N}}$ (the solid line) is a second-order one. The lines are guides for the eyes.

Figure 20

Temperature dependences of (a) the specific heat per site and (b) the entropy sum [Eq. (44)] for the effective orbital model (43). Black symbols show the results without the JT distortion. Gray symbols show the results in the presence of the tetragonal JT level splitting with $D∕J_{\mathrm{O}}=0.4$ in Eq. (47).

Figure 21

Reduction of the staggered moment due to quantum fluctuations (open symbols) estimated by the linear spin-wave calculation at $J_{\mathrm{F}}∕J_{\mathrm{AF}}=-0.113$. The staggered moment is also plotted (filled symbols), which is given by $M_{\mathrm{S}}=(S-\Delta S)g{\mu }_{\mathrm{B}}$ and by assuming $g=2$. The lines are guides for the eyes.

Figure 22

(Color) Orbital-ordering patterns (a), (b), (c), and (d) for the categories 1, 2, 3, and 4 specified by Eqs. (A7, A8, A9, A10), respectively. The ferro-type (antiferro-type) orbital bonds are shown by the blue solid (red dashed) lines.

Figure 23

The parameter region of Eqs. (A17, A18, A19) is shown by the shaded area. The dashed and solid lines denote the conditions of Eqs. (A18, A19), respectively.