Spin-induced symmetry breaking in orbitally ordered NiCr${}_2$O${}_4$ and CuCr${}_2$O${}_4$

At room temperature, the normal oxide spinels NiCr${}_2$O${}_4$ and CuCr${}_2$O${}_4$ are tetragonally distorted and crystallize in the $I{4}_1/amd$ space group due to cooperative Jahn-Teller ordering driven by the orbital degeneracy of tetrahedral Ni${}^{2+}$ ($t_2^4$) and Cu${}^{2+}$ ($t_2^5$). Upon cooling, these compounds undergo magnetic ordering transitions; interactions are somewhat frustrated for NiCr${}_2$O${}_4$ but not for CuCr${}_2$O${}_4$. We employ variable-temperature high-resolution synchrotron x-ray powder diffraction to establish that at the magnetic ordering temperatures there are further structural changes, which result in both compounds distorting to an orthorhombic structure consistent with the $Fddd$ space group. NiCr${}_2$O${}_4$ exhibits additional distortion, likely within the same space group, at a yet-lower transition temperature of $T=30$ K. The tetragonal to orthorhombic structural transition in these compounds appears to primarily involve changes in NiO${}_4$ and CuO${}_4$ tetrahedra.

Figure 1

Magnetic measurements of the spinel NiCr${}_2$O${}_4$. (a) Zero field cooled and field cooled temperature-dependent magnetic susceptibility measured under a 1000 Oe dc field show three anomalies at 310, 65, and 30 K. NiCr${}_2$O${}_4$ displays little change in the magnetism at 310 K, and is seen to order ferrimagnetically at 65 K, with an additional change in the magnetic structure at 30 K. (b) The isothermal field-dependent magnetization measured above the magnetic ordering temperature shows paramagnetic behavior. At 2 K, the coercive field and saturation magnetization are significantly larger than what is observed at 45 K.

Figure 2

Normalized inverse magnetic susceptibility of NiCr${}_2$O${}_4$ showing ideal Curie-Weiss paramagnetism above 310 K. Weak compensated interactions arise at 310 K and persist to about 65 K below which strong uncompensated interactions dominate. The subtle magnetic transition at 30 K is shown in the inset.

Figure 3

Magnetic measurements of the spinel CuCr${}_2$O${}_4$. (a) Magnetic susceptibility as a function of temperature under a 1000 Oe dc field shows an increase in susceptibility at the magnetic ordering temperature $\approx$130 K in both zero field cooled and field cooled measurements. This is a paramagnetic to ferrimagnetic transition. (b) Isothermal field-dependent magnetization measured above (200 K) and below (2 K) the magnetic ordering temperature.

Figure 4

Magnetostructural coupling in NiCr${}_2$O${}_4$. (a) NiCr${}_2$O${}_4$ orders ferrimagnetically at $T_N$ = 65 K, where the normalized inverse magnetic susceptibility deviates negatively from ideal CW paramagnetic behavior. (b) A structural transition occurs at the ferrimagnetic ordering temperature seen from the splitting of the tetragonal 440 diffraction peak into 080 and 800 orthorhombic peaks. Below 30 K, a subtle peak narrowing and intensity change is coincident with anomalies in magnetic and specific heat measurements. Diffraction patterns at 125 and 7 K are shown to the right and left of the central panel. During this structural transition, the 440 reflection splits in the $Q$ range 10${}^{-2}$Å, well within the instrumental resolution of 10${}^{-4}$Å.

Figure 5

Magnetostructural coupling in CuCr${}_2$O${}_4$. (a) Long-range ferrimagnetic order occurs at $T_N$ $=$ 130 K in CuCr${}_2$O${}_4$ where the normalized inverse magnetic susceptibility of CuCr${}_2$O${}_4$ deviates negatively from ideal CW behavior. (b) Concurrent with the onset of magnetic order is a structural transition seen in the splitting of the tetragonal 322 reflection into orthorhombic 206 and 260 reflections. Diffraction patterns at 288 and 7 K are shown to the right and left of the central plot, respectively. This structural distortion results in reflection splitting in the $Q$ range 10${}^{-2}$Å, well within the instrumental resolution of 10${}^{-4}$Å.

Figure 6

High-resolution synchrotron powder x-ray diffraction of NiCr${}_2$O${}_4$ and CuCr${}_2$O${}_4$. (a) The low-temperature diffraction pattern of NiCr${}_2$O${}_4$ is indexed to the orthorhombic space group $Fddd$. The lowering of average crystal symmetry in NiCr${}_2$O${}_4$ from tetragonal to orthorhombic symmetry is illustrated by the splitting of the (b) tetragonal (440) reflection into (c) orthorhombic 800 and 080 reflections. (d) Like NiCr${}_2$O${}_4$, the low-temperature diffraction data of CuCr${}_2$O${}_4$ is indexed to the orthorhombic space group $Fddd$ which is evident in the splitting of (e) (220) tetragonal reflections into (f) 004 and 040 orthorhombic reflections. Structural models are fit to the x-ray powder diffraction patterns using the Rietveld refinement method.

Figure 7

Changes in lattice parameters as a function of temperature in NiCr${}_2$O${}_4$. (a) A cubic to tetragonal structural transition occurs at 310 K where the $a$ lattice constant of the cubic phase diverges into $a$ and $c$ lattice parameters of the tetragonal phase. The $a$ lattice constant of the tetragonal cell is multiplied by $\sqrt{2}$ to clearly follow trends in the lattice parameters of NiCr${}_2$O${}_4$. In the tetragonal phase, the $a$ parameter decreases (b) while $c$ increases (a) with decreasing temperature. At 65 K, a tetragonal to orthorhombic structural distortion occurs resulting in three distinct lattice constants as shown in (a) and (b). (c) Variation of the cell volume normalized by the number of formula units ($Z$) in each cell. A further structural distortion of orthorhombic NiCr${}_2$O${}_4$ occurs at 30 K where there is a slight discontinuity of the lattice parameters (a) and (b) and cell volume (c); this is highlighted by the dashed line at $T$ $=$ 30 K. In (a), (b), and (c) the error bars are smaller than the data symbols.

Figure 8

(a) The thermal evolution of lattice parameters of CuCr${}_2$O${}_4$ reveals a tetragonal $I{4}_1/amd$ to orthorhombic $Fddd$ structural transition at $\sim$130 K. The tetragonal $a$ lattice parameter has been multiplied by $\sqrt{2}$ to match the low temperature $b$ and $c$ lattice values of the orthorhombic $Fddd$ cell. (b) Temperature dependence of the cell volume normalized by the number of formula units ($Z$) in each cell shows a steady decrease with temperature. In both (a) and (b), the error bars are smaller than the data symbols.

Figure 9

The variation in NiO${}_4$ polyhedra as a function of temperature. (a) The Ni-O bond length remains relatively constant in all the structural phases. (b) The single O-Ni-O angle of the cubic phase separates into a larger angle and a smaller angle in the tetragonal phase. Below the orthorhombic transition, there are three distinct O-Ni-O angles.

Figure 10

Changes in the CuO${}_4$ polyhedra as a function of temperature. (a) There is an overall decrease in the Cu-O bond distance, (b) an increase in the larger O-Cu-O angle, and (c) a decrease in the smaller O-Cu-O angle coupled with a splitting of this angle. These trends are obtained from Rietveld refinement of synchrotron x-ray diffraction data.

Figure 11

Low-temperature (10 K) orthorhombic crystal structures of (a) NiCr${}_2$O${}_4$ projected down the [101] direction and (b) CuCr${}_2$O${}_4$ projected down the [011] direction. Ni (grey) and Cu (red) are tetrahedrally coordinated by oxygen (orange). Chromium is shown in blue. The elongation of NiO${}_4$ tetrahedra along with the compression of CuO${}_4$ polyhedra is clearly seen in the low-temperature average structures.

Figure 12

Entropy changes in NiCr${}_2$O${}_4$ and CuCr${}_2$O${}_4$ resulting from structural and magnetic transformations. (a) The heat capacity of NiCr${}_2$O${}_4$ shows two anomalies at 65 and 30 K. (b) Fisher heat capacity of NiCr${}_2$O${}_4$ indicating release of magnetic entropy occurring at the same temperatures where changes in specific heat are observed. (c) CuCr${}_2$O${}_4$ also shows two transitions in the heat capacity at 155 and 130 K. Concurrent with these changes in heat capacity of CuCr${}_2$O${}_4$ are variations in magnetic structure as illustrated by Fisher heat capacity shown in (d).

Figure 13

Changes in magnetic order and heat capacity of NiCr${}_2$O${}_4$ at $T$ $=$ 30 K are accompanied by a structural change. (a) Zero field cooled and field cooled temperature-dependent magnetic susceptibility measurements of NiCr${}_2$O${}_4$ show a change in magnetic order at $T$ $=$ 30 K. Concurrent with this transition in magnetism is a change in entropy indicated by the anomaly in heat capacity. The central panel tracks changes in intensity of the orthorhombic 080 and 800 reflections at $T$ $=$ 30 K illustrating that a structural change takes place at $T$ $=$ 30 K. (c) This structural change is also reflected in the temperature-dependent lattice constants of NiCr${}_2$O${}_4$ which vary at this temperature. The structural distortion above occurs in the $Q$ range 10${}^{-2}$Å within the instrumental resolution of 10${}^{-4}$Å.