Critical size limits for collinear and spin-spiral magnetism in ${\mathrm{CoCr}}_2{\mathrm{O}}_4$

The multiferroic behavior of ${\mathrm{CoCr}}_2{\mathrm{O}}_4$ results from the appearance of conical spin-spiral magnetic ordering, which induces electric polarization. The magnetic ground state has a complex size-dependent behavior, which collapses when reaching a critical particle size. Here the magnetic phase stability of ${\mathrm{CoCr}}_2{\mathrm{O}}_4$ in the size range of 3.6–14.0 nm is presented in detail using the combination of neutron diffraction with XYZ polarization analysis and macroscopic magnetization measurements. We establish critical coherent domain sizes for the formation of the spin spiral and ferrimagnetic structure and reveal the evolution of the incommensurate spin spiral vector with particle size. We further confirm the presence of ferroelectric polarization in the spin spiral phase for nanocrystalline ${\mathrm{CoCr}}_2{\mathrm{O}}_4$.

Figure 1

BF TEM micrographs of cobalt chromite samples (scale bars: 50 nm).

Figure 2

(a–b) ZFC/FC (full/open symbols) magnetization measurements of cobalt chromite NPs with different particle sizes. Blue and brown vertical lines correspond to the transition temperatures $T_{\mathrm{s}}$ and $T_{\mathrm{b}}$, respectively. (c–d) The temperature dependence of the inverse susceptibility. Arrows indicate the linear extrapolation in the paramagnetic regime to obtain the Curie temperature.

Figure 3

(a) Temperature dependence of the magnetic scattering cross section (red dots) of cobalt chromite nanoparticles with coherent domain size of 14.0 nm (sample AA550). Purple and green vertical lines represent the position of fundamental and satellite reflections, respectively. (b) Magnetic scattering contribution of all studied samples recorded at 3.5 K (red points: experimental data). The black line is Le-Bail fit and the blue line is the residual between the fit and experimental data.

Figure 4

(a) Size dependence of the propagation vector $\tau$ at the transition temperature (bulk value is presented by black line [13]). (b) Size dependence of the period of spin-spiral ${\omega }_{\mathrm{spiral}}$, and the coherence length of spin-spiral ordering ${\xi }_{\mathrm{spiral}}$ and collinear structure ${\xi }_{\mathrm{col}}$. The grey and red dashed lines represent $\xi$ = $d_{\mathrm{XRD}}$ and the bulk value of ${\omega }_{\mathrm{spiral}}$ [10], respectively.

Figure 5

Size-temperature dependence of the magnetic phase transitions. The blue and green horizontal lines in the magnetic phase diagram correspond to the bulk value of the Curie ${\theta }_{\mathrm{c}}$ and $T_{\mathrm{s}}$ spin-spiral transition temperature. The Curie ${\theta }_{\mathrm{c}}$ (blue), blocking $T_{\mathrm{b}}$ (brown) and spin-spiral transition $T_{\mathrm{s}}$ (green) temperatures are described by Eq. (1). $d_{\mathrm{c},\mathrm{spiral}}$ and $d_{\mathrm{c},\mathrm{col}}$ denote the critical nanocrystalline size for the spin-spiral and ferrimagnetic ordering, respectively. Results obtained from magnetization and neutron diffraction measurements are described by open and full symbols, respectively.