Origin of magnetovolume effect in a cobaltite

The layered perovskite ${\mathrm{PrBaCo}}_2{\mathrm{O}}_{5.5+x}$ demonstrates a strong negative thermal expansion (NTE) which holds potential for being fabricated into composites with zero thermal expansion. The NTE was found to be intimately associated with the spontaneous magnetic ordering, known as magnetovolume effect (MVE). Here we report with compelling evidence that the continuouslike MVE in ${\mathrm{PrBaCo}}_2{\mathrm{O}}_{5.5+x}$ is intrinsically of discontinuous character, originating from a magnetoelectric transition from an antiferromagnetic insulating large-volume (AFILV) phase to a ferromagnetic less-insulating small-volume (FLISV) phase. Furthermore, the magnetoelectric effect (ME) shows high sensitivity to multiple external stimuli such as temperature, carrier doping, hydrostatic pressure, magnetic field, etc. In contrast to the well-known ME such as colossal magnetoresistance and multiferroic effect which involve symmetry breaking of crystal structure, the ME in the cobaltite is purely isostructural. Our discovery provides a pathway to realizing the ME as well as the NTE, which may find applications in new techniques.

Figure 1

Unit cell volume (a), magnetovolume (b),(c), Co-ion magnetic moments (d), transition model (e), and phase diagram (f) of ${\mathrm{PrBaCo}}_{\mathbf{2}}{\mathrm{O}}_{\mathbf{5}.\mathbf{5}+\mathbf{x}}$. All the results are derived from analysis of high-resolution NPD data. (a) Volume of sub unit cell ($2a_p\times 2a_p\times 2a_p$) as a function of temperature for various hole-doping fraction $x$ which shows that anomalous thermal expansion is enhanced with increasing $x$ until maximum at $x=0.24$ and drops with further hole doping. (b) Square of magnetic moment ${\left|M\right|}^2$ as a function of volumetric order parameter $\mathrm{\Delta }V$ for $x=0.24$. $M$ is determined from Rietveld refinement using a G-type antiferromagnetic structure model. $\mathrm{\Delta }V$ is extracted by subtracting the phonon contribution, which is calculated on the basis of the Debye-Gruneisen model as shown in the inset. The details of the calculations are described in the Appendix. The solid red line is the linear fit to the data. (c) Spontaneous magnetovolume contribution at 10 K as a function of hole-doping fraction $x$, derived in the same ways as that for $x=0.24$. (d) Magnetic moment of single Co ion at $T=10\mathrm{K}$ as a function of hole doping level $x$, which is determined from Rietveld refinement. The residual values ${\mathrm{R}}_{wp}$ is both below 10% and ${\mathrm{R}}_M$ below 20%. For the spin-state ordered (SSO) phase at $x=0$, where different crystallographic sites of Co bear different magnetic moments, the magnetic moment is averaged over all Co sites. For the coexisting long-range ordered ferromagnetic and antiferromagnetic phases at $x=0.25$, the magnetic moment is averaged over both phases. (e) The magnetoelectric transition model for NTE of $x=0.24$, i.e., the transition from an AFILV phase with antiferromagnetic (G-type) structure of Co spins (the upper-left drawing) to a FLISV phase with the ferromagnetic structure (the lower-right drawing). (f) Phase diagram for ${\mathrm{PrBaCo}}_2{\mathrm{O}}_{5.5+x}$, summarized from our previous [21] and present NPD experiments. AF(SSO), AF(G-type), and F denote the antiferromagnetic (spin-state ordering), antiferromagnetic (G-type), and ferromagnetic states, respectively. $T_{N1}$ (green squares) and $T_{N2}$ (green circles), $T_N$ (green triangles), and $T_C$ (black triangles) represent the transition temperatures for these long-range ordered magnetic structures. Strong phase fluctuations occur near the boundary between F and AF phases, as highlighted by the red shadows.

Figure 2

Hydrostatic-pressure NPD on ${\mathrm{PrBaCo}}_{\mathbf{2}}{\mathrm{O}}_{\mathbf{5}.\mathbf{5}+\mathbf{x}}$ ($\mathbf{x}=0.24$). All the indices of reflections are given under the framework of nuclear unit cell ($a_p\times 2a_p\times 2a_p$). There is no indication of symmetry breaking of the crystal structure under hydrostatic pressure up to $P=2.3$ GPa. (a),(b) Diffraction profiles showing antiferromagnetic (a) and ferromagnetic (b) peaks under ambient pressure and 0.8 GPa at 80 K. In the presence of hydrostatic pressure, the antiferromagnetic reflection $\frac{1}{2}11$ is fully suppressed while the ferromagnetic reflections are identified from the increase in integrated intensity of reflections 002, 100, and 020, indicating the pressure-induced antiferromagnetic-ferromagnetic transition. (c) Integrated intensity of the magnetic reflections $\frac{1}{2}11$ and 002 as a function of pressure at 80 K, suggesting the antiferromagnetic-ferromagnetic transition starts from below 0.4 GPa and completes at 0.8 GPa. (d) Volume of sub unit cell ($2a_p\times 2a_p\times 2a_p$), obtained from Rietveld refinement, as a function of temperature under ambient pressure and 1.4 GPa, showing the NTE under ambient pressure disappears at high pressure (in the pure F phase). Between 80 to 150 K, the pressure varies from 1.38 GPa to 1.45 GPa. Assuming linear compressibility, the error of volume was estimated by $\sigma =\frac{1.55\%}{2.3\text{GPa}}\times [P\left(T\right)-1.38\text{GPa}]\times 460{\text{Å}}^3$, where the volume decreases by 1.55% at 2.3 GPa based on the diffraction experiment. (e) Phase fraction for ferromagnetic (F) and antiferromagnetic (AF) phases, obtained from integrated intensity as shown in panel (c), as a function of hydrostatic pressure at 80 K of $x=0.24$. (f) Volume of sub unit cell ($2a_p\times 2a_p\times 2a_p$) obtained from Rietveld refinement, as a function of hydrostatic pressure at 80 K of the $x=0.24$ sample.

Figure 3

High-pressure NPD (a),(b) for Pb, high-pressure NPD (c)–(e), and magnetization (f) for ${\mathrm{PrBaCo}}_{\mathbf{2}}{\mathrm{O}}_{\mathbf{5}.\mathbf{5}+\mathbf{x}}$ ($\mathbf{x}=0.24$). (a) Diffraction profiles of reflection 111 of Pb, which was used for pressure calibration, under 0.08 GPa and 0.8 GPa at 80 K. (b) Integrated intensity of the reflection 111 of Pb as a function of pressure at 80 K. (c) Diffraction profiles of reflection 122 of $x=0.24$, under 0.08 GPa and 0.8 GPa at 80 K. (d) Full width at half maximum (FWHM) over $d$-spacing $\frac{\text{FWHM}}{d}$ (relative peak width) of Bragg reflections (002 and 122 of $x=0.24$, and 111 of Pb) as a function of pressure. The FWHMs are all derived from the Gaussian fitting to the NPD profiles. (e) Rietveld refinement on the data at $T=80\mathrm{K}$ and $P=0.4$ GPa using the model of double phases (AFILV and FLISV). The F and AF structure models are shown in Fig. 1. Experimental data points are shown by red dots, and the black line through them is the fit by Rietveld analysis. Since the peaks of high-pressure NPD are too broad to identify the peak splitting from coexistence of the AFILV phase and the FLISV phase, the nuclear structures approximate to the same one as shown by the blue bars. The green bars denote the indices from antiferromagnetic structure, and magenta bars represent those of the ferromagnetic structure. The red bars indicate the nuclear structure of Pb. The blue line shows the difference between experiment and calculation. The residual value ${\mathrm{R}}_{wp}$ is 16.30% and ${\mathrm{R}}_M$ is 38.4%. (f) Molar magnetization as a function of temperature ($\mathrm{M}\text{−}\mathrm{T}$ curve) measured under magnetic field of 0.1 T through zero-field cooling (ZFC) processes for various hydrostatic pressures. The $\mathrm{M}\text{−}\mathrm{T}$ curves at different pressures were measured in the direction of increasing pressure except for that at 0.19 GPa.

Figure 4

Magnetic-field NPD (a)–(d) and DC magnetization (e),(f) on ${\mathrm{PrBaCo}}_{\mathbf{2}}{\mathrm{O}}_{\mathbf{5}.\mathbf{5}+\mathbf{x}}$ ($\mathbf{x}=0.24$). All the indices of reflections are given under the framework of nuclear unit cell ($a_p\times 2a_p\times 2a_p$). No sign of symmetry breaking of nuclear crystal structure was observed under magnetic field up to 14 T. (a),(b) Diffraction profiles showing antiferromagnetic (a) and ferromagnetic (b) peaks under zero field and 14 T at 60 K. The antiferromagnetic reflection $\frac{1}{2}11$ is suppressed while the ferromagnetic intensity grows on top of nuclear reflections 022, 102, and 120 by applying magnetic field, indicating a field-induced antiferromagnetic-ferromagnetic transition. (c) Integrated intensity of the magnetic reflections 022 and 102 (upper panel) and $\frac{1}{2}11$ (lower panel) as a function of temperature under zero field and 14 T, respectively, showing that the magnetic field induces ferromagnetic ordering from 150 K while suppressing antiferromagnetic ordering from 120 K upon cooling. (d) Volume of sub unit cell ($2a_p\times 2a_p\times 2a_p$), obtained from Rietveld refinement, as a function of temperature under zero field and 14 T, showing the unit cell significantly contracts with applying magnetic field at low temperatures. (e) Molar magnetization as a function of temperature ($M\text{−}T$ curve) measured under magnetic field $H=14\mathrm{T}$ through both zero-field cooling (ZFC) and field cooling (FC) processes. (f) Molar magnetization as a function of magnetic field ($M\text{−}H$ curve) measured at 2 K and 80 K, respectively.

Figure 5

High-resolution NPD on ${\mathrm{PrBaCo}}_{\mathbf{2}}{\mathrm{O}}_{\mathbf{5}.\mathbf{5}+\mathbf{x}}$ ($\mathbf{x}=0.24$). All the indices of reflections are given under the framework of nuclear unit cell ($a_p\times 2a_p\times 2a_p$). (a) Diffraction profiles of the reflection 122 at different temperatures, showing the single peaks at all temperatures as well as the NTE. (b) Comparison of peak width at $T=100\mathrm{K}$ and $T=170\mathrm{K}$ for reflection 122 and reflection 004, respectively. The peaks are normalized to the same intensity and shifted to the same position for the better comparison. (c), (d) Full width at half maximum (FWHM) over $d$-spacing $\frac{\text{FWHM}}{d}$ (relative peak width) of nuclear reflection 122 and reflection 004 as a function of temperature. The FWHMs are all derived from the Gaussian fitting to the NPD profiles. Anomalous peak broadening occurs with decrease in temperature, indicating the transition between FLISV phase and AFILV phase. (e) Simulation of the $\frac{\text{FWHM}}{d}\mathrm{s}$ of reflection 122 in panel (c) with the discontinuous large-volume (LV)–small-volume (SV) phase transition model. The $\frac{\text{FWHM}}{d}$ of the single peak of LV (black line) or SV (blue line) is assumed to be 0.105% which is derived from the linear temperature dependence of $\frac{\text{FWHM}}{d}$ in SV phase (above 170 K) in panel (c). The sum of the LV and SV peaks simulates the profile (red line) observed by NPD. By assigning proper peak position separation $\mathrm{\Delta }{d}_{122}$ and relative volume fractions, the $\frac{\text{FWHM}}{d}\mathrm{s}$ at 70 K, 100 K, and 130 K are well reproduced. The temperature evolution of volumes fractions follows the LV-SV phase transition.

Figure 6

High-resolution NPD on ${\mathrm{PrBaCo}}_{\mathbf{2}}{\mathrm{O}}_{\mathbf{5}.\mathbf{5}+\mathbf{x}}$ ($\mathbf{x}=0.12$, 0.20, 0.26, 0.35, and 0.41) and transition model of $\mathbf{x}=0.12$. All the indices of reflections are given under the framework of nuclear unit cell ($a_p\times 2a_p\times 2a_p$). (a), (c)–(f), $\frac{\text{FWHM}}{d}$ (relative peak width) of nuclear reflection 122 as a function of temperature for hole-doping levels $x=0.12$, 0.20, 0.26, 0.35, and 0.41, respectively. The FWHMs are all derived from the Gaussian fitting to the NPD profiles. (b) The magnetoelectric transition model for NTE of $x=0.12$, i.e., the transition from an AFILV phase to a FLISV phase.

Figure 7

High-resolution NPD on ${\mathrm{PrBaCo}}_{\mathbf{2}}{\mathrm{O}}_{\mathbf{5}.\mathbf{5}+\mathbf{x}}$ ($\mathbf{x}=0.25$). All the reflection indices are given under the unit-cell framework ($a_p\times 2a_p\times 2a_p$). (a) Diffraction profiles of the reflection 122 at different temperatures. Peak splitting occurs in reflection 122 (and others) at low temperatures, indicating macroscopic phase separation of the AFILV phase and the FLISV phase. (b) Sub unit cell ($2a_p\times 2a_p\times 2a_p$) volume, mass ratio, and $\frac{\text{FWHM}}{d}$ of nuclear reflection 122 as a function of temperature. The $\frac{\text{FWHM}}{d}$ of the bottom panel was calculated from the single reflection 122 until it splits into two at about $T=50\mathrm{K}$ [panel (a)], below which the pattern was described by Rietveld refinement with the model of double phases [panel (d)], the FLISV phase and the AFILV phase. The resultant volume ($2a_p\times 2a_p\times 2a_p$) and mass ratio as a function of temperature are shown in top and middle panels. The averaged volume is calculated by $V_{\text{average}}=R_{\text{AFILV}}\times V_{\text{AFILV}}+R_{\text{FLISV}}\times V_{\text{FLISV}}$, where $R$ is the mass ratio. (c) Integrated intensities of antiferromagnetic reflection $\frac{1}{2}11$ and ferromagnetic reflection 002 as a function of temperature, showing the coexistence of antiferromagnetic and ferromagnetic phases. (d) Rietveld refinement on the data at $T=11\mathrm{K}$ using the model of double phases (FLISV and AFILV). The ferromagnetic and antiferromagnetic structure models are shown in Fig. 1. Experimental data points are shown by red dots, and the black line through them is a fit by Rietveld analysis. Red (magenta) bars denote the indices from nuclear (magnetic) structures of FLISV while blue (green) bars represent the indices from nuclear (magnetic) structures of AFILV phase. Blue line shows the difference between experiment and calculation. The residual value ${\mathrm{R}}_{wp}$ is 5.95% and ${\mathrm{R}}_M$ is 23.0%.

Figure 8

Resistivity measurement of ${\mathrm{PrBaCo}}_{\mathbf{2}}{\mathrm{O}}_{\mathbf{5}.\mathbf{5}+\mathbf{x}}$. (a) Resistivity as a function of temperature measured under zero magnetic field at various hole doping levels, which shows a sharp decrease in resistivity with increasing hole doping. The relatively lower value in $\rho$ for $x=0.26$ and 0.35 than other hole doping $x$ is possibly due to the lower boundary resistance of polycrystal grains compared with that in samples with other hole-doping fractions. (b)–(f) Resistivity as a function of temperature measured under zero magnetic field and $H=9\mathrm{T}$ for $x=0.06, x=0.24, x=0.26, x=0.35$, and $x=0.41$, respectively. The insets in (b)–(d) show the resistivity under zero field in a focused temperature window. (g) Magnetoresistance calculated from the relative resistivity difference between $H=0\mathrm{T}$ and 9 T as a function of temperature at various hole-doping level $x$, which shows that the strongest magnetoresistance occurs near the doping level of the AF-F phase boundary.

Figure 9

Muon spin relaxation ($\mu \mathrm{SR}$) measurements of ${\mathrm{PrBaCo}}_{\mathbf{2}}{\mathrm{O}}_{\mathbf{5}.\mathbf{5}+\mathbf{x}}$ ($\mathbf{x}=0.24$, 0.25, 0.35, and 0.41). (a)–(d), $\mu \mathrm{SR}$ spectra of $x=0.24$ at $T=4\mathrm{K}$ (a), $x=0.25$ at 6 K (b), $x=0.35$ at 6 K (c), and $x=0.41$ at 4 K (d) under various longitudinal fields (LFs). The spectrum of zero field at long $t$ is significantly lifted up with application of a weak LF 0.01 T, which arises from the decoupling of muon from the nuclear moments or spin glass component. The fitting equation for describing the zero-field (ZF) spectrum is $A_s{G}_z(H,t)=G_{\text{GKT}}\left(t\right)\{A_1\text{exp}[-\mathrm{\Lambda }t]+A_2[\frac{1}{3}+\frac{2}{3}\text{cos}\left({\gamma }_{\mu }Bt\right)]\}$ for $x=0.24$, 0.25, and 0.35, $A_s{G}_z(H,t)=G_{\text{LKT}}\left(t\right)\{A_1\text{exp}[-\mathrm{\Lambda }t]+A_2[\frac{1}{3}+\frac{2}{3}\text{cos}\left({\gamma }_{\mu }Bt\right)]\}$ for $x=0.41$, where $G_{\text{GKT}}\left(t\right)$ is the Gaussian Kubo-Toyabe relaxation function (nuclear moments distribution), $G_{\text{LKT}}\left(t\right)$ the Lorentzian Kubo-Toyabe relaxation function (spin glass component), and the other terms are described in the main text for Eq. (1). In all samples, the spectra under LFs are fitted to Eq. (1) because the nuclear moment component [$G_{\text{GKT}}\left(t\right)$] and spin glass component [$G_{\text{LKT}}\left(t\right)$] are fully decoupled by LFs. All the fittings are shown as the solid lines. (e)–(h) $\mu \mathrm{SR}$ spectra of $x=0.24$ at $T=130\mathrm{K}$ (e), $x=0.25$ at 180 K (f), $x=0.35$ at 178 K (g), and $x=0.41$ at 180 K (h) under various longitudinal fields (LFs). The zero-field (ZF) spectra indicate that the background is negligible. Since all spectra were collected above magnetic ordering temperatures, only the dynamic spin fluctuation and nuclear spin distribution were taken into account and all the spectra are fitted to the equation $A_s{G}_z(H,t)=A_1\text{exp}[-\mathrm{\Lambda }t]G_{\text{GKT}}\left(t\right)$, where $G_{\text{GKT}}\left(t\right)$ amounts to 0 under LFs since $G_{\text{GKT}}\left(t\right)$ is fully decoupled by LFs. The solid lines show the fittings.

Figure 10

Muon spin relaxation ($\mu \mathrm{SR}$) measurements of ${\mathrm{PrBaCo}}_{\mathbf{2}}{\mathrm{O}}_{\mathbf{5}.\mathbf{5}+\mathbf{x}}$ ($\mathbf{x}=0.24$ and 0.41). (a),(b) Magnetic volume fractions as a function of temperature for $x=0.24$ (a) and $x=0.41$ (b), derived from fraction (dynamic phase) = $\frac{A_1}{A_1+A_2}$ and fraction (static phase) = $\frac{A_2}{A_1+A_2}$, where $A_1$ and $A_2$ are obtained from the fittings of the spectra under LF of 0.01 T to Eq. (1). (c),(d) Temperature dependence of muon spin-lattice-relaxation rate $\mathrm{\Lambda }$ for $x=0.24$ (c) and $x=0.41$ (d), derived from fittings of the spectra under LF of 0.01 T to Eq. (1). The $\mathrm{\Lambda }$ for $x=0.41$ reduces to 0 upon cooling down to about 80 K, which is commonly observed in those materials showing magnetic ordering. However, the $\mathrm{\Lambda }$ for $x=0.24$ shows unusual behavior, i.e., it stays nonzero even down to the base temperature. The results imply that the strong phase fluctuations occur at $x=0.24$ so that the related dynamic spin fluctuations survive until the base temperature.

Figure 11

DFT calculations of ${\mathrm{PrBaCo}}_{\mathbf{2}}{\mathrm{O}}_{\mathbf{5}.\mathbf{5}+\mathbf{x}}$ ($\mathbf{x}=0.25$). (a) Sub unit cell ($2a_p\times 2a_p\times 2a_p$) used in DFT calculations. Numbers indicate Co site index. An apical oxygen atom located between Co1 and Co5 is removed to mimic the oxygen vacancy. (b) Spin density (blue: up-spin and red: down-spin state) of $\mathrm{Co}-3d$ orbital states in an energy range of $E_F-0.5\text{eV}<E<E_F$ in the antiferromagnetic (AF) state. (c) Density of states for antiferromagnetic (AF, upper panel) and ferromagnetic (F, lower panel) states of $x=0.25$, showing the antiferromagnetic state has an open gap while the ferromagnetic phase is gapless. The dashed line denotes Fermi level. The black shadowed lines show the total density of states and the colored lines indicate the contribution from each of the eight Co sites in the nuclear subcell ($2a_p\times 2a_p\times 2a_p$). (d) Calculated total energies for the AF and F states as a function of cell volume, fitted to a polynomial function to derive the energy minima as shown by dashed lines.