High-field phase transitions in the orbitally ordered multiferroic $\mathrm{Ge}{\mathrm{V}}_4{\mathrm{S}}_8$

The high-field (H,T) phase diagram of the multiferroic lacunar spinel $\mathrm{Ge}{\mathrm{V}}_4{\mathrm{S}}_8$ has been studied by ultrasound, magnetization, and pyrocurrent experiments in magnetic fields up to 60 T. The title compound consists of molecular building blocks, with vanadium ${\mathrm{V}}_4$ clusters characterized by a unique electron density. These vanadium tetrahedra constitute a Jahn-Teller active entity, which drive an orbital-ordering transition at 30 K with the concomitant appearance of ferroelectricity. Ultrasound and magnetization experiments reveal sharp anomalies in magnetic fields of 46 T, which are associated with a first-order phase transition into an orbitally disordered state characterized by significant field and temperature hystereses. We report a sequence of complex magnetic, polar, and orbitally ordered states, i.e., the appearance of two orbitally ordered phases OO1 and OO2 for ${\mu }_{0}H<45\mathrm{T}$ and $T<30\mathrm{K}$. Beyond the paraelectric phase we further evidenced three ferroelectric phases, FE1, FE2, and FE3. Finally, antiferromagnetic (AFM) order $(T<15\text{\hspace{0.17em}K})$ and fully polarized ferromagnetic order $\left({\mu }_{0}H>60\mathrm{T}\right)$ have been observed in $\mathrm{Ge}{\mathrm{V}}_4{\mathrm{S}}_8$. At low temperatures and for fields below 40 T, AFM order coexists with the polar phase FE3 identifying a multiferroic state. Our results demonstrate a fascinating competition of the different orders, which the material manifests in high magnetic fields and at low temperatures.

Figure 1

(a) Crystallographic structure of the high-temperature phase of the lacunar spinel $\mathrm{Ge}{\mathrm{V}}_4{\mathrm{S}}_8$. The constituting ions are colored as indicated in the figure. Yellow bars indicate the ${\mathrm{V}}_4{\mathrm{S}}_4$ cubane clusters, black bars the $\mathrm{Ge}{\mathrm{S}}_4$ tetrahedra, which are light gray colored. The solid black lines, which constitute a cube, indicate the fcc high-temperature structure of the lacunar spinels. a, b, and c correspond to the crystallographic axes of the cubic cell. (b) Level scheme of the ${\mathrm{V}}_4$ clusters with a unique electron and spin density. The figure has been adapted from Ref. [3]. The eight electrons of the vanadium cluster are distributed among the singlet, the doublet, and the triplet, with two electrons in the excited triplet constituting a spin $S=1$, and hence, these clusters are Jahn-Teller active.

Figure 2

Relative change of the temperature-dependent sound velocity Δv/v (left scale) and of the sound attenuation Δα (right scale) of the transverse acoustic mode propagating along the $\langle 111\rangle$ direction in zero magnetic field in single-crystalline $\mathrm{Ge}{\mathrm{V}}_4{\mathrm{S}}_8$. Sound velocity and damping were normalized at the lowest temperatures. The structural $T_{\mathrm{s}}$ and magnetic $T_{\mathrm{N}}$ phase-transition temperatures are indicated. The inset shows the anomalous damping at the phase transitions with hysteretic behavior on an enlarged scale. Here the arrows indicate ordering temperatures as well as heating and cooling directions.

Figure 3

Relative change of the sound velocity Δv/v of the longitudinal acoustic mode propagating along the $\langle 111\rangle$ direction vs temperature in external magnetic fields of 0, 5, and 10 T, in single crystalline $\mathrm{Ge}{\mathrm{V}}_4{\mathrm{S}}_8$. The arrows indicate heating and cooling direction. All data were normalized to zero relative change for the measurements on heating in zero external magnetic fields. For clarity, the curves at 5 and 10 T are shifted with respect to the vertical axis.

Figure 4

Relative change of the sound velocity Δv/v of the transverse acoustic mode along $\langle 111\rangle$ vs magnetic field for $\mathrm{Ge}{\mathrm{V}}_4{\mathrm{S}}_8$ at a series of temperatures crossing the structural and magnetic phase transitions: (a) below and close to the zero-field AFM phase transition at $T_{\mathrm{N}}$ and (b) in the zero-field paramagnetic, but still orbitally ordered phase. Arrows indicate measurements for up and down-field sweeps. In all experiments, at the structural phase transitions, extreme hysteretic behavior occurs, indicating that the experiments may not always have been conducted in thermal equilibrium (see text).

Figure 5

Relative change of the sound velocity Δv/v of the longitudinal acoustic mode along $\langle 111\rangle$ vs magnetic field for $\mathrm{Ge}{\mathrm{V}}_4{\mathrm{S}}_8$ at a series of temperatures crossing the structural and magnetic phase transitions: (a) below and close to the zero-field antiferromagnetic (AFM) transition at $T_{\mathrm{N}}$ and (b) in the zero-field paramagnetic (PM) phase.

Figure 6

Magnetization $M$ vs magnetic field $H$ for single crystalline $\mathrm{Ge}{\mathrm{V}}_4{\mathrm{S}}_8$ measured along the $\langle 111\rangle$ direction at a series of temperature between 1.5 and 40 K. For clarity, the curves are shifted by approximately by $0.2{\mu }_{\mathrm{B}}$ along the vertical axis. Arrows indicate the field-sweep directions. The inset shows a magnetization measurement up to 100 T at 7 K to document the transition into the polarized ferromagnetic phase just above 60 T.

Figure 7

Ferroelectric polarization $P$ vs magnetic field $H$ measured along $\langle 111\rangle$ for different temperatures for $\mathrm{Ge}{\mathrm{V}}_4{\mathrm{S}}_8$. In all experiments shown, the external field was applied parallel to the electric field direction. (a) Polarization experiments below 15 K in the magnetically ordered phase. (b) FE polarization in the PM phase. In all panels, the arrows indicate the field sweep directions. In all field-dependent measurements, it was assumed that a paraelectric (PE) phase with zero polarization is reached above 60 T.

Figure 8

(H,T) phase diagram of single crystalline $\mathrm{Ge}{\mathrm{V}}_4{\mathrm{S}}_8$ with the external magnetic field applied along the $\langle 111\rangle$ direction. The anomalies detected during up sweeps are used to construct the phase diagram. The anomalies as determined in static (s) and pulsed (Us) ultrasound experiments utilizing longitudinal (L) and transverse (T) sound waves are indicated together with anomalies from magnetization $\left(M\right)$ and pyrocurrent $\left(P\right)$ experiments. Anomalies from heat-capacity experiments $\left(C_{\mathrm{p}}\right)$ by Widmann et al. [15] are indicated as well. Magnetic phases: antiferromagnetic (AFM), paramagnetic (PM), and field-induced ferromagnetic (FM) phases were determined. In addition, a zoo of polar phases, paraelectric (PE), ferroelectric (FE1–FE3), as well as orbitally disordered (OD), and orbitally ordered (OO1–OO2) phases, were identified. Solid lines serve as guides to the eye. As documented in the field-dependent experimental results, the measurements were not always performed in thermal equilibrium. For this reason, only critical temperatures in up-field cycles are shown in this phase diagram.

Figure 9

Adiabatic temperature changes of $\mathrm{Ge}{\mathrm{V}}_4{\mathrm{S}}_8$ measured in pulsed magnetic fields for (a) $\mathbf{H}\left|\right|<111>$ and (b) $\mathbf{H}\left|\right|<100>$. The results for the selected initial temperatures are shown. The arrows show the field-sweep directions.