Quantum tricritical point in the temperature-pressure-magnetic field phase diagram of ${\mathrm{CeTiGe}}_3$

We report the temperature-pressure-magnetic-field phase diagram of the ferromagnetic Kondo-lattice ${\mathrm{CeTiGe}}_3$ determined by means of electrical resistivity measurements. Measurements up to $\sim 5.8\mathrm{GPa}$ reveal a rich phase diagram with multiple phase transitions. At ambient pressure, ${\mathrm{CeTiGe}}_3$ orders ferromagnetically at $T_{\text{C}}=14$ K. Application of pressure suppresses $T_{\text{C}}$, but a pressure-induced ferromagnetic quantum criticality is avoided by the appearance of two new successive transitions for $p>4.1$ GPa that are probably antiferromagnetic in nature. These two transitions are suppressed under pressure, with the lower-temperature phase being fully suppressed above 5.3 GPa. The critical pressures for the presumed quantum phase transitions are $p_1\cong 4.1$ GPa and $p_2\cong 5.3$ GPa. Above 4.1 GPa, application of magnetic field shows a tricritical point evolving into a wing-structure phase with a quantum tricritical point at 2.8 T at 5.4 GPa, where the first-order antiferromagnetic-ferromagnetic transition changes into the second-order antiferromagnetic-ferromagnetic transition.

Figure 1

(a) Temperature dependence of the in-plane resistivity $\rho \left(T\right)$ of a ${\mathrm{CeTiGe}}_3$ single crystal under various pressures $p$ up to 5.76 GPa on a semilog plot. The resistivity at 300 K linearly increases with the pressure at a rate of 7.$4\mu \mathrm{\Omega }$ cm ${\mathrm{GPa}}^{-1}$ from 0 to 5.76 GPa, as shown in the inset. (b) Low-temperature resistivity at various pressures. Data are offset by increments of $10\mu \mathrm{\Omega }$ cm for clarity.

Figure 2

Low-temperature, in-plane resistivity (left axis) and its corresponding temperature derivative (right axis) of ${\mathrm{CeTiGe}}_3$ for several representative pressure regions: (a) and (b) $p<4.1$ GPa, (c)–(e) 4.1 GPa $<p<5.3$ GPa, and (f) and (g) 5.3 GPa $<p$. The solid symbols mark the characteristic temperatures that are associated with phase transitions: black square, PM to FM; green triangle, PM to MP1/MP1'; and green star, MP1 to MP2. The insets of (c)–(e) show the observed hysteretic behavior at their representative pressures. However, no hysteretic behavior is observed above 5.3 GPa, as shown in inset of (f).

Figure 3

Evolution of the temperature derivative of the resistivity at low temperature for representative pressures. The data are vertically offset by $28\mu \mathrm{\Omega }$ cm ${\mathrm{K}}^{-1}$ to reduce overlap. Solid symbols represent the criteria described in Fig. 2. At 5.29 GPa there is an additional anomaly in $d\rho /dT$, as shown by the orange circle.

Figure 4

(a) $T-p$ phase diagram of ${\mathrm{CeTiGe}}_3$ in zero applied field. Transition temperatures are determined from the anomalies in $d\rho /dT$, as shown in Figs. 2 and 3. The values of critical pressure $p_1$ and $p_2$ are 4.1 and 5.3 GPa, respectively. Solid lines are a guide to the eye, and dashed lines are suggested extrapolations of phase boundaries. The red and blue lines represent the second- and first-order phase transitions. The vertical and horizontal arrows represent the Lifshitz and tricritical points, respectively. (b) Maximum in resistivity $T_{\text{max}}$ [shown in Fig. 1] as a function of pressure increase as an exponential function. (c) Pressure dependence of $\rho$ at 1.8 K.

Figure 5

(a) Temperature dependence of the resistivity at various fixed fields for $p=4.48$ GPa and $H\parallel c$. The data are vertically shifted by an integer of $25\mu \mathrm{\Omega }$ cm to avoid overlapping. The insets show the observed hysteretic behavior in the temperature scan. Solid and dashed lines represent the temperature increasing and decreasing, respectively. (b) Corresponding temperature derivative $d\rho /dT$ of (a). The data are vertically shifted by an integer of $15\mu \mathrm{\Omega }$ cm ${\mathrm{K}}^{-1}$ to avoid overlapping. Solid symbols represent the criteria used to obtain the transition temperatures at various magnetic fields. (c) Field dependence of the resistivity at fixed temperatures. For these data the sample was cooled in zero field, and then $\rho \left(H\right)$ data were collected for increasing field ${\rho }_{\mathrm{up}}$ and then decreasing field ${\rho }_{\mathrm{down}}$. Then the temperature was increased to the desired value, and data were collected for increasing and decreasing field. Solid and dashed lines represent the field increasing and decreasing, respectively. Insets show the observed hysteretic behavior and the criteria used to obtain the transition fields. Above 7 K no hysteretic behavior is observed.

Figure 6

$T-H$ phase diagrams for various pressures, (a) 4.48, (b) 4.86, (c) 5.15, and (d) 5.46 GPa, determined by tracking various anomalies in temperature and field derivatives of the resistivity measurement as shown in Fig. 5. Solid and open symbols represent transition temperatures determined by $T$ sweeps and transition fields determined by $H$ sweeps (as described in Fig. 5), respectively. Dashed blue and solid red lines indicate the first-order and second-order transitions, respectively. (e) Temperature dependence of hysteresis widths for the transition at H1 at various pressures. The data are vertically offset by 0.03 T to avoid overlap. Vertical arrows represent the estimated tricritical points for each pressure. Zero for each data set is shown on the right-hand axis.

Figure 7

$T-H$ phase diagrams, including those shown in Figs. 6–6, at various increasing applied pressures. At 5.29 GPa, $T-H$ phase diagrams show a complex behavior with additional metamagnetic transitions (gray and brown open triangles) in $\rho \left(H\right)$ data (raw data are not shown). H1, H2, and H3 are the anomalies observed in $\rho \left(H\right)$ data as shown in Figs. 5, 8, and 8.

Figure 8

(a) Field dependence of $\rho$ at 1.8 K for various pressures. Solid and dashed lines represent the field increasing and decreasing, respectively. Representative $\rho \left(H\right)$ data for (b) 4.1 GPa $<p<5.3$ GPa and (c) $p>5.3$ GPa and the criteria used to obtain the transition fields at 1.8 K. The open symbols represent the corresponding transition fields. The gray star represents the shoulderlike anomaly that appeared at 5.76 GPa (Fig. 7). (d) and (e) Representative derivative $d\rho /dH$ data for the H1 transition at 1.8 K. The blue lines represent the Gaussian+linear-background fitted curves which are used to obtained the full width of the H1 transition. (f) The left axis shows the pressure dependence hysteresis width of transition H1 at 1.8 K. The right axis shows the pressure dependence of the full width of H1 obtained with $d\rho /dH$ [(d) and (e)] at 1.8 K. The vertical dashed line represents the tricritical pressure $\sim 5.3\mathrm{GPa}$ at 1.8 K. (g) $H-p$ phase diagram at 1.8 K based on the criterion shown in (b) and (c). Blue dashed and red solid lines represent the first- and second-order transitions. Red open circle represents the extrapolated QTCP.

Figure 9

Projection of the tricritical points (TCPs) in the (a) $T-H$, (b) $T-p$, and (c) $H-p$ planes. Red solid circles represent the TCP determined by Fig. 6. Red solid squares were obtained from Fig. 8 (f). Dashed lines are guides to the eyes, and open red circles represent the extrapolated QTCP.

Figure 10

The constructed, partial $T-p-H$ phase diagram of ${\mathrm{CeTiGe}}_3$ based on resistivity measurements. Blue surfaces represent the first-order planes, and the green surface represents the second-order MP1,2,3,4 phase boundary. Solid red and dashed blue lines represent the second- and first-order transitions, respectively. The open circle represents the extrapolated QTCP.