1Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan2RIKEN, Center for Emergent Matter Science, Wako, Saitama 352-0198, Japan3Institute for Materials Research, Tohoku University, Oarai, Ibaraki 311-1313, Japan4Department of Physics, Kobe University, Kobe 657-8501, Japan
Received January 24, 2024; Accepted July 16, 2024; Published August 22, 2024
We succeeded in growing high-quality single crystals of RGa6 (R: rare earth metals from Y to Ho except Eu) with the PuGa6-type tetragonal structure by the Ga-self flux method, and measured the electrical resistivity, specific heat, magnetic susceptibility, and magnetization, together with the de Haas–van Alphen (dHvA) experiment for SmGa6. All the magnetic RGa6 compounds order antiferromagnetically and the corresponding magnetic easy-axes are mainly directed along the [001] direction, except CeGa6 and SmGa6. Characteristic metamagnetic transitions were observed in PrGa6, NdGa6, TbGa6, DyGa6, and HoGa6 in the magnetization curves. The main Fermi surface of SmGa6 was detected in the dHvA experiment, with a cyclotron effective mass of about 1 m0 (m0: rest mass of an electron), revealing an antiferromagnet with a localized magnetic moment of Sm3+.
There exist several R (R: rare earth metals) - Ga compounds in the alloy phase diagram, such as La3Ga, La5Ga3, LaGa, LaGa2, and LaGa6 in the La–Ga compounds.1) Among them, RGa6 compounds crystallize in the PuGa6-type tetragonal structure (\(P4/nbm\), No. 125), as shown in Fig. 1. The nearest neighbor rare earth distance in the (001) plane is 4.319 Å, which is shorter than the distance of the nearest neighbor along the [001] direction, 7.694 Å, in LaGa6. Characteristic is that four Ga layers are stacked between two La layers, indicating a short distance between Ga ions both along the [001] direction and the (001) plane.
Figure 1. (Color online) Tetragonal crystal structure of LaGa6.
The unit cell contains two molecules of RGa6, revealing a compensated metal with equal volumes of electron and hole Fermi surfaces. In the previous reports on RGa6,2–6) all the RGa6 compounds are metallic and most likely order antiferromagnetically. The antiferromagnetic easy-axis in the magnetic structure can not be uniquely determined in the (001) plane because of the dense Ga layers between two La layers. The previous studies were, however, carried out by using the polycrystalline samples. This is because RGa6 compounds react peritectically and the peritectic temperature is very low.
We tried to grow single crystals of RGa6 and succeeded in growing very small single crystals of R = Y to Ho, except Eu. We measured the electrical resistivity, specific heat, magnetic susceptibility, and magnetization. Among these RGa6 compounds, the residual electrical resistivity is far below 1 µΩ·cm in SmGa6 and measured the de Haas–van Alphen (dHvA) effect to clarify the Fermi surface properties.
2. Experimental
We tried to grow single crystals of RGa6 (R: Y to Lu) by the Ga-self flux method. For example, starting materials of 3N (99.9% pure)-La and 5N-Ga with the constitution of \(1:99\) were inserted into an alumina crucible in the case of LaGa6. The crucible was encapsulated in a quartz tube, which was heated to 1000 °C over 1 day, kept at this temperature for 1 day, cooled quickly to 480 °C, and cooled slowly to room temperature over 12 days. The excess Ga was removed by spinning the ampoule in a centrifuge at 60 °C. As-grown single crystals of RGa6 were shown in Fig. 2.
YGa6 does not exist in the reported alloy phase diagram, but it was grown as very small single crystals, as shown in Fig. 2(a). Unfortunately, the sample of YGa6 was very small, 0.1 mm in size, and the measurements were not carried out for a superconductor YGa6.3) Moreover, we could not grow single crystals of ErGa6, TmGa6, and YbGa6. This is because RGa6 compounds form peritectically. The peritectic temperature is 477 °C in LaGa6, but decreases gradually with increasing the number of \(4f\) electrons, for example, 282 °C in YbGa6. EuGa6 and LuGa6 do not exist, as reported in the alloy phase diagram.
The crystal structure was confirmed using a single-crystal X-ray diffractometer (XtaLAB Mini, Rigaku) with graphite monochromated Mo-Kα radiation at room temperature. A small single crystal with a size of \(0.1^{3}\) mm3 was selected to minimize the absorption as well as a secondary extinction effect. Each single crystal was mounted on a glass fiber with epoxy. The structure parameters were refined using the program SHELEX,7) as shows in Table I.
Table I. Atomic coordinates and thermal parameters of RGa6 (R = Y, La–Ho) at room temperature determined by single-crystal X-ray measurements. The number of formula unit cell is \(\mathrm{Z}= 2\). \(B_{\text{eq}}\) is the equivalent isotropic atomic displacement parameter. \(R_{1}\) and \(wR_{2}\) are reliability factors. Standard deviations in the positions of the least significant digits are given in parentheses.
The lanthanide contraction in RGa6 is observed, as shown in Fig. 3. The lattice parameter of the c-value is large in YbGa6, which is due to the divalent nature.3) The directions of single crystalline samples were determined by the X-ray Laue method.
Figure 3. (Color online) Lattice parameters \((a, c)\) and volumes V (\(= a^{2} c\)) in RGa6 shown by red solid circles (●), where the data shown by squares (□) are cited from Ref. 3.
The electrical resistivity ρ was measured using a conventional four-probe DC/AC method. The specific heat (C) was measured using a thermal relaxation method with a physical property measurement system (PPMS; Quantum Design) down to 2 K and using a dilution refrigerator down to 0.15 K. The magnetic susceptibility χ and magnetization M were measured using a commercial superconducting quantum interference device (SQUID) magnetometer at temperatures down to 2 K and magnetic fields up to 7 T. The dHvA experiment was carried out by the standard field modulation method.
3. Experimental Results and Analyses
Electronic properties in RGa6
LaGa6
First we show the electronic properties in LaGa6. Figures 4(a) and 4(b) show the temperature dependence of electrical resistivity ρ and specific heat C, respectively. The resistivity for the current J along the [110] direction decreases monotonically with decreasing temperature. The residual resistivity \(\rho_{0}\) is extremely large; \(\rho_{0} = 18\) µΩ·cm. The present single crystalline sample is not good in quality.
Figure 4. (Color online) Temperature dependences of (a) the electrical resistivity ρ along \(J\parallel [110]\), and (b) specific heat C, where the inset shows the \(T^{2}\)-dependence of the specific heat in the term of \(C/T\) in LaGa6.
Figure 4(b) shows the temperature dependence of specific heat C, where the inset shows the \(T^{2}\)-dependence of \(C/T\) and a solid line represents a conventional fitting line of \(C/T =\gamma +\beta T^{2}\), where the specific heat C consists of electronic (\(C_{\text{e}} =\gamma T\)) and lattice (\(C_{\text{ph}} =\beta T^{3}\)) contributions. The electronic specific heat coefficient γ is estimated to be \(\gamma = 3.1\) mJ/(K2·mol) and the Debye temperature is \(\theta_{\text{D}} = 170\) K from the β-value of 2.2 mJ/(K4·mol).
CeGa6
Next we show the temperature dependences of electrical resistivity, specific heat, magnetic susceptibility in the form of \(M/H\), and magnetization M in Fig. 5. As shown in the inset of Fig. 5(a), the electrical resistivity decreases slightly below 2.5 K, where the measured lowest temperature was 2.0 K. This temperature \(T_{\text{mag}}\) corresponds to the onset of the magnetic ordering. The magnetic ordering temperature \(T_{\text{mag}}\) might be close to 2.0 K. The Kondo effect is not observed in resistivity because of no existence of \(-{\log T}\) dependence of the resistivity. A shoulder-like anomaly around 30 K might be attributed to the magnetic resistivity based on the crystalline electric field (CEF) of \(4f\) electrons.
Figure 5. (Color online) Temperature dependence of (a) the electrical resistivity ρ along \(J\parallel [110]\). Inset shows the low-temperature resistivity. Temperature dependences of (b) specific heat C, (c) magnetic specific heat \(C_{\text{mag}}\) and magnetic entropy \(S_{\text{mag}}\), and (d) magnetic susceptibility in the form of \(M/H\) and inverse susceptibility under \(\mu_{0} H = 1\) T. Inset of (b) shows the low-temperature specific heat under magnetic fields up to 1 T. (e) Magnetization curves at 2 K for \(H\parallel [100]\) and [001] in CeGa6. (f) Magnetic phase diagram for \(H\parallel [100]\), constructed by the specific heat data in CeGa6.
We measured the specific heat C down to 1.2 K and found the magnetic ordering, as shown in Fig. 5(b). The magnetic ordering temperature is \(T_{\text{mag}} = 1.8\) K, which is consistent with the result of electrical resistivity measurement. To clarify the nature of the present magnetic ordering, we measured the specific heat under magnetic field, as shown in the inset of Fig. 5(b). The magnetic ordering temperature \(T_{\text{mag}}\) is found to shift to lower temperatures with increasing magnetic fields. The corresponding magnetic phase diagram is shown in Fig. 5(f). This indicates that \(T_{\text{mag}}\) corresponds to the antiferromagnetic transition temperature or the Néel temperature \(T_{\text{N}}\).
We roughly estimated the magnetic specific heat \(C_{\text{mag}}\); \(C_{\text{mag}} = C (\text{CeGa$_{6}$}) - C (\text{LaGa$_{6}$})\). To estimate the magnetic entropy \(S_{\text{mag}}\), the low-temperature specific heat C of CeGa6 was estimated by drawing a straight line from a datum at 1.2 K. Here, the specific heat C in the magnetic compound is written as the sum of electronic (\(C_{\text{e}}\)), lattice (\(C_{\text{ph}}\)), magnetic (\(C_{\text{mag}}\)), and nuclear (\(C_{\text{nuc}}\)) contributions. The nuclear specific heat appears below approximately 1–2 K, and then the magnetic specific heat can be simply obtained as \(C_{\text{mag}} = C(\text{CeGa$_{6}$}) - C(\text{LaGa$_{6}$})\). Figure 5(c) represents the temperature dependences of \(C_{\text{mag}}\) and \(S_{\text{mag}}\), where \(S_{\text{mag}}\) is obtained by integrating \(C_{\text{mag}}/T\) over temperature. \(S_{\text{mag}}\) reaches \(R\ln 2\) at around 10 K, which corresponds to the doublet ground state in the CEF scheme. The short-range magnetic ordering might start below about 10 K.
Next we show in Figs. 5(d) and 5(e) the magnetic susceptibility χ in the form of \(M/H\) and the magnetization M, respectively. From the temperature dependence of \(H/M\) above 200 K under the magnetic field of \(\mu_{0}H = 1\) T, the effective magnetic moment \(\mu_{\text{eff}}\) and the paramagnetic Curie temperature are estimated to be \(\mu_{\text{eff}} = 2.4\)\(\mu_{\text{B}}\)/Ce and \(\theta_{\text{p}} = 6\) K for \(H\parallel [001]\) and 2.3 \(\mu_{\text{B}}\)/Ce and \(\theta_{\text{p}} = 16\) K for \(H\parallel [100]\), respectively, close to \(\mu_{\text{eff}} = 2.54\)\(\mu_{\text{B}}\)/Ce for Ce3+. The magnetization curve at 2 K is highly anisotropic between \(H\parallel [100]\) and [001]. We cannot definitely conclude the nature of the present magnetic ordering from the data of magnetic susceptibilities and magnetizations. This is because the \(\theta_{\text{p}}\) values for \(H\parallel [100]\) and [001] are positive, suggesting the ferromagnetic exchange interaction. The corresponding ferromagnetic ordering was suggested in Ref. 5. On the other hand, all the others RGa6 order antiferromagnetically. Moreover, the \(\theta_{\text{p}}\) values are positive in the next compound PrGa6, which orders antiferromagnetically. We conclude from the magnetic phase diagram in Fig. 5(f) obtained from the specific heat measurements that CeGa6 orders antiferromagnetically below \(T_{\text{N}} = 1.8\) K.
PrGa6
Figure 6(a) shows the temperature dependence of the electrical resistivity in the current along the \(J\parallel [110]\) direction. The antiferromagnetic ordering is not clear in the resistivity measurement. A clear λ-type specific heat jump is, however, observed at 9.6 K, as shown in Fig. 6(b). We calculated the magnetic specific heat and magnetic entropy as in the case of CeGa6. The magnetic entropy in Fig. 6(c) reaches \(R\ln 3\) at \(T_{\text{N}}\), indicating that a doublet state and a singlet state or three singlet states in the \(4f\) levels are involved in the magnetic ordering.
Figure 6. (Color online) Temperature dependence of (a) the electrical resistivity ρ along \(J\parallel [110]\). Temperature dependences of (b) the specific heat C, (c) magnetic specific heat \(C_{\text{mag}}\) and magnetic entropy \(S_{\text{mag}}\), and (d) magnetic susceptibility in the form of \(M/H\) and inverse susceptibility under \(\mu_{0} H = 1\) T. (e) Magnetization curves at 2 K for \(H\parallel [100]\) and [001]. (f) Magnetic phase diagram for \(H\parallel [001]\), constructed by the magnetization and specific heat data in PrGa6.
We measured the magnetic susceptibility in the form of \(M/H\) under \(\mu_{0}H = 1\) T, as shown in Fig. 6(d). A peak of the susceptibility for \(H\parallel [001]\) is observed at 8.8 K under 1 T, and decreases below 8.8 K, indicating the antiferromagnetic ordering. The \(\mu_{\text{eff}}\) and \(\theta_{\text{p}}\) values are obtained from the slope of the inverse susceptibility in the temperature range from 150 and 300 K; \(\mu_{\text{eff}} = 3.4\)\(\mu_{\text{B}}\)/Pr and \(\theta_{\text{p}} = 0.7\) K for \(H\parallel [100]\), and \(\mu_{\text{eff}} = 3.3\)\(\mu_{\text{B}}\)/Pr and \(\theta_{\text{p}} = 11.0\) K for \(H\parallel [001]\), respectively, approximately consistent with \(\mu_{\text{eff}} = 3.58\)\(\mu_{\text{B}}\)/Pr for Pr+3. Both the \(\theta_{\text{p}}\) values are positive, but PrGa6 is found to order antiferromagnetically below \(T_{\text{N}} = 9.6\) K.
From the results of magnetic susceptibilities and magnetization curves in Figs. 6(d) and 6(e), the magnetic easy-axis is found to be the [001] direction and the hard-axis in the [100] direction. Three metamagnetic transitions are observed at \(H_{\text{m}_{1}} = 1.02\) T, \(H_{\text{m}_{2}} = 1.35\) T, and \(H_{\text{m}_{3}} = 1.54\) T at 1.8 K and the magnetization saturates above \(H_{\text{m}_{3}}\), revealing an ordered moment \(\mu_{\text{s}} = 1.7\)\(\mu_{\text{B}}\)/Pr. The magnetic phase diagram is shown in Fig. 6(f).
NdGa6
The similar experimental results are obtained in an antiferromagnet NdGa6 with \(T_{\text{N}} = 10.4\) K, as shown in Fig. 7. The present single crystalline sample is also in high-quality as in PrGa6, indicating a residual resistivity \(\rho_{0}\simeq 1\) µΩ·cm, as shown in the inset of Fig. 7(a). The magnetic entropy at \(T_{\text{N}}\) is \(R\ln 4\), as shown in Fig. 7(c), indicating that the doublet ground state and the next exciting doublet state are involved in the antiferromagnetic ordering. There exists another broad peak at around 5 K. This is not the magnetic transition because of no change of the magnetic susceptibility at this temperature, suggesting that the anomaly around 5 K is the Schottky anomaly due to split CEF levels by an internal magnetic field in the antiferromagnetic ordering state.
Figure 7. (Color online) Temperature dependence of (a) the electrical resistivity ρ along \(J\parallel [110]\). Inset shows the low-temperature resistivity. Temperature dependences of (b) the specific heat C, (c) magnetic specific heat \(C_{\text{mag}}\) and magnetic entropy \(S_{\text{mag}}\), and (d) magnetic susceptibility in the form of \(M/H\) under \(\mu_{0} H = 1\) T. (e) Magnetization curves at 2 K for \(H\parallel [100]\) and [001]. (f) Magnetic phase diagram for \(H\parallel [001]\) in NdGa6.
We show in Fig. 7(d) the low-temperature susceptibility. The susceptibility in the form of \(M/H\) under 1 T for \(H\parallel [001]\) decreases with decreasing temperature below \(T_{\text{N}} = 10.4\) K. The magnetic easy-axis corresponds to the [001] direction. We have to mention that we could not obtain reliable data on the magnetic susceptibility at higher temperatures because of a very tiny sample, as shown in Fig. 2.
On the other hand, a clear metamagnetic transition is observed at \(H_{\text{m}} = 3.1\) T in the magnetization curve at 1.8 K for \(H\parallel [001]\), as shown in Fig. 7(e). The saturated magnetic moment is \(\mu_{\text{s}} = 2.4\)\(\mu_{\text{B}}\)/Nd. We constructed the magnetic phase diagram from the magnetization measurements, as shown in Fig. 7(f).
SmGa6
Figure 8(a) shows the temperature dependence of the electrical resistivity in the current along \(J\parallel [110]\) in SmGa6. The resistivity decreases below \(T_{\text{N}} = 4.0\) K and the residual resistivity is also far below 1 µΩ·cm, suggesting \(\rho_{0} = 0.1\) µΩ·cm, and then the residual resistivity ratio, RRR \((=\rho_{\text{RT}}/\rho_{0})\simeq 180\). A very high-quality single crystal is obtained in the present study. A clear λ-type magnetic transition is observed in the specific heat at \(T_{\text{N}} = 4.1\) K, as shown in Fig. 8(b). The temperature dependence of the specific heat in a magnetic field near the transition temperature is shown in the inset. The transition temperature remains almost unchanged in a magnetic field of 9 T. The magnetic entropy at \(T_{\text{N}}\) is \(R\ln 2\), as shown in Fig. 8(c), indicating the doublet ground state.
Figure 8. (Color online) Temperature dependence of (a) the electrical resistivity ρ along \(J\parallel [110]\). Inset shows the low-temperature resistivity. Temperature dependences of (b) the specific heat C, (c) magnetic specific heat \(C_{\text{mag}}\) and entropy \(S_{\text{mag}}\), and (d) magnetic susceptibility in the form of \(M/H\) and inverse susceptibility under \(\mu_{0} H = 3\) T. (e) Magnetization curves at 2 K for \(H\parallel [100]\) and [001]. (f) Magnetic phase diagram for \(H\parallel [100]\) in SmGa6.
The temperature dependence of the magnetic susceptibility is shown in 8(d). The decrease in the magnetic susceptibility below \(T_{\text{N}}\) suggests that the transition at \(T_{\text{N}}\) is an antiferromagnetic transition. The magnetic susceptibility is approximately explained as \begin{equation} \chi = N \left(\frac{\mu^{2}_{\text{eff}}}{3k_{\text{B}}(T-\theta_{\text{p}})} + \frac{20\,\mu^{2}_{\text{B}}}{7\Delta E} \right), \end{equation} (1) where \(\theta_{\text{p}}\) is the paramagnetic Curie temperature and \(\Delta E\) is the average energy between the \(J = 5/2\) ground state multiplet and the \(J = 7/2\) multiplet. The first term in Eq. (1) shows a Curie–Weiss contribution from the \(J = 5/2\) ground state multiplet, and the second term is a temperature-independent Van Vleck term susceptibility based on the \(J = 5/2\) and 7/2 multiplets. The value of \(\Delta E = 1800\) K is roughly estimated from the susceptibility data for \(H\parallel [100]\) in the temperature range from 150 and 300 K, together with \(\mu_{\text{eff}} = 0.5\)\(\mu_{\text{B}}\)/Sm. The present \(\mu_{\text{eff}}\) value is smaller than 0.85 \(\mu_{\text{B}}\)/Sm for Sm3+. If we use the susceptibility data in the temperature range from 50 to 200 K, \(\mu_{\text{eff}} = 0.8\)\(\mu_{\text{B}}\)/Sm is obtained but \(\Delta E = 3000\) K. The \(\Delta E\) value is much higher than \(\Delta E = 1500\) K for Sm3+. These are mainly due to small susceptibility data including the experimental errors. Moreover, we cannot obtain reliable data above 100 K for \(H\parallel [001]\).
The magnetization at 1.8 K in Fig. 8(e) is small in value in the present magnetic fields up to 7 T and then the antiferromagnetic phase boundary is almost unchanged, as shown in Fig. 8(f), which was obtained from the susceptibility and specific heat experiments.
GdGa6
The similar magnetic properties in GdGa6 with \(T_{\text{N}} = 12.5\) K are shown in Fig. 9. The present sample is also in high quality, revealing a residual resistivity \(\rho_{0} = 1\) µΩ·cm, as shown in the inset of Fig. 9(a). The specific heat indicates the λ-type anomaly at \(T_{\text{N}} = 12.5\) K. A broad anomaly around 5 K in Fig. 9(b) is due to a Schottky peak of the \(4f\) spins with \(S=7/2\), where the Zeeman splitting of the degenerated spins is realized due to a large magnetic exchange field in the magnetically ordered GdGa6. The magnetic entropy at \(T_{\text{N}}\) is close to \(R\ln 8\), as shown in Fig. 9(c).
Figure 9. (Color online) Temperature dependence of (a) the electrical resistivity ρ along \(J\parallel [110]\). Inset shows the low-temperature resistivity. Temperature dependences of (b) the specific heat C and (c) magnetic specific heat \(C_{\text{mag}}\) and magnetic entropy \(S_{\text{mag}}\), and (d) magnetic susceptibility in the form of \(M/H\) and the inverse one under \(\mu_{0} H = 1\) T. (e) Magnetization curves at 1.8 K for \(H\parallel [100]\) and [001], where the inset shows the low-field magnetization curves. (f) Magnetic phase diagram for \(H\parallel [001]\) in GdGa6.
The magnetic susceptibility in the form of \(M/H\) under \(\mu_{0}H = 1\) T is shown in Fig. 9(d), indicating an antiferromagnetic ordering at \(T_{\text{N}} = 12.5\) K. The \(\mu_{\text{eff}}\) and \(\theta_{\text{p}}\) values are \(\mu_{\text{eff}} = 7.9\)\(\mu_{\text{B}}\)/Gd and \(\theta_{\text{p}} = -39.2\) K for \(H\parallel [100]\), and \(\mu_{\text{eff}} = 7.8\)\(\mu_{\text{B}}\)/Gd and \(\theta_{\text{p}} = -43.1\) K for \(H\parallel [001]\), respectively, being in good agreement with \(\mu_{\text{eff}} = 7.94\)\(\mu_{\text{B}}\)/Gd for Gd+3. The magnetic easy-axis is \(H\parallel [001]\), indicating a metamagnetic transition at 1.2 T. The saturation field is roughly estimated from the \(T_{\text{N}}\) and \(\theta_{\text{p}}\) values for the hard-axis of \(H\parallel [100]\),8) following the relation of \(H_{\text{s}} = 5 (T_{\text{N}} -\theta_{\text{p}}) = 5\times (12.5 + 39)\ [\text{kOe}] = 260\,\text{kOe}= 26\,\text{T}\), which is in good agreement with the value of 26 T, which corresponds to the magnetic field showing 7 \(\mu_{\text{B}}\)/Gd, extrapolated from a straight line of the hard-axis magnetization. The magnetic phase diagram obtained from the specific heat and magnetization measurements is shown in Fig. 8(f).
TbGa6
We show in Fig. 10 the magnetic properties of an antiferromagnet TbGa6 with \(T_{\text{N}} = 17.9\) K. This compound is also in high quality, revealing a residual resistivity \(\rho_{0} = 0.5\) µΩ·cm. The resistivity decreases below 18 K, but the next magnetic transition at \(T_{\text{N}}' = 14.6\) K is not clear in the resistivity, as shown in the inset of Fig. 10(a). The former temperature corresponds to the Néel temperature, and the second one corresponds to the first-order magnetic transition, which is clearly represented in the specific heat in Fig. 10(b) and the magnetic susceptibility in Fig. 10(d). The magnetic entropy \(S_{\text{mag}}\) in Fig. 10(c) exceeds \(R\ln 3\) at \(T_{\text{N}}\) and is close to \(R\ln 5\) in magnitude, indicating that five CEF states (5 singlets or 1 singlet + 2 doublets or 3 singlets + 1 doublet) are involved in this antiferromagnetic ordering.
Figure 10. (Color online) Temperature dependence of (a) the electrical resistivity ρ along \(J\parallel [110]\). Inset shows the low-temperature resistivity. Temperature dependences of (b) the specific heat C and (c) magnetic specific heat \(C_{\text{mag}}\) and magnetic entropy \(S_{\text{mag}}\), and (d) magnetic susceptibility in the form of \(M/H\) and the inverse one under \(\mu_{0} H = 1\) T. (e) Magnetization curves at 1.8 K for \(H\parallel [100]\) and [001]. (f) Magnetic phase diagram for \(H\parallel [001]\) in TbGa6.
The magnetic easy-axis corresponds to the \(H\parallel [001]\) direction, as shown in Fig. 10(d). The \(\mu_{\text{eff}}\) and \(\theta_{\text{p}}\) values are \(\mu_{\text{eff}} = 10.2\)\(\mu_{\text{B}}\)/Tb and \(\theta_{\text{p}} = -30.3\) K for \(H\parallel [100]\), and \(\mu_{\text{eff}} = 9.7\)\(\mu_{\text{B}}\)/Tb and \(\theta_{\text{p}} = -13.9\) K for \(H\parallel [001]\), which are in agreement with \(\mu_{\text{eff}} = 9.72\)\(\mu_{\text{B}}\)/Tb for Tb+3. Interestingly, the magnetization curve at 1.8 K for \(H\parallel [001]\) indicates the metamagnetic transition, indicating a magnetization of 2.5 \(\mu_{\text{B}}\)/Tb at \(H_{\text{m}_{1}} = 3.5\) T and a magnetization of 4.8 \(\mu_{\text{B}}\)/Tb at \(H_{\text{m}_{2}} = 5\) T, as shown in Fig. 10(e). A much higher field is needed to reach 9 \(\mu_{\text{B}}\)/Tb of Tb+3. Figure 10(f) represents the magnetic phase diagram for \(H\parallel [001]\).
DyGa6
The similar magnetic properties are obtained for an antiferromagnet DyGa6 with \(T_{\text{N}} = 11.5\) K, as shown in Fig. 11. The present sample is also in high quality, revealing a residual resistivity \(\rho_{0} = 0.3\) µΩ·cm, as shown in Fig. 11(a). The resistivity decreases rather steeply below \(T_{\text{N}} =11.5\) K. The magnetic susceptibility in the form of \(M/H\) in Fig. 11(b) indicates a steep decrease below \(T_{\text{N}} =11.5\) K for \(H\parallel [001]\). The \(\mu_{\text{eff}}\) and \(\theta_{\text{p}}\) values are \(\mu_{\text{eff}} = 10.7\)\(\mu_{\text{B}}\)/Dy and \(\theta_{\text{p}} = -22.2\) K for \(H\parallel [100]\), and \(\mu_{\text{eff}} = 11.5\)\(\mu_{\text{B}}\)/Dy and \(\theta_{\text{p}} = -2.6\) K for \(H\parallel [001]\), are fairly in agreement with \(\mu_{\text{eff}} = 10.65\)\(\mu_{\text{B}}\)/Dy for Dy+3. Several metamagnetic transitions are observed in the magnetization curve at 1.8 K for \(H\parallel [001]\), indicating step-like increases of magnetization, and reaching a saturation moment of 10 \(\mu_{\text{B}}\)/Dy for Dy+3. The magnetic phase diagram is shown in Fig. 11(d). Note that we could not obtain the reliable data of the specific heat for DyGa6. It was very difficult to grow single crystals and moreover, the sample was very small in size.
Figure 11. (Color online) Temperature dependence of (a) the electrical resistivity ρ along \(J\parallel [110]\). Inset shows the low-temperature resistivity. (b) Temperature dependences of magnetic susceptibility in the form of \(M/H\) and the inverse one under \(\mu_{0} H = 1\) T. (c) Magnetization curves at 1.8 K for \(H\parallel [100]\) and [001]. (d) Magnetic phase diagram for \(H\parallel [001]\) in DyGa6.
HoGa6
The magnetic properties of an antiferromagnet HoGa6 with \(T_{\text{N}} = 6.4\) K are shown in Fig. 12. The present sample is also in high quality, revealing a residual resistivity \(\rho_{0} = 0.1\) µΩ·cm, as shown in the inset of Fig. 12(a). The specific heat in Fig. 12(b) represents a λ-type antiferromagnetic ordering at \(T_{\text{N}} = 6.4\) K and a broad peak based on the Schottky peak. The magnetic specific heat \(C_{\text{mag}}\) in Fig. 12(c) is close to \(R\ln 5\) at \(T_{\text{N}}\), revealing that five CEF states (5 singlets or 1 singlet + 2 doublets or 3 singlets + 1 doublet) as in the case of TbGa6 are involved in the antiferromagnetic ordering.
Figure 12. (Color online) Temperature dependence of (a) the electrical resistivity ρ along \(J\parallel [110]\). Inset shows the low-temperature resistivity. Temperature dependences of (b) the specific heat C, (c) magnetic specific heat \(C_{\text{mag}}\) and magnetic entropy \(S_{\text{mag}}\), and (d) magnetic susceptibility in the form of \(M/H\) and the inverse one under \(\mu_{0} H = 1\) T. (e) Magnetization curves at 1.8 K for \(H\parallel [100]\) and [001]. (f) Magnetic phase diagram for \(H\parallel [001]\) in HoGa6.
The \(\mu_{\text{eff}}\) and \(\theta_{\text{p}}\) values are \(\mu_{\text{eff}} = 10.6\)\(\mu_{\text{B}}\)/Ho and \(\theta_{\text{p}} = -18.4\) K for \(H\parallel [100]\), and \(\mu_{\text{eff}} = 10.3\)\(\mu_{\text{B}}\)/Ho and \(\theta_{\text{p}} = 4.3\) K for \(H\parallel [001]\), respectively. The antiferromagnetic easy-axis corresponds to the \(H\parallel [001]\) direction, as shown in Fig. 12(d), and the corresponding magnetization in Fig. 12(e) indicates several metamagnetic transitions. The saturation moment is 9.4 \(\mu_{\text{B}}\)/Ho, which roughly corresponds to 10 \(\mu_{\text{B}}\)/Ho for Ho3+.
de Haas–van Alphen effect in SmGa6
We carried out the dHvA effect in SmGa6. Figure 13 shows the typical dHvA oscillations for \(H\parallel [001]\) in the field range from 5 to 14.5 T and the corresponding fast Fourier transformation (FFT) spectrum in SmGa6. The dHvA frequency F (\(= c\hbar S_{\text{F}}/2\pi e\)) is proportional to the maximum or minimum cross-sectional area \(S_{\text{F}}\) of the Fermi surface and is expressed as a unit of magnetic field \(\mu_{0}H\). The dHvA frequency F is in the range from 0.125 to 9.490 kT.
Figure 13. (a) Typical dHvA oscillations for \(H\parallel [001]\) and (b) the corresponding FFT spectrum in SmGa6.
We determined the cyclotron effective mass \(m^{*}_{\text{c}}\) from the temperature dependence of the dHvA amplitude. The cyclotron masses are, for example, 0.95 and 1.18 \(m_{0}\) (\(m_{0}\): rest mass of an electron), for a main dHvA branch named \(\alpha_{1}\) (\(F= 9.490\) and 9.345 kT, respectively). The dHvA frequencies and the corresponding cyclotron masses for \(H\parallel [001]\) and [100] are summarized in Tables II and III.
Table II. dHvA frequencies F and the corresponding cyclotron masses \(m^{*}_{\text{c}}\) for \(H\parallel [001]\) in SmGa6 together with theoretical ones (\(F_{\text{b}}\) and \(m_{\text{b}}\)) in LaGa6.
Table III. dHvA frequencies F and the corresponding cyclotron masses \(m^{*}_{\text{c}}\) for \(H\parallel [100]\) in SmGa6 together with theoretical ones (\(F_{\text{b}}\) and \(m_{\text{b}}\)) in LaGa6.
We measured the angular dependences of the dHvA frequencies by rotating the sample, as shown in Fig. 14(a). Figures 14(b) and 14(c) are the theoretical angular dependences of dHvA frequencies in a non-\(4f\) reference compound LaGa6 and the corresponding Fermi surfaces, respectively.
Figure 14. (Color online) Angular dependences of (a) experimental dHvA frequencies in SmGa6 and (b) the theoretical ones in the non-\(4f\) reference compound LaGa6, together with (c) the theoretical Fermi surfaces.
Here, the band calculations are based on the density functional theory by local density approximation (LDA) and the full potential linear augmented plane wave (FLAPW) method using the lattice parameters and the atomic positions of SmGa6 in Table I. In the present band calculations, the scaler relativistic effect is taken into account for all the valence electrons and then the spin–orbit coupling is included self-consistently for all the valence electrons in a second variational procedure. Here, \(5p^{6} 5d^{1} 6s^{2}\) electrons for La and \(3d^{10} 4s^{2} 4p^{1}\) for Ga are treated as valence electrons in the band calculations. As a primitive cell contains 2 molecules of LaGa6, the present compound is a compensated metal with equal volumes of electron and hole Fermi surfaces, which is reflected in the Fermi surfaces in Fig. 14(c).
Main dHvA branches are branches \(\alpha_{i}\) and \(\beta_{i}\). The present angular dependences of dHvA frequencies are approximately explained by the theoretical ones in LaGa6. This means that the \(4f\) electrons in SmGa6 is localized. From a viewpoint of dHvA experiments, even in the localized-\(4f\) electrons compound, the presence of the \(4f\) electrons alters the Fermi surface through the \(4f\) electrons contribution to the crystal potential and through the introductions of new magnetic Brillouin zone boundaries and the magnetic energy gaps which occur when \(4f\) electrons order magnetically. The latter effect is, simply thinking, approximated by a band-folding procedure where the paramagnetic Fermi surface, which is similar to the Fermi surfaces of the corresponding La compound, is folded into a small Brillouin zone based on the large antiferromagnetic unit cell. Therefore, two kinds of Fermi surfaces, namely Fermi surfaces in the paramagnetic and antiferromagnetic states are observed in some antiferromagnets. See, for example, the dHvA frequencies in RIn3 (R: rare earth) including SmIn3.9) We observed the band-87 hole Fermi surface with the largest dHvA frequency named \(\alpha_{1}\). This is based on the magnetic breakthrough effect.
Branch \(\alpha_{1}\) with the largest dHvA frequency (\(F_{\text{b}} = 9.1067\) kT, \(m_{\text{b}} = 0.707 \,m_{0}\)) corresponds to an inner orbit of the “square lattice” in the band-87 hole Fermi surface, which was observed in the present dHvA experiments. The next largest one is branch \(\beta_{1}\), which corresponds to an orbit of the “cushion”-like Fermi surface centered at Γ in the band-88 electron Fermi surface. This branch was not observed experimentally, which might be due to the damping of the dHvA amplitude based on the curvature factor of the Fermi surface.
The energy band, and partial and total densities of states are shown in Figs. 15(a) and 15(b), respectively. The density of states at Fermi energy \(E_{\text{F}}\) are mainly due to La-\(5d\), Ga-\(4p\), and Ga-\(4s\). The theoretical γ value is \(\gamma_{\text{b}} = 2.50\) mJ/(K2·mol).
Figure 15. (Color online) (a) Energy band structure, and (b) total and partial densities of states in LaGa6.
We show in Fig. 16 the relation between detected dHvA frequencies and the corresponding cyclotron masses in SmGa6 vs theoretical ones in LaGa6 for \(H\parallel [001]\) and [100]. The mass versus dHvA frequency is plotted in a logarithmic scale. Solid lines are guidelines. Figure 16 implies that the mass increases as a function of the dHvA frequency and the mass enhancement factor of \(m^{*}_{\text{c}}/m_{\text{b}}\) is approximately equal for all the conduction electrons. The cyclotron masses are roughly 1.6 and 2.1 times larger than the theoretical ones for \(H\parallel [001]\) and [100], respectively, leading to \(\gamma = 4.0{\text{–}}5.3\) mJ/(K2·mol) in SmGa6, using a relation of the mass enhancement factor λ: \begin{equation} \frac{m_{\text{c}}^{*}}{m_{\text{b}}} = \frac{\gamma}{\gamma_{\text{b}}} = 1 + \lambda. \end{equation} (2) Origins for λ are ascribed to be the many-body effects which cannot be taken into account in the usual band theory. As the most probable origins, the electron–phonon interaction and the magnetic interaction are considered. The \(4f\) electrons in the lanthanide compounds are localized at the lanthanide ions and then spin fluctuations enhance the effective mass of the conduction electrons via c–f interactions such as the RKKY interaction and many-body Kondo effect, where c stands for conduction electrons and f for \(4f\) electrons. For example, the Fermi surface of CeB6 is quite similar to that of LaB6, although the large cyclotron mass of 20 \(m_{0}\) is observed.10) CeB6 indicates a \(-{\log T}\) dependence of the resistivity at low temperatures on the basis of the Kondo effect, together with the octupole ordering at \(T_{\text{O}} = 3.15\) K and antiferromagnetic ordering at \(T_{\text{N}} = 2.3\) K.11,12) In the case of antiferromagnet PrB6, the \(4f\) electrons are localized, but do not indicate the Kondo effect. Therefore, only a small mass enhancement is observed.10,13) SmGa6 is similar to PrB6 in mass enhancement. The present result means that SmGa6 is the antiferromagnet with the localized magnetic moment of SmGa6, which is consistent with the effective magnetic moment of Sm3+ and no indication of \(-{\log T}\) dependence in the electrical resistivity based on the Kondo effect. Very recently, we detected a main Fermi surface with a large cyclotron mass of 26 \(m_{0}\) in SmTi2Al20 with the \(-{\log T}\) dependence of the electrical resistivity.14)
Figure 16. (Color online) Relation between dHvA frequencies F and the corresponding cyclotron masses \(m_{\text{c}}^{*}\) in SmGa6 and theoretical ones \((F_{\text{b}}, m_{\text{b}})\) in LaGa6.
de Gennes scaling of Néel temperature
The fundamental magnetic properties in RGa6 is summarized in Table IV. The Néel temperature \(T_{\text{N}}\) in RGa6 roughly follows the so-called de Gennes scaling of \((g_{\text{J}} - 1)^{2} J (J+1)\) based on the RKKY interaction, as shown in Fig. 17, where \(g_{\text{J}}\) is the Landé \(g_{\text{J}}\)-factor and J is the total angular momentum. A solid line in Fig. 17 indicates the calculated Néel temperature based on the de Gennes scaling normalized at \(T_{\text{N}}\) in TbGa6. The Néel temperature \(T_{\text{N}}\) in R = Pr, Nd, and Gd deviates from the present de Gennes scaling. CeGa6 and SmGa6 are found to order antiferromagnetically with the doublet ground state. On the other hand, in the other RGa6, some CEF levels are involved in the antiferromagnetic ordering via the magnetic exchange interaction. This indicates that the splitting energies of \(4f\) levels in the CEF scheme are rather small in magnitude. In the case of RCu2Si2, the quadrupolar interaction is clearly related with the high Néel temperature in PrCu2Si2.15,16) It is not clear whether the quadrupolar interaction is related to the Néel temperature of PrGa6, which is left to the future study, including \(T_{\text{N}}\) of NdGa6. A low Néel temperature in GdGa6 is not explained by the CEF scheme. This is because of \(J = S = 7/2\) and \(L = 0\) in GdGa6. The magnetic properties are thus highly different between GdGa6 and the other magnetic RGa6. In the case of RCu2Si2, the Néel temperature of GdCu2Si2 is also deviated from the de Gennes scaling and is rather a small value.15,16)
Figure 17. (Color online) de Gennes scaling normalized at \(T_{\text{N}} = 17.8\) K in TbGa6. Present data are shown by solid circles (●), and the data shown by squares (□) are cited from Ref. 3.
Table IV. Magnetic properties of Néel temperature \(T_{\text{N}}\), magnetic transition \(T_{\text{N}}'\), easy-axis of the magnetization, parallel to the [001] direction: || or perpendicular to [001]: \(\bot\), effective magnetic moment \(\mu_{\text{eff}}\), paramagnetic Curie temperature \(\theta_{\text{p}}\), and saturation moment in the magnetization curve \(\mu_{\text{s}}\) in RGa6.
4. Summary
We succeeded in growing single crystals of RGa6 (R: Y to Ho except Eu) by the Ga-self flux method and measured the electrical resistivity, specific heat, magnetic susceptibility, and magnetization, except YGa6, together with the dHvA experiment for SmGa6 Experimental results are summarized as follows;
1) All the magnetic RGa6 compounds order antiferromagnetically, and the corresponding easy-axes are directed along the [001] direction in the magnetization curves, except CeGa6 and SmGa6.
2) The Néel temperature roughly follows the de Gennes scaling normalized at \(T_{\text{N}} = 17.9\) K in TbGa6, except PrGa6, NdGa6, and GdGa6.
3) Magnetization curves in PrGa6, NdGa6, TbGa6, DyGa6, and HoGa6 exhibit sharp metamagnetic transitions, for example, indicating five step-like increases of magnetization and reaching a saturate in moment of 10 \(\mu_{\text{B}}\)/Dy in DyGa6.
4) RGa6 compounds are found to grow in single crystalline sample with high quality, except LaGa6 and CeGa6, where the residual resistivity is 1 µΩ·cm or less than 1 µΩ·cm, for example, less than 0.5 µΩ·cm in SmGa6.
5) A main Fermi surface of SmGa6 was detected in the dHvA experiment, with a cyclotron mass \(m^{*}_{\text{c}}\) of about 1 \(m_{0}\) (\(m_{0}\): rest mass of an electron). The mass versus dHvA frequency is plotted on a logarithmic scale. The mass enhancement factor of \(m^{*}_{\text{c}}/m_{\text{b}}\) is approximately equal for all the conduction electrons. Namely, the cyclotron masses \(m^{*}_{\text{c}}\) are roughly 1.6 and 2.1 times larger than the theoretical ones \(m_{\text{b}}\) for \(H\parallel [001]\) and [100], respectively, leading to an electronic specific heat coefficient \(\gamma = 4.0{\text{–}}5.3\) mJ/(K2·mol) in SmGa6, where the theoretical \(\gamma_{\text{b}}\) value is 2.50 mJ/(K2·mol). This is consistent with the fact that SmGa6 is an antiferromagnet with a localized magnetic moment of Sm3+ with no existence of \(-{\log T}\) dependence of the electrical resistivity based on the Kondo effect.
Acknowledgments
This work was partly supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan Society for the Promotion of Science (JSPS) KAKENHI Grant-in-Aid (Numbers: JP19H01839, JP22K03517, JP22K03522, JP23K03332, JP23H04870) and the Tokyo Metropolitan Government Advanced Research Grant Number (H31-1).
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