Fermi-surface topology of the heavy-fermion system Ce$_{2}$PtIn$_{8}$

TL;DR

This study investigates the Fermi-surface topology of the heavy-fermion compound Ce$_{2}$PtIn$_{8}$ using quantum oscillation measurements and band-structure calculations, revealing itinerant Ce-$4f$ electrons and large effective masses.

Contribution

It provides the first detailed Fermi-surface analysis of Ce$_{2}$PtIn$_{8}$, demonstrating the itinerant nature of Ce-$4f$ electrons through experimental and theoretical comparison.

Findings

Ce$_{2}$PtIn$_{8}$ exhibits strongly enhanced effective masses up to 26 times the electron mass.

The Fermi surface topology suggests itinerant Ce-$4f$ electrons.

Comparison with Pr$_{2}$PtIn$_{8}$ supports the itinerant electron conclusion.

Abstract

Ce$_{2}$PtIn$_{8}$ is a recently discovered heavy-fermion system structurally related to the well-studied superconductor CeCoIn$_{5}$. Here, we report on low-temperature de Haas-van Alphen-effect measurements in high magnetic fields in Ce$_{2}$PtIn$_{8}$ and Pr$_{2}$PtIn$_{8}$. In addition, we performed band-structure calculations for localized and itinerant Ce-$4f$ electrons in Ce$_{2}$PtIn$_{8}$. Comparison with the experimental data of Ce$_{2}$PtIn$_{8}$ and of the $4f$-localized Pr$_{2}$PtIn$_{8}$ suggests the itinerant character of the Ce-$4f$ electrons. This conclusion is further supported by the observation of effective masses in Ce$_{2}$PtIn$_{8}$, which are strongly enhanced with up to 26 bare electron masses.

Tables

Table 1. Table 1: Experimental and calculated (with shifted 4 f 𝑓 f states) dHvA frequencies and effective masses of Ce 2 PtIn 8  for Θ 010 = 2 ∘ subscript Θ 010 superscript 2 \Theta_{010}=2^{\circ} . Branch assignments refer to Fig.  7 .

	Experiment		Calculation
Branch	$F$ (kT)	$m^{\ast }$ ($m_e$)	$F$ (kT)	$m_b$ ($m_e$)
			0.227	0.67
			0.337	0.35
			0.355	0.25
$\gamma$	0.55	3.3(1)
$\delta$	0.73	15.0(3)
			0.983	1.46
$\epsilon$	1.24	4.6(1)
			2.222	2.14
${\alpha }_2$	3.34	19.2(1)	3.357	0.61
${\alpha }_1$	4.62	10.5(2)	4.719	1.00
			7.676	1.64
$\kappa$	8.99	25.7(10)

Full text

I Introduction

Unconventional superconductivity in the vicinity of quantum-critical points (QCPs) is an intensely-studied phenomenon in Ce-based heavy-fermion (HF) systems. Among these materials, CeCoIn5 exhibits the highest known superconducting transition temperature $T_{c}=2.3$ K Petrovic et al. (2001); Ikeda et al. (2001). Several theoretical works suggest that the presence of two-dimensional (2D) Fermi surface (FS) sheets enhances antiferromagnetic (AFM) fluctuations which are believed to be responsible for Cooper-pair formation in HF compounds Moriya et al. (1990); Mathur et al. (1998); Monthoux and Lonzarich (1999); Monthoux (2003); Monthoux et al. (2007); Shishido et al. (2010). Indeed, CeCoIn5 possesses almost cylindrical FS sheets at low temperatures, where Ce-$4f$ electrons become itinerant Settai et al. (2001); Hall et al. (2001); Nomoto and Ikeda (2014).

Structurally, CeCoIn5 belongs to the family of Ce${}_{n}T_{m}$In3n+2m ($T$: transition metal, $n=1$, 2, 3, $m=0$, 1, 2). Unit cells within this family are composed of $n$ layers of conducting CeIn3 separated by $m$ layers of insulating $T$In2, as shown in Fig. 1. By increasing the ratio $m/n$, the spacing between the CeIn3 building blocks becomes larger, which is expected to augment 2D behavior. Therefore, controlling the $m/n$ ratio enables a systematic study of the relation between FS dimensionality and superconducting properties.

Here, we focus on the platinum members of the family, i.e., CenPtmIn3n+2m. So far, CeIn3, Ce3PtIn11, Ce2PtIn8, Kratochvilova et al. (2014) and CePt2In7 Kurenbaeva et al. (2008) have been successfully synthesized. Inside this group, CeIn3 has the lowest $m/n$ ratio and a cubic crystal structure. Thus, it is expected to host the most three-dimensional (3D) FS sheets, which is supported by both the band-structure calculations Settai et al. (2005) and de Haas-van Alphen (dHvA) experiments Ebihara et al. (1993); Endo et al. (2005). CeIn3 becomes superconducting with $T_{c}=0.2$ K at a critical pressure $P_{c}=2.5$ GPa Mathur et al. (1998). Increasing the pressure beyond $P_{c}$ changes the character of the Ce-$4f$ electrons from localized to itinerant Settai et al. (2005).

On the other end, CePt2In7 has the highest $m/n$ ratio and is, therefore, expected to be the most 2D compound in the CenPtmIn3n+2m series. This assumption was ascertained by quantum-oscillation measurements and band-structure calculations Altarawneh et al. (2011); Miyake et al. (2015); Götze et al. (2017). Measurements employing a tunnel-diode-oscillator technique in pulsed magnetic fields gave evidence for the existence of localized $4f$ electrons only above an anomaly at $B_{m}=45$ T Altarawneh et al. (2011). Subsequent experiments clarified that the localized character also persists at lower fields Miyake et al. (2015); Götze et al. (2017). Superconductivity in CePt2In7 emerges around a pressure-induced suppression of the AFM ground state at $P_{c}=\text{3.2-3.5}$ GPa with $T_{c}=2.1$ K Bauer et al. (2010); Sidorov et al. (2013); Kurahashi et al. (2015).

Intermediate $m/n$ ratios are realized in Ce3PtIn11 and Ce2PtIn8, which have been synthesized for the first time only recently Kratochvilova et al. (2014). Ce3PtIn11 exhibits AFM order below $T_{N}=2.2$ K and, unlike CeIn3 and CePt2In7, becomes superconducting already at ambient pressure with $T_{c}=0.32$ K, which increases to $T_{c}=0.7$ K at $P_{c}=1.3$ GPaProkleška et al. (2015). In addition, its large Sommerfeld coefficient $\gamma_{el}=520$ mJ/mol K2 indicates the HF nature of this compound Prokleška et al. (2015). Due to the novelty of Ce2PtIn8, there are, to the best of our knowledge, no publications addressing its ground state so far. However, the isostructural non-magnetic compound Ce2PdIn8 is known to become superconducting at $T_{c}=0.7$ K Kaczorowski et al. (2009); Uhlířová et al. (2010a); Kaczorowski et al. (2010a); Uhlířová et al. (2010b); Kaczorowski et al. (2010b). The same crystal structure is also realized in Ce${}_{2}T$In8 with $T=$ Co, Rh, Ir. Two of these compounds are HF superconductors: Ce2CoIn8 ($T_{c}=0.4$ K, $\gamma_{el}=1$ J/mol K2)Chen et al. (2002) and Ce2RhIn8 ($T_{c}=2$ K at $P_{c}\approx 0.4$ GPa Nicklas et al. (2003), $\gamma_{el}=800$ mJ/mol K2 Ueda et al. (2004)). The third compound, Ce2IrIn8, is also a HF system ($\gamma_{el}=1.4$ J/mol K2). In this material, no superconducting or magnetic order was found down to 50 mK Thompson et al. (2001), but a non-Fermi-liquid behavior was observed in applied magnetic fields Kim et al. (2004). For all three compounds, quantum-oscillation measurements and/or angle-resolved photoemission spectroscopy revealed quasi-2D FSs Ueda et al. (2004); Raj et al. (2005); Souma et al. (2008); Jiang et al. (2015).

The most direct way to establish the FS topology are quantum-oscillation measurements. In this paper, we report on high-field dHvA measurements of Ce2PtIn8. Comparing the results to band-structure calculations and to our dHvA data of the $4f$-localized counterpart Pr2PtIn8 evidences itinerant Ce-$4f$ electrons in Ce2PtIn8. Furthermore, our data reveal the existence of a large, 2D FS sheet in this compound. Albeit being less corrugated than its FS counterpart in the isostructural superconductor Ce2PdIn8, there is no signature of superconductivity in the torque data of Ce2PtIn8. We also determined the effective masses of Ce2PtIn8, yielding values between 3.3$m_{e}$ and 25.7$m_{e}$, where $m_{e}$ is the bare electron mass.

II Experimental

Single crystals of Ce2PtIn8 and Pr2PtIn8 were grown using the In self-flux method. A detailed description of the growth conditions can be found in Ref. [Shishido et al., 2015]. Both samples used in our study were investigated by Laue diffraction. All the spots we observed in the diffraction images originate from Ce2PtIn8 and Pr2PtIn8, respectively. However, it was previously demonstrated that Ce2PtIn8 single crystals grown by In self-flux technique often contain Ce3PtIn11 and CeIn3 impurities, the amount of which depends on the growing conditions Kratochvilova et al. (2014). We have, therefore, measured the specific heat of our Ce2PtIn8 sample by relaxation technique in the temperature range from 1.8 to 30 K. The total heat capacity, $C$, divided by temperature, $T$, of the 125 $\mu$g sample is shown in Fig. 2. We have indeed observed two anomalies at 2 and 10 K, which coincide with the AFM transitions in Ce3PtIn11 Kratochvilova et al. (2014); Prokleška et al. (2015) and CeIn3 Settai et al. (2005), respectively. The size of these anomalies, $\Delta C/T\simeq 1.4\times 10^{-8}$ J/K2, is small compared to the large specific-heat jumps at $T_{N}$ in pure Ce3PtIn11 ($\Delta C/T=2.4$ J/mol K2) Kratochvilova et al. (2014); Prokleška et al. (2015) and CeIn3 ($\Delta C/T=1.1$ J/mol K2) Settai et al. (2005). We, thus, estimate that our Ce2PtIn8 sample contains a mass fraction of approximately 8 % Ce3PtIn11 and 5 % CeIn3 impurities. This is taken into account during the data analysis, and does not affect our main conclusions. After subtracting impurity contributions, we estimated a Sommerfeld coefficient $\gamma=0.5$ J/mol K2.

Angle-dependent quantum-oscillation measurements were performed on the same sample using capacitive torque magnetometry, employing 25 $\mu$m and 50 $\mu$m thick CuBe cantilevers. The experiments were conducted in a dilution refrigerator with a base temperature of about 30 mK. High magnetic fields up to 34 T were provided by a resistive magnet in the LNCMI-Grenoble.

III Results and Discussion

III.1 dHvA measurements of Ce2PtIn8

Figure 3(a) shows a typical torque signal for Ce2PtIn8 taken at an angle $\Theta_{010}=2^{\circ}$ after subtracting the non-oscillatory background. Throughout this paper, all angles are measured from the crystallographic $c$ axis. The quantum oscillations are clearly visible. By performing a Fourier transformation, we obtained the corresponding frequency spectrum shown in Fig. 3(b). There are six fundamental frequencies, denoted as $\alpha_{1}$, $\alpha_{2}$, $\gamma$, $\delta$, $\varepsilon$, and $\kappa$. Details of the angular dependence of the dHvA frequencies will be discussed in Sec. III.3, together with the results of the band-structure calculations. No signatures of phase transitions were observed in any of the torque signals.

III.2 Band-structure calculations

In order to determine whether the $4f$ electrons are itinerant or localized, we performed band-structure calculations using the kansai code, based on an FLAPW (full potential linear augmented plane wave) method with a local density approximation (LDA). The relativistic effect is considered by using the technique proposed by Koelling and Harmon Koelling and Harmon (1977). The spin-orbit coupling is included as the second variational procedure for valence electrons.

In the calculations, we used the lattice parameters of $a=4.6893$ Å and $c=12.1490$ Å, taken from Ref. [Kratochvilova et al., 2014]. Structure parameters are 0.30795 for the $2d$ site of Ce, 0.12241 for the $4i$ site of In, and 0.3072 for the $2h$ site of In. Core electrons (Xe core minus $5s^{2}5p^{6}$ electrons for Ce/La, Xe core minus $5p^{6}$ plus $4f^{14}$ for Pt, Kr core for In) are calculated inside the Muffin-Tin spheres in each self-consistent step. $5s^{2}5p^{6}$ electrons on Ce/La, $5p^{6}$ electrons on Pt, and $4d^{10}$ electrons on In are calculated as valence electrons by using a second energy window. The LAPW basis functions are truncated at $|k+G_{i}|\leq 4.20\times 2\pi/a$, corresponding to 821 LAPW functions. The sampling points are 220 $k$-points uniformly distributed in the irreducible 1/16th of the Brillouin zone, which are divided by (18, 18, 6).

In order to improve the agreement between the calculations and our Ce2PtIn8 dHvA data, the Ce-$4f$ states are shifted upward by 0.2 Ry from LDA. To compare the non-$f$ reference or localized $4f$ electrons model, the calculation for La2PtIn8 was also performed by using the structural parameters of Ce2PtIn8 as crystals of La2PtIn8 are currently unavailable. For both itinerant and localized $4f$ electrons, the resulting densities of states and band structures are shown in Figs. 4 and 5, respectively. The corresponding Fermi surfaces are shown in Fig. 6.

III.3 Experimental data of Ce2PtIn8 vs. band-structure calculations

The comparison of the experimentally determined dHvA frequencies in Ce2PtIn8 and the calculated frequencies for La2PtIn8 and Ce2PtIn8 is shown in Fig. 7. Overall, we observed eight dHvA frequency branches, denoted by $\alpha$, $\gamma$, $\delta$, $\varepsilon$, $\zeta$, $\eta$, $\kappa$, and $\omega$. One of these branches, $\alpha$, could be traced up to 70∘ away from the $c$ axis. Roughly following a $1/\cos(\Theta)$ behavior, it indicates the existence of a quasi-2D corrugated cylindrical FS sheet. In addition, a multitude of frequencies lower than 2 kT suggests the existence of several small FS pockets. Two of these frequencies, $\varepsilon$ and $\eta$, were observed over the entire angular range. Surprisingly, we also observed a fairly large and almost spherical FS sheet, evidenced by the branch $\omega$ with the dHvA frequency of about 3 kT. Considering the tetragonal crystal structure with a $c/a$ ratio of about 2.6, near-spherical FS pockets of this size are unexpected. The highest frequency, $\kappa$, appears only close to the $c$ axis. It was observed only for the rotation from [001] to [100] direction, because we used a thinner and, therefore, more sensitive cantilever for small angles within this plane.

Due to the layered structure of La2PtIn8 and Ce2PtIn8, both the calculations for localized and for itinerant $4f$ electrons suggest the existence of quasi-2D FSs. The main difference, however, is that the calculation for localized $4f$ electrons predicts the existence of two quasi-2D FSs, opposed to only one in the calculation for itinerant $4f$ electrons (see Fig. 6). Since 2D FSs feature a favorable curvature factor and a $1/\cos(\Theta)$ angular dependence of their extremal cross section, they are easily detectable by torque magnetometry. Therefore, the experimental observation of only one 2D FS sheet with extremal areas $\alpha_{1}$ and $\alpha_{2}$ strongly supports the calculation for itinerant $4f$ electrons. Moreover, the direct comparison of the experimental data with the La2PtIn8 calculations, shown in Fig. 7(a), yields a very poor agreement. In particular, none of the observed smaller dHvA frequencies $\gamma\ldots\eta$ can be attributed to a calculated branch, especially near the $c$ axis. For $\alpha$, there is a calculated branch exhibiting a similar angular dependence, but the frequencies differ by about 1 kT.

In contrast, most of the high- and low-frequency experimental branches are reproduced by the Ce2PtIn8 calculations, as shown in Fig. 7(b). All the low frequencies $\gamma\ldots\eta$ are in good qualitative agreement with calculated ones. More importantly, the frequency associated with the almost cylindrical FS sheet, $\alpha$, is in very good qualitative agreement with the corresponding calculated branch. We could further improve the quantitative agreement of the quasi-2D FS sheet by shifting the $4f$ states by $0.2$ Ry, as depicted in Fig. 7(c). However, there are also a few mismatches. Several calculated branches originating from band 72 were not observed experimentally. Presumably, this is due to an unfavorable curvature factor. We note, that the calculated frequency branches from band 72 were not observed in the isostructural compound Ce2PdIn8 as well Götze et al. (2015).

The frequencies $\omega$ and $\kappa$ are not explained by theory. Assuming an epitaxial growth of the CeIn3 impurities in our crystal, $\omega$ coincides with the frequency branch $d$ of CeIn3 Ebihara et al. (1993), rendering CeIn3 impurities to be the most likely source of $\omega$. On the other hand, the origin of $\kappa$ is yet unknown. Since the angular dependence of this branch does not reflect the cubic symmetry of CeIn3 and does not coincide with any of the known frequency branches of CeIn3, it appears unlikely for $\kappa$ to originate from the CeIn3 impurity. However, the Ce3PtIn11 impurity phase, also present in our sample, cannot be ruled out as a possible source of the branch $\kappa$. Quantum-oscillation or ARPES measurements of Ce3PtIn11 are required to verify this hypothesis. Apart from the aforementioned branches, the band-structure calculations for Ce2PtIn8 with shifted $4f$-electron bands yield an excellent agreement with the experimental data, strongly suggesting itinerant Ce-$4f$ electrons in this compound.

III.4 Comparison with Pr2PtIn8

It would be desirable to directly compare the dHvA oscillations in Ce2PtIn8 with those measured in La2PtIn8 in order to establish the localized or itinerant character of the Ce-$4f$ electrons beyond any doubts. However, single crystals of La2PtIn8 are currently unavailable, rendering a direct comparison impossible. Since the Pr-based analogues of Ce-based HF compounds are known to have localized $4f$ electrons, e.g. in PrCoIn5 Elgazzar et al. (2008), Pr2PtIn8 poses an alternative candidate for comparison. We, therefore, performed dHvA measurements on Pr2PtIn8. In addition, we conducted specific-heat measurements on our Pr2PtIn8 single crystal, revealing a Sommerfeld coefficient of $\gamma=90$ mJ/mol K2. This evidences greatly reduced electronic correlations in Pr2PtIn8 compared to Ce2PtIn8, as expected for localized $4f$ electrons.

Figure 8(a) shows a typical torque signal for Pr2PtIn8, after subtracting the non-oscillatory background, taken at the same angle, $\Theta_{010}=2^{\circ}$, as shown in Fig. 3 for Ce2PtIn8. The corresponding dHvA frequency spectrum in Fig. 8(b) is clearly different from the one in Fig. 3(b), suggesting different topologies of the FSs of Ce2PtIn8 and Pr2PtIn8. In order to verify the consistency, Fig. 9 compares calculated frequencies for $4f$-localized La2PtIn8 with the experimental data of Pr2PtIn8. Indeed, there is a very good agreement between the two data sets. All the large FS sheets calculated for bands 71, 72, and 73 are qualitatively supported by the experimental data. Thus, both the results of the La2PtIn8 band-structure calculations and the Pr2PtIn8 dHvA measurements are a good reference for $4f$-localized Ce2PtIn8.

When juxtaposing the angular dependences of Ce2PtIn8 and Pr2PtIn8, clear FS differences become even more obvious, as shown in Fig. 10. Apart from parts of the branch $\eta$, none of the branches of Ce2PtIn8 can be found in Pr2PtIn8. Consequently, the electronic states in Ce2PtIn8 differ from those in the $4f$-localized Pr2PtIn8, further proving the itinerant character of the Ce-$4f$ electrons in Ce2PtIn8. Although Ce2PtIn8 is an intermediate compound between CeIn3 and CePt2In7 from a structural point of view, it does not share their localized Ce-$4f$ character.

III.5 Effective masses of Ce2PtIn8

We determined the effective masses of Ce2PtIn8 from the temperature dependence of the dHvA oscillation amplitudes. These amplitudes were measured in the field range of 20 to 34 T, for temperatures between 50 and 500 mK, at the same angle as the data shown in Fig. 3. According to the Lifshitz-Kosevich formula, this dependence is proportional to $x/\sinh x$, where $x=\alpha Tm^{\ast}/B$ and $\alpha=14.69$ T/K Shoenberg (1984). Here, $m^{\ast}$ represents the effective mass given in multiples of the bare electron mass $m_{e}$. Figure 11 shows the fit of the temperature dependence of the oscillatory amplitudes by this formula, yielding $m^{\ast}$ as fit parameter. The results are summarized in Table 1, where the band masses, $m_{b}$, obtained from band-structure calculations with shifted $4f$ states by taking the derivative $dF/dE$, are also shown.

The effective masses are fairly large, ranging from 3.3$m_{e}$ for $\gamma$ to 25.7$m_{e}$ for $\kappa$. The higher the mass of a specific orbit, the stronger the damping of the corresponding oscillation by impurity scattering Shoenberg (1984). Given the strongly enhanced effective mass of the $\kappa$ branch, it seems unlikely that it originates from the Ce3PtIn11 impurity, because it would require a very high crystal quality of the impurity itself. Similarly, we observed the impurity frequency $\omega$ only for field directions where its effective mass is as small as $2m_{e}$ Ebihara et al. (1993).

The measured effective masses exceed the calculated band masses by about an order of magnitude or more. This implies that there is a significant mass enhancement due to many-body interactions, which are not included in the LDA calculations. Effective masses much higher than calculated were observed in other non-magnetic HF compounds with itinerant $4f$ electrons, such as CeCoIn5 Settai et al. (2001) and Ce2PdIn8 Götze et al. (2015). For the $\alpha_{1}$ and $\alpha_{2}$ orbits, both lying on the same FS sheet, the mass-enhancement factors are 9.5 and 30, respectively. Such largely different mass enhancements on a single FS sheet were also found in CeCoIn5 Settai et al. (2001) and CeIrIn5 Haga et al. (2001). In CeCoIn5, the effective masses were reported to strongly decrease with magnetic field Settai et al. (2001). Similarly, the Sommerfeld coefficient of the specific heat in Ce2PdIn8 is strongly suppressed by magnetic field Tokiwa et al. (2011). This is due to a close proximity to a QCP of these two compounds. In Ce2PtIn8, on the contrary, we have not observed any appreciable field dependence of the effective masses over the field range 16–34 T of our measurements. This suggests that Ce2PtIn8 is located further away from a QCP.

In comparison to AFM CeIn3 (m^{\ast}=2\ldots 12$$m_{e} Umehara et al. (1990); Ebihara et al. (1993)) and CePt2In7 (m^{\ast}=1.3\ldots 6.2$$m_{e} Altarawneh et al. (2011); Götze et al. (2017)) with localized $4f$ electrons, the effective masses in Ce2PtIn8 are significantly higher. Again, this reflects the itinerant character of the Ce-$4f$ electrons, leading to an increased density of states at the Fermi level and stronger many-body interactions.

III.6 Fermi-surface dimensionality

In order to quantify the degree of deviation from ideal two-dimensionality, we previously introduced the value $\Delta=(S_{max}-S_{min})/S_{avg}$ Götze et al. (2015). Here, $S_{min}$, $S_{avg}$, and $S_{max}$ are the minimum, average, and maximum cross sections of a quasi-2D FS sheet, respectively. Due to the fact that the observed dHvA frequencies are proportional to the extremal cross sections, we can calculate $\Delta$ from our experimental data.

As shown in Fig. 6, one of the calculated FS sheets of Ce2PtIn8 is nearly cylindrical. The observation of the experimental branch $\alpha$ proves the existence of this 2D sheet [see Fig. 7(b)]. For this FS sheet, we find a value of $\Delta=0.32$. This compares to 0.04, 0.03, and 0.07 for the three cylindrical sheets of CePt2In7 Götze et al. (2017). Also in the prototypical compound, CeCoIn5, the value $\Delta=0.21$ (averaged from Refs. [Settai et al., 2001], [Hall et al., 2001], and [Polyakov et al., 2012]) is smaller. Consequently, the FS dimensionality of Ce2PtIn8 lies between those of CeIn3 and CeCoIn5, as expected from the intermediate two-dimensionality of the crystal structure.

IV Conclusions

In summary, we conducted measurements of the dHvA effect on the HF compound Ce2PtIn8 and its $4f$-localized counterpart Pr2PtIn8. By contrasting the data of these two compounds and by comparing the Ce2PtIn8 data to band-structure calculations, we find clear evidence for itinerant Ce-$4f$ electrons in this material. In addition, the effective masses are strongly enhanced, varying between 3.3$m_{e}$ and 25.7$m_{e}$ for the different FS sheets.

Structurally, Ce2PtIn8 can be considered as an intermediate compound between CeIn3 and CePt2In7. The FSs of Ce2PtIn8 follow the structural trend, being less 2D than those of CePt2In7, but more than those of CeIn3. Both CeIn3 and CePt2In7 have localized Ce-$4f$ electrons, in contrast to Ce2PtIn8. It is, therefore, not surprising that the effective masses in Ce2PtIn8 are considerably enhanced.

There are a lot of similarities between the electronic structures of Ce2PtIn8 and the isostructural Ce2PdIn8 Götze et al. (2015). All the FS sheets of Ce2PtIn8 correspond to almost identical sheets in Ce2PdIn8. The only exception are very small FS pockets originating from band 74 of Ce2PdIn8 Götze et al. (2015), which cannot be found in Ce2PtIn8. Both compounds feature enhanced effective masses and itinerant $4f$ electrons.

In the generic temperature-pressure phase diagram of Ce-based HF compounds, the AFM ordering temperature $T_{N}$ decreases upon increasing pressure Flouquet et al. (2011), as shown in Fig. 12. At $T=0$ and the critical pressure $P_{c}$, a QCP separates the AFM and the paramagnetic phases. The $f$ electrons of Ce are usually found to be localized in the former, and itinerant in the latter Araki et al. (2001); Shishido et al. (2005); Settai et al. (2005). The superconducting dome is located around this critical pressure. In Ce2PdIn8, for example, the observation of a non-Fermi-liquid behavior at low temperature Dong et al. (2011); Tran et al. (2011); Gnida et al. (2012); Tokiwa et al. (2011); Matusiak et al. (2011); Fukazawa et al. (2012); Tran et al. (2012) suggests the close proximity to a QCP.

Our data reveal the $4f$-itinerant ground state of Ce2PtIn8, and no signatures of a superconducting transition. This is not surprising, given that Ce2PtIn8 has a smaller unit cell volume than Ce2PdIn8Kratochvilova et al. (2014), but very similar FSs. Consequently, Ce2PtIn8 can be considered a compressed version of Ce2PdIn8. Altogether, this suggests that in the temperature-pressure phase diagram, Ce2PtIn8 is located outside the superconducting dome at pressures larger than $P_{c}$ (see Fig. 12). Thus, we assume that a negative pressure induced by chemical substitution would tune Ce2PtIn8 to an AFM QCP, similar to CeCu6-xAux Germann et al. (1988); Lees and Coles (1988); Germann and Löhneysen (1989) or Ce1-xLaxRu2Si2 Besnus et al. (1987).

In Ref. [Götze et al., 2015], we found a trend between the FS dimensionality and the superconducting $T_{c}$ of several Ce-based HF compounds. The corrugated cylindrical FS sheet of Ce2PtIn8 ($\Delta=0.32$) deviates less from ideal two-dimensionality than the corresponding sheet in Ce2PdIn8 ($\Delta=0.39$) Götze et al. (2015). Therefore, one could expect to observe superconductivity in Ce2PtIn8 with a higher $T_{c}$ than in Ce2PdIn8. However, we have not observed any sign of superconductivity in Ce2PtIn8 in our torque measurements. Since Ce2PtIn8 is likely to be located further away from a QCP, we conclude that the proximity to a QCP is essential for superconductivity. The other known members of the CenPtmIn3n+2m family can be tuned to a QCP by applying pressure. With increasing structural two-dimensionality, their respective $T_{c}$’s at the QCP increase from 0.2 K (CeIn3 Mathur et al. (1998)) to 0.7 K (Ce3PtIn11Prokleška et al. (2015)) to 2.1 K (CePt2In7 Bauer et al. (2010); Sidorov et al. (2013); Kurahashi et al. (2015)). For CeIn3 Settai et al. (2005); Ebihara et al. (1993); Endo et al. (2005), Ce2PtIn8 (this work), and CePt2In7Altarawneh et al. (2011); Miyake et al. (2015); Götze et al. (2017), the FS dimensionality follows the structural dimensionality. Assuming that the FSs of Ce3PtIn11 also follow this trend, this indicates that quasi-2D FSs may enhance the $T_{c}$ observed at the QCP. An investigation of the Ce3PtIn11 FSs would be desirable to verify this trend.

V Acknowledgments

We acknowledge the support of the LNCMI-CNRS, and the HLD-HZDR, members of the European Magnetic Field Laboratory (EMFL), ANR-DFG grant “Fermi-NESt”, JSPS KAKENHI Grants No. JP15H05882, No. JP15H05884, No. JP15H05886, No. JP15K21732 (J-Physics), and JP16H02450. K.G. acknowledges support from the DFG within GRK 1621.