de Haas-van Alphen measurement of the antiferromagnet URhIn$_5$

TL;DR

This study uses de Haas-van Alphen measurements to explore the Fermi surface of the antiferromagnet URhIn$_5$, revealing multiple Fermi surface pockets and highlighting discrepancies with theoretical calculations, indicating complex electronic behavior.

Contribution

First dHvA measurement of URhIn$_5$ revealing its Fermi surface topology and masses, showing the need for advanced calculations beyond simple LDA+SO.

Findings

Fermi surface consists of four distinct pockets.

Measured cyclotron masses range from 1.9 to 4.3 times the electron mass.

Simple theoretical models do not match the experimental Fermi surface topology.

Abstract

We report on the results of a de Haas-van Alphen (dHvA) measurement performed on the recently discovered antiferromagnet URhIn$_5$ ($T_N$ = 98 K), a 5\textit{f}-analogue of the well studied heavy fermion antiferromagnet CeRhIn$_5$. The Fermi surface is found to consist of four surfaces: a roughly spherical pocket $\beta$, with $F_\beta \simeq 0.3$ kT; a pillow-shaped closed surface, $\alpha$, with $F_\alpha \simeq 1.1$ kT; and two higher frequencies $\gamma_1$ with $F_{\gamma_1} \simeq 3.2$ kT and $\gamma_2$ with $F_{\gamma_2} \simeq 3.5$ kT that are seen only near the \textit{c}-axis, and that may arise on cylindrical Fermi surfaces. The measured cyclotron masses range from 1.9 $m_e$ to 4.3 $m_e$. A simple LDA+SO calculation performed for the paramagnetic ground state shows a very different Fermi surface topology, demonstrating a need for more advanced electronic structure calculations.

Figures

Tables

Table 1. Table 1 : Summary of the experimental dHvA frequencies and fitted cyclotron masses m ∗ superscript 𝑚 m^{*} . Missing entries mean that the oscillation were only observed with the applied field along the c -axis.

	B $\parallel$ [100]		B $\parallel$ [110]		B $\parallel$ [001]
Orbit	$F$ (kT)	$m^{\ast }\left(m_e\right)$	$F$ (kT)	$m^{\ast }\left(m_e\right)$	$F$ (kT)	$m^{\ast }\left(m_e\right)$
$\beta$					0.3	4.3$\pm$0.4
$\alpha$	1.1	1.5$\pm$0.02	1.1	2.0$\pm$0.2	1.1	1.9$\pm$0.01
${\gamma }_1$					3.2	2.3$\pm$0.1
${\gamma }_2$					3.5	3.2$\pm$0.1

Table 2. Table 2 : dHvA frequencies found using SKEAF, for B ∥ B\parallel [100], B ∥ B\parallel [110] and B ∥ B\parallel [001].

Band 47	$B$ Direction	$F$(kT)	$m^{\ast }\left(m_e\right)$
	[100]	1.1034	0.73
		1.1516	0.96
	[110]	0.1309	0.83
	[001]	0.2110	0.47
		0.2117	1.02
		0.3034	0.44
		0.5469	0.65
Band 48	[100]	1.0037	1.37
		1.5812	1.42
	[110]	1.1677	1.45
	[001]	1.0745	2.16

Equations

$$A=\frac{C{m}^{\ast }T/B}{\sinh C{m}^{\ast }T/B}$$

Full text

I Introduction

Uranium based compounds exhibits a wide range of exotic properties, from the unconventional superconductivity in heavy fermion UPt3 McMullan et al. (2008) to the enigmatic hidden order in URu2Si2Mydosh and Oppeneer (2014), and all the way to weakly-correlated metallic behaviour in the uranium metal itself Mydosh and Oppeneer (2014). U atoms possess 5f electrons, whose spatial extent is greater than that of the 4f valence orbitals of the rare earths, but less than that of 3d orbitals of the row four transition metals Bartha et al. (2015). Depending on their environment, the 5f electrons can be localized or itinerant (i.e. they may or may not be included in the Fermi volume) Zwicknagl et al. (2003). Moreover, uranium can have 5f valence ranging from 5f3 to 5f0, presenting further opportunities for inter- as well as intra-orbital correlations and complexity.

For some uranium compounds such as the heavy fermion UPt3 McMullan et al. (2008), the Fermi surface has been mapped and the 5f electrons are understood to be itinerant in character. On the other hand, the 5f electrons are fully localized in the related compound UPd3 Tokiwa et al. (2001a), while quantum oscillation data on the heavy fermion superconductor UPd2Al3 Inada et al. (1999) has been explained by band structure calculations in which some 5f electrons are localized on the U site, while others are itinerant band electrons Zwicknagl et al. (2003). Whether and when the 5f electrons are localized or itinerant, or partially localized, is still not well understood at present.

The focus of this paper is URhIn5, a member of the Un**TX3n+2 (n = 0,1,2; T = transition metal; X = In, Ga) compounds. They are isostructural with the Cen**TX3n+2 family Bartha et al. (2015). In addition to the open question of the character of the 5f electrons, these compounds are also generally interesting because the system has a layered tetragonal structure, which means that adding layers of TX2 can change the dimensionality from 3D to 2D Bartha et al. (2015).

URhIn5 has been well characterized along with its bilayer cousin, U2RhIn8 Bartha et al. (2015); Matsumoto et al. (2014). It is tetragonal with HoCoGa5-type structure (P4/mmm). It orders antiferromagnetically at a Néel temperature of $T_{N}=98$ K Bartha et al. (2015). The Sommerfeld coefficient is given as $\gamma=60.7$ mJ mol*-1K-2* with Debye temperature $\Theta_{D}=165$ K in Ref. Bartha et al., 2015, whereas $\gamma=50$ mJ mol*-1K-2*, $\Theta_{D}=187$ K are reported by Ref. Matsumoto et al., 2013. The Sommerfeld coefficient is similar to that of UNiGa5 Tokiwa et al. (2001b), and rather large considering the high Néel temperature. The magnetic moment, $\mu_{\rm eff}=3.6\mu_{B}/U$ Bartha et al. (2015), is close to the U4+ (5f2) moment of 3.58 $\mu_{B}/U$ Tokiwa et al. (2001b). These are again properties very similar to UNiGa5 Tokiwa et al. (2001b).

Zero field NMR and NQR measurements have been performed on URhIn5 Sakai et al. (2013). The nuclear relaxation rate $1/T_{1}T$ satisfies the Korringa relation just below $T^{*}\approx 150$ K, which has led to the suggestion that the 5f electrons are localized above $T^{*}$ and itinerant below Sakai et al. (2013). The AFM propagation vector is commensurate, with propagation vector $\mathbf{Q}=(1/2,1/2,1/2)$ as determined unambiguously by recent neutron diffraction study Bartha et al. . Interestingly, UNiGa5 also exhibits the same ordering vector Sakai et al. (2013).

Matsumoto et al. Matsumoto et al. (2014) have previously reported dHvA quantum oscillations in URhIn5. They only observed two frequencies, 1.1 kT and 720 T along [100], with angle dependence suggesting that they both arise on a small ellipsoidal pocket. The presence of small pockets is consistent with the non-magnetic, 5f itinerant semi-metal URhGa5 Ikeda et al. (2002), but the authors note that the large specific heat for the antiferromagnetic state of URhIn5 cannot be explained by this pocket alone, suggesting the likely existence of larger Fermi surfaces undetected by their study. In comparison, quantum oscillation measurements on UNiGa5 show one cylindrical surface (1.3 kT) along with two larger ellipsoidal surfaces (2.6 kT and 1.6 kT), but additional heavier Fermi surfaces would again be needed to explain the large specific heat Tokiwa et al. (2001b). On the other hand, a larger Fermi surface (4 kT) has been detected, in addition to smaller pockets, in UIn3, the $n=\infty$ member of the Un**TX3n+2 family, although additional, multi-connected surfaces are still needed for $\gamma$ to match experimental values Tokiwa et al. (2001c).

Recently, magnetoresistance and Hall resistance also have been measured on URhIn5 Haga et al. (2017). The magnetoresistance follows the $H^{2}$ behaviour at higher temperatures but deviates from it for temperatures below 30 K Haga et al. (2017). At low temperature the magnetoresistance is positive, but the angle dependence, which could reveal the presence of open orbits, has not been reported. The Hall resistance $R_{H}$ is negative below the Néel temperature ($T_{N}$), suggesting that the electron-like carriers dominate electronic transport Haga et al. (2017).

Motivated by these previous measurements, we performed a quantum oscillation measurement on high quality URhIn5 samples. In this paper, we present these results in detail.

II Samples and Experimental Method

Three high quality crystals of URhIn5, prepared using the self-flux method Bartha et al. (2015) were used. The samples were X-rayed and characterized using specific heat before the experiment was performed, and they had extremely high RRR (exceeding 200) Bartha et al. (2015), ideal for quantum oscillation measurements.

Modulation field de Haas-van Alphen measurements were carried out on these samples, with field ranges from 9-16 T and at temperatures between 65-2000 mK. The samples were rotated with respect to the applied field in the [001] to [100] plane and the [100] to [110] plane.

III Results

Fig. 2(a) shows an example of our raw data from 10 to 16 T at 65 mK, with field tilted at 4∘ from [001], and Fig. 2(b) shows the Fourier transform of the same set of raw data. A very strong peak at 1.1 kT and its three harmonics can be easily seen. This frequency is labelled $\alpha$. In addition, expending the vertical scale in Fig. 2(c), other sharp peaks are present as well: at 0.3 kT, 3.2 kT and 3.5 kT. These are labelled $\beta$, $\gamma_{1}$ and $\gamma_{2}$ respectively.

The effective mass $m^{*}$ associated with each extremal orbit can be extracted using the Lifshitz-Kosevich (LK) formula:

[TABLE]

where $C=\frac{2\pi^{2}k_{B}m_{e}}{e\hbar}$. Typical fits are shown in Fig. 3. The cyclotron masses found for other directions are summarized in Table 1.

The angle dependence of the dHvA frequencies is shown in Fig. 4. This angle dependence suggests that the $\beta$ branch, which has very little angle dependence, arises from a nearly spherical Fermi surface. However, the lack of any significant angle dependence may also be a sign of an impurity phase from, say, the inclusion of flux used to grow the sample, or perhaps a spherical pocket in an impurity phase such U2RhIn8. The $\alpha$ branch splits into two frequencies around 40∘. This suggest a distorted sphere that has only one extremal orbit at $c$-axis, but two extremal orbits beyond a critical angle. The two branches $\gamma_{1}$ and $\gamma_{2}$ are close in frequency and it is natural to think that they arise on the same Fermi surface sheet, geometrically this would be certain only if they converge as the angle is rotated away from the c-axis. But the frequency of $\gamma_{1}$ shows only a small upward curvature before it disappears. Thus we cannot rule out the possibility that they are distinct surfaces. $\gamma_{1}$ and $\gamma_{2}$ may arise from cylindrical surfaces, as shown from their upward curvatures. This is supported by the fact that we do not observe these frequencies at [100] or [110], which we would expect if this were a closed surface like $\alpha$. However, as $\gamma_{1}$ and $\gamma_{2}$ disappear relatively quickly when we rotate away from the c-axis, the evidence is not conclusive.

IV Analysis and Discussion

IV.1 Paramagnetic Ground State Calculation

We performed an LDA+SO calculation of URhIn5’s ground state, using the linearized augmented plane wave (LAPW) scheme that was implemented in the WIEN2k package Blaha et al. (2011). Spin-orbit coupling was turned on, and RKmax was set to 7.0. The resulting Fermi surface is shown in Fig. 5. Using the Supercell K-space Extremal Area Finder (SKEAF) Rourke and Julian (2012) program, dHvA frequencies and approximate effective masses were extracted and are given in Table 2. These values do not generally match the experimentally obtained values in Table 1, however, band 48 does have a 1.1 kT frequency that is similar to our $\alpha$ branch.

To study this 1.1 kT frequency further, we used SKEAF to find the angle dependence of the frequencies, shown in Fig. 6. Here we can see that the 1.1 kT frequency is a closed ellipsoidal surface similar to our $\alpha$ branch. However, the difference from experiment is that a second, higher frequency of 1.5 kT branches out from the main frequency whereas in experimental results, a lower frequency branches out. The topology of the two Fermi surfaces are therefore not very similar. While the $\gamma_{1}$ and $\gamma_{2}$ frequencies in our experiment could be coming from cylindrical surfaces,they are almost 10 times larger than the frequencies found by SKEAF in the paramagnetic WIEN2k calculation. It is extremely unlikely that antiferromagnetism could produce larger Fermi surfaces, so our conclusion is that the paramagnetic, all-itinerant Fermi surface is not a good starting point for explaining URhIn5.

IV.2 Fermi Volume Estimation

The total Brillouin zone (BZ) volume is $1.6\times 10^{30}$ m*-3*. If antiferromagnetism doubles the unit cell in the c-axis, the AFM BZ shrinks to $7.8\times 10^{29}$ m*-3*. If we assume all the Fermi surface sheets are spherical so the volume of each sheet is given by $V=4/3\pi k_{F}^{3}$, then given the antiferromagnetic BZ, $\alpha$ occupies 3.2% of the BZ; $\beta$ occupies 0.4%; $\gamma_{1}$ occupies 16% and $\gamma_{2}$ occupies 19% of the Brillouin zone. If we assume $\gamma_{1}$ and $\gamma_{2}$ are cylindrical surfaces and their volumes are given by $S_{F}\times 2\pi/(2c)$ where $S_{F}$ is the area enclosed by the cylinder and given by the dHvA frequency, c is the c-axis lattice parameter, and $2\pi/(2c)$ is the height of the AFM BZ, the volumes of $\gamma_{1}$ and $\gamma_{2}$ become 17% and 18% of the total BZ volume, respectively. In total these Fermi surface sheets occupy nearly 40% of the total AFM BZ, if we assume there is only one copy of each Fermi surface in the BZ. As noted in the next section, however, it is probable that there are multiple copies of at least some Fermi surfaces.

IV.3 Sommerfeld Coefficient Estimation

We can estimate the Sommerfeld coefficient $C/T=\gamma$ from the Fermi surfaces obtained. If we make the simplifying assumption that the frequencies at [001] correspond to spherical Fermi surfaces, so that the extremal area is given by $S_{F}=\pi k_{F}^{2}$, then $\alpha$ would contribute 1.7 mJ mol*-1K-2*; $\beta$ would contribute 2.1 mJ mol*-1K-2*; $\gamma_{1}$ would contribute 3.6 mJ mol*-1K-2* and $\gamma_{2}$ would contribute 5.2 mJ mol*-1K-2*. This sums up to 12.6 mJ mol*-1K-2* if we assume one sheet each, whereas the experimental value is around 50-60 mJ mol*-1K-2* Bartha et al. (2015); Matsumoto et al. (2013).

If, however, we assume that $\gamma_{1}$ and $\gamma_{2}$ are distinct, cylindrical, Fermi surfaces. Then their specific heat contributions are no longer $k_{F}$ dependent and they rise to 7.6 mJ mol*-1K-2* for $\gamma_{1}$ and 10.6 mJ mol*-1K-2* for $\gamma_{2}$. The total $C/T$ contribution then rises to 22 mJ mol*-1K-2* if we assume one copy of each sheet. This is still much too small. Thus we assume multiple copies of each sheet. For example, $4\times\alpha$, $4\times\beta$, $2\times\gamma_{1}$ and $2\times\gamma_{2}$ gives 52 mJ mol*-1K-2*, in the range of the observed Sommerfeld coefficient, and the total BZ occupation will be around 84%. If, as seems more likely, $\gamma_{1}$ and $\gamma_{2}$ are extremal orbits on the same cylindrical Fermi surface, then we can have $4\times\gamma$ instead of $2\times\gamma_{1}$ and $2\times\gamma_{2}$ with an average contribution of 9.1 mJ mol*-1K-2* per sheet gives the same result. This scenario would account for all of the specific heat, but the BZ would be very packed. The only other possibility is that there are heavy orbits (i.e. hot spots) on the Fermi surface sheets that we have observed, or else other heavy surfaces that we have not observed. A better estimate of $\gamma$ will be obtained once we have reliable band-structure calculations.

IV.4 Comparisons to Previous Studies and Other Related Compounds

Comparing our results to the earlier dHvA study on URhIn5 by Matsumoto et al. Matsumoto et al. (2014), the frequency $\alpha$ and its splitting at higher angles were seen in both experiments. The angle dependence, which is indicative of an ellipsoidal Fermi surface, agrees well. In our experiment however, we also observed two larger Fermi surface sheets, $\gamma_{1}$ and $\gamma_{2}$, as well as one additional small spherical pocket $\beta$. While the total contribution from the observed sheets to specific heat remains well under the experimental value of $\gamma=50-60$ mJ mol*-1K-2*, we can at least consider the possibility of some combination of these sheets as mentioned in the section above, that will make up the missing specific heat contribution. The observation of these larger sheets also suggests that URhIn5 is very different from the non-magnetic, 5f-itinerant semi-metal URhGa5 which has only small pockets and small specific heat Ikeda et al. (2002); Matsumoto et al. (2014).

In Ref. Matsumoto et al., 2014, a comparison was made to a reference compound ThRhIn5, which has the same crystal structure but does not have 5f electrons. ThRhIn5 has very large Fermi surfaces (up to 7 kT) and with strong angle dependence that would be expected for quasi-cylindrical surfaces Matsuda et al. (2007); Matsumoto et al. (2014). These are considerably larger than our observations. However, as $\gamma_{1}$ and $\gamma_{2}$ are suggestive of being cylindrical surfaces, it may be that the electronic structure of URhIn5 is not as drastically different from ThRhIn5 as previously thought Matsumoto et al. (2014).

UNiGa5 is another antiferromagnet in this family, with a Néel temperature $T_{N}=85.5$ K and a Sommerfeld coefficient $\gamma=30$ mJ mol*-1K-2* Tokiwa et al. (2001b). Its Fermi surface has also been mapped using dHvA. In URhIn5, our $\alpha$ surface has a very different angular dependence from that of any of the ellipsoidal Fermi surfaces in UNiGa5. The largest cylindrical Fermi surface in UNiGa5, on the other hand, also has much smaller cross-section for field along the c-axis than our $\gamma_{1}$ and $\gamma_{2}$ surfaces. Whereas Tokiwa et al. see clear evidences of cylindrical Fermi surfaces Tokiwa et al. (2001b), we were unable to follow our $\gamma_{1}$ and $\gamma_{2}$ frequencies far enough to confirm the 1/$\cos\theta$ dependence expected for a cylindrical Fermi surface, but on balance it seems very likely that they arise on a cylindrical surface. The effective masses of UNiGa5 at around 1.4 to 3.1 $m_{e}$ are, however, similar to those found for URhIn5 Tokiwa et al. (2001b) There are no calculations available for the antiferromagnetic ground state of UNiGa5, and the paramagnetic state calculation does not match the experimental results Tokiwa et al. (2001b).

As mentioned in the introduction, another related compound to URhIn5 is UIn3. This compound is also an antiferromagnet and orders at $T_{N}=88$ K Tokiwa et al. (2001c). It has a Sommerfeld coefficient of $\gamma=40$ mJ mol*-1K-2*. The Fermi surface has been mapped, and no cylindrical Fermi surfaces were found Tokiwa et al. (2001c). The angle dependence does not resemble our findings for URhIn5, and the effective masses of UIn3 range from 9.8 to 33 $m_{e}$, which are significantly larger than those of URhIn5. However, as is the case of UNiGa5, the lack of reliable band structure calculations for URhIn5 and UIn3 Tokiwa et al. (2001c) makes these comparisons quite speculative.

V Conclusion

We remark that there are major discrepancies between the calculated 5f all-itinerant Fermi surface and the experimentally obtained dHvA frequencies, and the cyclotron masses also differ. The present results cannot be fully explained by the paramagnetic calculation and it is therefore highly desirable that a detailed band structure calculation be carried out for the antiferromagnetic ground state.

VI Acknowledgements

J. C. would like to acknowledge many helpful discussions with M. Diviš. This work was generously supported by NSERC and CIFAR of Canada, Canada Research chair, the Czech Science Foundation Grant No. P203/12/1201 and the Charles University project CA UK Grant No. 128317.