Main

First we present the bulk magnetic and electronic properties of β-YbAlB4, a new morphology of the previously known α-YbAlB4 (refs 7, 8). Shown in Fig. 1a is the orthorhombic crystal structure of β-YbAlB4 and the temperature dependence of the d.c. magnetic susceptibility χ=M/H. Here, M and H represent the magnetization and external field, respectively. The magnetic susceptibility shows the strong uniaxial anisotropy of an Ising system with moments aligned along the c axis. Above 100 K the c-axis susceptibility has a Curie–Weiss form χc(T)=C/(T−θW), with θW∼−210 K and a Curie constant C corresponding to an effective Ising moment μeff=gJJZ∼3.1 μB, where gJ is the Landé g factor and JZ is the c-axis component of the total angular momentum. The in-plane susceptibility, on the other hand, is almost temperature independent, showing a weak peak around 200 K.

Figure 1: Physical properties of β-YbAlB4.
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a, Temperature dependence of the d.c. susceptibility M/H measured in a field of 100 mT along the a b plane and c axis. The c-axis susceptibility changes in slope around 40 K. This is most likely due to a crystal-electric-field splitting Δ, separating the ground-state Kramers doublet and the excited states. Inset, the crystallographic unit cell of β-YbAlB4. It has an orthorhombic (Cmmm) space group with lattice parameters a=0.73080(4) nm, b=0.93150(5) nm, c=0.34980(2) nm and can be viewed as an interleaving of planar B nets and Yb/Al layers7. Interestingly, the closest Yb–Yb contact is 0.34980(2) nm, corresponding to c, which is slightly more than twice the metallic radius of Yb3+ (0.174 nm). The underlying structural unit for magnetism may well be a one-dimensional chain of Yb3+ penetrating the B net. b, Temperature dependence of zero-field in-plane resistivity ρa b (open circle) and its 4f-electron contribution ρm (solid line). The latter was estimated by subtracting the temperature dependence of ρa b for β-LuAlB4, the non-magnetic isostructural analogue of β-YbAlB4 (ref. 7). No superconductivity is found for β-LuAlB4 down to T=35 mK. A slight change in slope of ρm around 40 K is attributable to a crystal-electric-field effect. Insets, Low-temperature part of ρa b(T) versus T (top left) and T1.5 (bottom right).

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Shown in Fig. 1b is the temperature dependence of the in-plane resistivity, ρa b, along with the estimated 4f-electron contribution ρm (defined in the figure caption), which shows a coherence peak at about 250 K. The low residual resistivity ρa b(0)∼0.4 μΩ cm and correspondingly high residual resistivity ratio, ρa b(300 K)/ρa b(0)∼300, suggest that the electronic mean free path is of the order of 0.1 μm.

In contrast to most other heavy-fermion compounds, the resistivity does not show a Fermi liquid (FL) regime characterized by a T2 temperature variation (Fig. 1b). As shown in Fig. 1b insets, ρa b is linear between 4 and 1 K and varies as T1.5 below T0∼1 K down to 80 mK. Below 80 mK our highest-purity samples are superconducting (Fig. 1b, insets). We shall return to this key finding presently.

The effect of an applied magnetic field on the temperature dependence of the in-plane resistivity is shown in Fig. 2. It is seen that the FL form of ρa b is rapidly restored in a magnetic field (blue region in Fig. 2b) and that the non-FL form discussed above with a temperature exponent of 1.5 (yellow region) exists only below about 0.1 T.

Figure 2: Crossover from non-FL to FL behaviour in the resistivity.
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a, a b-plane resistivity ρa b versus T2 at various magnetic fields along the c axis. b, Contour plot of the resistivity exponent α defined by Δ ρ=(ρ(T)−ρ(0))∼Tα in the temperature–field phase diagram.

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As shown in Fig. 3a,b, non-FL behaviour is also seen in the susceptibility χc and the magnetic part of the specific heat CM (defined in the figure caption). In zero magnetic field χc∼T−1/3 and CM/T∼(S*/T*)ln(T*/T) up to crossover temperatures T0 of the order of 2 K and 3 K, respectively. The values of T0 for the resistivity, susceptibility and specific heat are thus similar. A fit to the specific-heat data gives S*∼5.1 J/(mol Yb K), a value close to Rln2. T*∼200 K∼θW may be interpreted as the temperature at which the entropy of a ground state doublet is recovered.

Figure 3: Crossover from non-FL to FL behaviours in the susceptibility and heat capacity.
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a, Inverse susceptibility H/M versus T1/3 at various magnetic fields applied along the c axis. The linear fit indicates a T−1/3 dependence for the susceptibility. b, Magnetic part of the specific heat divided by temperature, CM/T, versus the logarithm of the temperature. Here, CM is obtained by subtracting the specific heat CP of β-LuAlB4 from that for β-YbAlB4. The linear fit indicates a ln(T*/T) dependence with T*∼200 K. Above 10 K, CM/T shows the tail of a Schottky peak due to the crystal-electric-field splitting Δ to the first excited state. Because the peak should be located at T>30 K, Δ must be more than 60 K and must be the origin of the change in slope observed around 40 K in χc(T) and ρa b(T) in Fig. 1. In keeping with the large temperature scale T*∼200 K, the entropy obtained through integration of CM/T from 0.4 to 30 K reaches only 50% of R ln2. c, Log–log plots of A versus γ and A versus χ for various fields, where χ=M/H. The solid lines represent A/γ2∼3×10−5 μΩ cm K−2(mol K2 mJ−1)2 and A/χ2∼5×102 μΩ cm K−2 (mol emu−1)2. d, Field dependence of A. The solid line represents the best fit to the form B−β, which yields β=0.50(2).

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As in the cases of ρa b, we find that an applied field rapidly restores FL behaviour, that is, both χc and CM/T saturate at low T. CM/T saturates at above 150 mJ/(mol Yb K2), a value consistent with that expected for a heavy-fermion system.

The FL state can be characterized by the low-T limits of A=(ρa b(T)−ρa b(0))/T2, χc and γ=CM/T. In the following the saturation values of χc and CM/T are taken at 80 mK and 400 mK, respectively. From Fig. 3c we see that in order of magnitude our data roughly agree with the relations A∝γ2∝χ2, so γ∝χ in the FL regime between 0.5 and 4 T. The ratio A/γ2, the Kadowaki–Woods ratio, is of order 3×10−5 μΩ cm K−2(mol K2 mJ−1)2 and similar to that found in other heavy-fermion compounds9. The ratio χ/γ gives a Wilson ratio, RW=π2kB2/(μ0μeff2)(χ/γ), of the order of 5–6, which is characteristic of a system with strong magnetic correlations. At the low-T limit, A shows a diverging behaviour as B vanishes, following the form B−1/2 in the critical regime below ∼4 T (Fig. 3d).

Thus, all our observations indicate that as the magnetic field decreases to zero the FL state becomes unstable and A, χc and CM/T all become singular at low T. This is the behaviour expected of a system with a zero-field quantum critical point (QCP), characterized by divergences of A, χc and CM/T of the form T−x, where x is 1/2,1/3 and 0+, respectively (0+ stands for logarithmic divergence).

We now return to the observation of superconductivity, which emerges at very low temperatures from the non-FL state in samples with residual resistivity ratio RRR>100. Figure 4a shows the in-plane resistivity of two out of a dozen samples studied, one normal and one superconducting, with RRR of 70 and 300, respectively. Figure 4b shows d.c. magnetization of a sample consisting of a dozen crystals with an average RRR∼240 and Tc∼68 mK. The samples are thin plates with the c-axis normal to the plate surface, typically 1 mm across and 0.01 mm thick. Significant shielding of the applied field is evident in both the zero-field-cooled (ZFC) and field-cooled (FC) data. The fraction of the sample volume undergoing a superconducting transition has been estimated by comparing the measurements in β-YbAlB4 with those of Al plates of similar geometry and dimensions. At the lowest temperature of 25 mK we find that the superconducting volume fractions, f, in percentage terms are as follows: for a field in the a b plane fa b=15% (ZFC), 6% (FC), and for a field along the c axis fc=45% (ZFC), 5% (FC). These fractions are large enough, given the strong sensitivity of superconductivity to sample purity and probable variation in purity over a typical sample, to indicate that the superconductivity is essentially a bulk phenomenon.

Figure 4: Superconductivity in high-purity single crystals of β-YbAlB4.
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a, Temperature dependence of the low-temperature in-plane resistivity ρa b(T) for the higher-quality sample A with RRR ∼300 and for the lower-quality sample B with RRR ∼70. Only the higher-quality sample A shows superconductivity. b, Temperature dependence of the FC and ZFC data of the d.c. susceptibility M/H in a field of 0.1 mT along the a b plane for a dozen thin platelike crystals with RRR>200 and an average RRR∼240. By measuring the superconducting shielding effect of Al plates with similar geometry and size, the volume fraction of superconductivity at 25 mK is estimated to be 15 % (ZFC) for the field along the a b plane. The reduction of the volume fraction from 100% is probably because the sample thickness along the c axis (∼10 μm) is of the same order as the penetration depth λc along the c axis; λc is estimated to be ∼6 μm by our preliminary experiments to measure the superconducting critical fields.

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We now turn to a qualitative discussion of our experimental findings, starting with the non-FL normal state and the crossover to an FL state with applied magnetic field. The emergence of non-FL behaviour in heavy-fermion systems has been discussed in terms of the itinerant-electron and localized-electron models10,11,12,13,14. In the itinerant-electron model the conduction and f electrons hybridize to form a coherent state characterized by heavy quasiparticles on a Fermi surface enclosing conduction and f-electron states. In this model non-FL behaviour and superconductivity can arise, for example, when the system is tuned by some control parameter (doping, pressure or magnetic field) to the border of long-range magnetic order. When the magnetic transition as a function of the ‘quantum’ tuning parameter is continuous, we have a QCP characterized by a set of non-FL exponents for A, χc and CM/T. In the localized-electron model it is supposed that the coherence between the conduction and f-electron states fails to form completely or otherwise is different from that of the itinerant-electron model. In this model, too, we arrive at non-FL behaviour at a QCP characterized by a set of non-FL exponents.

In both the itinerant- and localized-electron models the non-FL exponents can depend sensitively on a number of details such as the effective dimensionality and nature of the magnetic correlations. We find that the three exponents characterizing A, χc and CM/T in β-YbAlB4 (namely, x=1/2,1/3 and 0+, respectively) cannot be understood in a consistent way in terms of either the itinerant- or localized-electron models in their present forms. As an example, we consider the predictions of the itinerant-electron model for three dimensions on the border of antiferromagnetism10,11. In the presence of residual disorder this model predicts x=1/2, 0 and 0 for A, χ and C/T, respectively, in only partial agreement with our results for β-YbAlB4 (x=1/2,1/3 and 0+). In the same model, but on the border of ferromagnetism, we expect x=1/3,4/3 and 0+ for A, χ and C/T, respectively, which is again in only partial agreement with our results for β-YbAlB4.

Systems in which the itinerant-electron model may be applicable are discussed, for example, in refs 1, 2, 4. Systems in which the localized-electron model is thought to be more appropriate are discussed in refs 15, 16 for Au-doped CeCu6 and in refs 6, 17, 18, 19 for YbRh2Si2. Here we compare our findings in β-YbAlB4 to those in the latter ytterbium compound. The quantum critical exponents characterizing these two ytterbium materials are different. However, in other respects we find compelling similarities. In both YbRh2Si2 and β-YbAlB4 the Wilson ratios RW are high and RW, A/χ2 and A/γ2 are weakly magnetic field dependent in the quantum critical regime. The high value of the Wilson ratio and the weak field dependence of A/χ2 have been interpreted to suggest that there is a softening of both ferromagnetic as well as antiferromagnetic spin fluctuations18,19. We also comment that the crossover temperatures T0 and T* are about one order of magnitude higher in β-YbAlB4 than in YbRh2Si2 (refs 17, 18). This suggests that β-YbAlB4 provides a wider accessible T range to identify the asymptotic non-FL exponents and has a more accessible electron-pairing energy scale than in YbRh2Si2.

Remarkably, we observe a non-FL state in β-YbAlB4 without chemical doping, in zero applied magnetic field and at ambient pressure, a rare example of quantum criticality emerging without external ‘quantum’ tuning. Furthermore, as in cerium-based compounds close to a QCP (refs 1, 2, 3, 4, 5, 6), the superconductivity appears precisely in the non-FL region of the phase diagram where spin fluctuations are expected to be the strongest, suggesting that the pairing mechanism in β-YbAlB4, the first Yb-based heavy-fermion superconductor, is based on magnetic interactions.

The extraordinarily high sensitivity of superconductivity to sample purity in β-YbAlB4 is also consistent with the idea that the Cooper pair state in this system is of an unconventional, non-s-wave, character. Superconductivity tends to be suppressed when the electronic mean free path falls below the superconducting coherence length. The latter can be quite short in some cases, such as in the high-temperature superconductors, but long (∼50 nm) in a system such as β-YbAlB4 with a very low Tc.

Heavy-fermion superconductivity might be more fragile in the ytterbium than in the corresponding cerium systems because the relevant intersite spin–spin interaction in the former tends to be weaker than in the latter, reducing the superconducting transition temperature while increasing the sensitivity of anisotropic pairing to disorder. Superconductivity in the cerium compounds has been variously described in terms of intersite spin–spin interactions as well as in some cases in terms of effects that arise on the border of density or valence instabilities1,4,20. How such effects play out and are relevant in the ytterbium-based systems are questions that we can now begin to address.

Methods

Single crystals were grown by using the Al-flux method as described in the literature7. The excess Al flux was etched by a water solution of sodium hydroxide. To remove any possible impurities attached to the surface during the etching process, single crystals were washed with dilute nitric acid before measurements. X-ray powder diffraction patterns show single-phase samples. Inductively coupled plasma analysis confirms stoichiometry of the compound. Single crystallinity was confirmed by transmission electron microscopy and by single-crystal four-axis X-ray diffraction7. Magnetization M above 1.8 K was measured with a commercial superconducting quantum interference device magnetometer whereas M between 0.08 and 2.5 K and under fields up to 7 T was measured by the Faraday method in a dilution refrigerator21. To detect the Meissner effect, magnetization measurements were made at 0.1 mT using a superconducting quantum interference device magnetometer in a dilution refrigerator, and a Nb superconducting shield covered with a μ-metal tube was used to eliminate the Earth’s magnetic field. Four-terminal resistivity measurements were made by using an a.c. method down to 25 mK. Specific heat CP was measured by a thermal relaxation method down to 0.4 K.