Quantum Criticality Without Tuning in the Mixed Valence Compound β-YbAlB4

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Science  21 Jan 2011:
Vol. 331, Issue 6015, pp. 316-319
DOI: 10.1126/science.1197531


Fermi liquid theory, the standard theory of metals, has been challenged by a number of observations of anomalous metallic behavior found in the vicinity of a quantum phase transition. The breakdown of the Fermi liquid is accomplished by fine-tuning the material to a quantum critical point by using a control parameter such as the magnetic field, pressure, or chemical composition. Our high-precision magnetization measurements of the ultrapure f-electron–based superconductor β-YbAlB4 demonstrate a scaling of its free energy that is indicative of zero-field quantum criticality without tuning in a metal. The breakdown of Fermi liquid behavior takes place in a mixed-valence state, which is in sharp contrast with other known examples of quantum critical f-electron systems that are magnetic Kondo lattice systems with integral valence.

Quantum phase transitions occur at zero temperature as a consequence of quantum rather than thermal correlations. Generally, a quantum critical point (QCP) can be reached by driving a finite-temperature critical point (1, 2) or a first-order critical end-point (3) to absolute zero. The breakdown of Fermi liquid (FL) behavior in metals observed near a magnetic QCP challenges our current understanding of strongly correlated electrons. Although the mechanism of unconventional quantum criticality is actively debated, there is a growing consensus that the underlying physics involves a jump in the Fermi surface volume associated with a partial electron localization (48). To date, the FL breakdown has only been observed by fine-tuning a material to a QCP through use of a control parameter such as magnetic field, pressure, or chemical composition.

Recent work reporting the discovery of superconductivity in an ytterbium (Yb)–based heavy fermion material β-YbAlB4 has raised the interesting possibility that this system may exhibit quantum criticality without tuning (9). In this compound, signatures of quantum criticality were observed to develop above a tiny superconducting (SC) dome, with a SC transition temperature of Tc ~ 80 mK and an upper critical field μ0Hc2 ≈ 30 mT (9, 10). Although this observation motivated the possibility of a zero-field QCP, it did not rule out a QCP located near the upper critical field Hc2, as observed in the heavy fermion superconductor CeCoIn5 (11).

In this report, we present clear and quantitative evidence that quantum criticality develops at zero field without tuning in β-YbAlB4, buried deep inside the SC dome. Moreover, we report a simple T/B–scaling form of the free energy spanning almost four decades in magnetic field, revealing that the signatures of the putative quantum critical point extend up to temperatures T and fields B more than 100 times larger than Tc and μ0Hc2, respectively.

To quantify the free energy F(T,B), we used high-precision measurements of the magnetization M = –∂F/∂B. Measurements were made on ultrahigh-purity single crystals with a mean free path exceeding 1000 Å and residual resistivity of less than 0.6 microhm·cm, which were carefully etched to fully remove surface impurities (12). Our measurements revealed a simple T/B scaling over a wide range of temperature and field, governed by a single quantum-critical (QC) scaling exponent previously masked (9) by a limited experimental resolution and the impurity effects caused by surface and bulk impurities (12). The T/B scaling leads to the following meaningful consequences. First, the QC physics is self-similar over four decades of T/B, with no intrinsic energy scale. Second, the field-induced FL is characterized by a Fermi temperature that grows linearly with the field, determined by the Zeeman energy of the underlying critical modes. Lastly, the scaling allowed us to determine an upper bound on the magnitude of the critical field |Bc| < 0.2 mT, which is well inside the SC dome and comparable with Earth’s magnetic field; this indicates that β-YbAlB4 is intrinsically quantum critical, without tuning the magnetic field, pressure, or composition.

These results are surprising given the fluctuating valence nature of this material, with valence Yb+2.75 significantly far from integral, revealed by recent experiments (13). All QC heavy-fermion intermetallics known to date have an almost integral valence that stabilizes the local moments (1, 2). Such so-called Kondo lattice systems are characterized by a small characteristic scale T0, below which the moments are screened to form a paramagnetic heavy FL. Various types of order, such as superconductivity and antiferromagnetism (AFM), compete with the heavy FL, leading to quantum criticality, as seen in, for example, CeCu5.9Au0.1 (T0 = 6.2 K) (14) and YbRh2Si2 (24 K) (2). In contrast, mixed-valence compounds display a much larger T0, below which they typically behave as stable FLs with moderate quasiparticle effective masses and no competing order. For example, YbAl3, with nonintegral valence Yb+2.71, is characterized by T0 ~ 300 K (15).

A remarkable feature of β-YbAlB4 (Fig. 1A) is that it is quantum critical (9), yet the scale T0 ~ 250 K, obtained from the resistivity coherence peak, is one or two orders of magnitude larger than in other known QC materials. This is confirmed by the scaling behavior of the magnetic specific heat: CM/T=S0T0ln(T0/T), where S0 is a constant (Fig. 1B, inset). The −lnT dependence of CM/T in the three QC materials CeCu5.9Au0.1 (14), YbRh2Si2 (2, 16), and β-YbAlB4 collapse onto one curve after setting T0 for β-YbAlB4 ~ 200 K. The recent observation of intermediate valence (Yb+2.75) in β-YbAlB4 at 20 K by use of hard x-ray photoemission spectroscopy (13) is consistent with this large T0.

Fig. 1

(A) Crystal structures of β-YbAlB4 and α-YbAlB4, which are formed from straight and zigzag arrangements of distorted hexagons of Yb atoms (shaded in red for the α phase), respectively (17). The crystallographic unit cells of both phases are orthorhombic and can be viewed as an interleaving of planar B nets and Yb/Al layers. (B) Magnetic part (f-electron contribution) of the specific heat CM plotted as CM/T versus T for both β-YbAlB4 (solid circles) and α-YbAlB4 (open squares) (12). CM/T at B = 0 for the β phase shows a lnT dependence for 0.2 K < T < 20 K. T0 ~ 200 K was determined from the fit to CM/T = S0/T0ln(T0/T). The upturn in the lowest T may contain a nuclear contribution. (Inset) CM/T scaled by T0 compared with quantum critical systems CeCu5.9Au0.1 (T0 = 6.2 K) (14) and YbRh2(Si1−xGex)2 (T0 = 24 K) (2, 16). The lnT dependence of the three QC materials collapse on top of each other using nearly the same coefficient S0 ~ 4 J/mole K, indicating a common meaning of T0 as the T scale below which ~70% of the ground doublet entropy R ln2 is released.

The quantum criticality in this valence-fluctuating state is accompanied by several properties reminiscent of an integral-valence Kondo lattice. To understand their origin, it is useful to compare them with those of α-YbAlB4 (Fig. 1A) (17), which is a locally isostructural polymorph of β-YbAlB4 and a FL with a similarly intermediate valence (Yb+2.73) (13). Instead of Pauli paramagnetism normally seen in a valence-fluctuating material, the magnetic susceptibilities of both materials display Curie-Weiss behavior with Weiss temperature ΘW, χ = C/(T + ΘW), indicating the existence of local moments (Fig. 2A). In addition, both materials have a maximum in –dM/dT at T* ~ 8 K, signaling a crossover from local moment behavior (fig. S1). Below T*, CM/T of the α phase levels off to a constant characteristic of heavy FL behavior, whereas that of the β phase continues to diverge (Fig. 1B). Thus, the fate of local moments found above T* is different in these locally isostructural systems: Yb spins are fully screened in the α phase but may well survive down to lower temperatures in the β phase and produce the quantum criticality. In both phases, strong correlation effects are manifest—for example, in the strongly enhanced CMT|T0>˜130 mJ/mol K2, which is two orders of magnitude larger than the band calculation estimate (~ 6 mJ/mol K2) (18).

Fig. 2

(A) Temperature dependence of the magnetic susceptibility M/B of both β-YbAlB4 (solid circles) and α-YbAlB4 (open squares). The Curie-Weiss fit above 150 K yields a Weiss temperature ΘW ~ −110 K and an effective moment of ~2.2 μB for both systems. (Inset) The quantum critical B-T range where the scaling applies (solid circles in the blue shaded region) and the superconducting (SC) phase under the upper critical fields [open circles and triangles (12)]. (B) Scaling observed for the magnetization at T ≲ 3 K and B ≲ 2 T. The data was fitted to the empirical Eq. 1 with scaling function ϕ(x) = Λ x (A + x2)n, a form chosen to satisfy the appropriate limiting behavior in the FL regime (12). (Right inset) Pearson’s correlation coefficient R for the fit with finite Bc. R reaches a maximum value of 1 if the fit quality is perfect. The best fit is obtained with n = 1.25 ± 0.01 and Bc = −0.1 ± 0.1 mT (light blue line), corresponding to α = 3/2 in the scaling form of the free energy (Eq. 2) (12) and |Bc| < 0.2 mT. (Left inset) The B-T phase diagram of β-YbAlB4 in the low T and B region. The filled circles are determined from the peak temperatures of −dM/dT, below which the FL ground state is stabilized. At low field, the thermodynamic boundary between the FL and NFL regions is on a kBT ~ gμBB line (broken line). The open circles are the temperature scale TFL, below which the T2 dependence of the resistivity is observed (9).

These signatures indicate that both phases are governed by two distinct energy scales: a high-energy valence fluctuation scale T0 ~ 200 K and a low-energy scale T* ~ 8 K, characterizing the emergence of Kondo lattice physics. A possible origin of this behavior is the presence of ferromagnetic (FM) interactions between Yb moments, manifested by the large Wilson ratio RW between χ and CM/T, observed in both α- and β-phases (minimum estimate RW ≵7) (9) and further corroborated by the observation of an electron spin resonance signal (19), which is generally only seen in the presence of FM correlations (20). Such FM interactions are known to give rise to Kondo resonance narrowing (21) in d-electron systems in which Hund’s coupling causes a marked reduction in the Kondo temperature (21, 22).

In the present case, the role of Hund’s coupling is played by FM intersite RKKY interactions, probably along the short Yb-Yb bonds that form chains along the c axis. The moments of a few n neighboring Yb ions may thus become aligned, forming a fluctuating “block” spin S = n J. The observed valence Yb+2.75 could then be understood in terms of Yb3+ >⇌ Yb2+ fluctuations because at any one time, approximately 1/4 of the Yb atoms along the chains are in a singlet Yb2+ configuration, forming the ferromagnetic blocks of approximately n ~ 3 spins. The effect of these block spins is to exponentially suppress the characteristic spin fluctuation scale (21), resulting in localized-moment behavior. The absence of long-range magnetic order in the α or β phase points to the presence of competing magnetic interactions. Indeed, the Weiss temperature ΘW ~ −110 K (Fig. 2A), which is characteristic of an AFM, indicates the importance of magnetic frustration. The competing interplay of FM interactions and valence fluctuations thus leads to Kondo lattice–like behavior in a mixed-valent material, setting the stage for quantum criticality to emerge at lower temperatures in β-YbAlB4.

A prominent feature of the quantum criticality in β-YbAlB4 is the divergence of the magnetic susceptibility χ as T → 0. By examining the field evolution of magnetization M = −∂F/∂B as a function of both T and B (12), we can accurately probe the free-energy F near quantum criticality. Figure 2A shows the T dependence of χ(B) = M/B for different values of Bc. Spanning four orders of magnitude in T and B, the data show a systematic evolution from a non-FL (NFL) metal with divergent susceptibility at zero field (χ ~ T−1/2) to a FL with finite χ in a field gμBBkBT, where kB is the Boltzmann constant.

Intriguingly, the evolution of M/B found in the region T ≲ 3 K and B ≲ 2 T (Fig. 2A, inset) can be collapsed onto a single scaling function of the ratio T/B:dMdT=B1/2ϕ(TB) (1) as shown in Fig. 2B. The peak of the scaling curve lies at kBT /gμBB ~ 1, marking a cross-over between the FL and NFL regions, showing that kBTF ~ gμBB plays the role of a field-induced Fermi energy, as shown in the Fig. 2B inset. Integrating both parts of Eq. 1, one obtains the following scaling law for the free energy (12):FQC=B3/2f(TB) (2)where f is a scaling function of the ratio T/B with the limiting behavior f(x) ∝ x3/2 in the NFL regime (x >> 1) and f(x) ∝ const + x2 in the FL phase (x << 1). Indeed, the observed scaling of dM/dT in Eq. 1 is best fitted with ϕ(x)=Λx(A+x2)α22 (12), resulting in a particularly simple form of the free energy:FQC=1(kBT˜)1/2[(gμBB)2+(kBT)2]3/4 (3)with the best fit obtained with effective moment gμB = 1.94 μB and the energy scale kBT˜ ≈ 6.6 eV of the order of the conduction electron bandwidth (12). This means that the free energy depends only on the distance from the origin in the (T,B) phase diagram, similarly to the T/B scaling established in the Tomonaga-Luttinger liquids in one-dimensional metals (23, 24). Equation 3 implies that the effective mass of the quasi-particles diverges as m* ~ B−1/2 at the QCP (12). This divergence in a three-dimensional material, together with the T/B scaling, cannot be accounted for by the standard theory based on spin-density-wave fluctuations (25, 26). Instead, it indicates a breakdown of the FL driven by unconventional quantum criticality.

The T/B scaling suggests that the critical field Bc of the quantum phase transition is actually zero. A finite Bc would require that the argument of the scaling functions f(x) and ϕ(x) is the ratio x = T/|BBc|, as seen for instance in YbRh2Si2 (16). To place a bound on Bc, we substituted this form for x into Eq. 1, seeking the value of Bc that would best fit the experimental data. The Pearson’s correlation coefficient R obtained for this fit (Fig. 2B, inset), indicates that Bc is optimal at −0.1 ± 0.1 mT. The uncertainty is only a few times larger than Earth’s magnetic field (~0.05 mT). More importantly, it is two orders of magnitude smaller than μ0Hc2 = 30 mT and six orders of magnitude smaller than valence fluctuation scale T0 ~ 200 K. Thus, β-YbAlB4 provides an example of essentially zero-field quantum criticality.

Further evidence for zero-field quantum criticality is obtained from an analysis of the magnetocaloric ratio, ΓH1TS/BS/T=M/TC (Fig. 3). Here, C is the total specific heat (12). Our results show a clear divergence of ΓH/B as T → 0 in the NFL regime, which is a strong indicator of quantum criticality (27). From the NFL regime, we can extract the critical field |Bc| < 0.2 mT, which is consistent with the estimate of Bc obtained from the scaling behavior of M in Eq. 1 (fig. S2).

Fig. 3

Temperature dependence of the magnetocaloric effect divided by B, ΓH/B. See also fig. S2 for field dependence of ΓH.

The simple T/B scaling in the thermodynamics enables us to characterize the QC excitations of β-YbAlB4. In particular, the collapse of all magnetization data in terms of the dimensionless ratio r = kBT/(gμBB) between the Boltzmann energy kBT and the Zeeman energy gμBB indicates an absence of scale in the zero-field normal state. Furthermore, the appearance of a field-induced Fermi energy—linear over more than three decades in B—shows that the underlying critical modes are magnetic in character.

Using the Heisenberg energy-time uncertainty principle (ΔtΔEћ, where ћ is Planck’s constant h divided by 2π), we can reinterpret the T/B scaling in the time domain, visualizing the field-induced FL as a kind of “quantum soda” of bubbles of quantum critical matter of finite duration τQ = ћ/gμBB, immersed in a FL. At finite temperatures, thermodynamics averages the physics over a thermal time scale τT = ћ/kBT; thus, the quantity r = τQ / τT ~ T/B in the scaling is the ratio of the correlation time τQ to the thermal time-scale τT. At low temperatures r << 1 (τT >> τQ), thermodynamics probes the FL exterior of the bubbles, but when r >> 1 and τT << τQ, it reflects the QC interior of the bubbles. This accounts for the cross-over between FL and QC behaviors at r ~ 1. Moreover, T/B scaling over a wide range r ~ 10−1 to ~103 indicates that the quantum fluctuations in the ground state are self-similar down to 1/1000th of the correlation time τQ.

The observation of zero-field quantum criticality in valence-fluctuating β-YbAlB4 cannot naturally be interpreted as a conventional QCP, which would require a fortuitous combination of structure and chemistry to fine-tune the critical field Bc to within 0.2 mT of zero. A more natural interpretation of the results is that β-YbAlB4 forms a quantum critical phase that is driven into a FL state by an infinitesimal magnetic field. The T/B scaling requires that the critical modes are Zeeman-split by a field, and as such, various scenarios—such as critical Fermi surfaces (28) or local quantum criticality with E/T scaling (6)—may be possible contenders for the explanation, provided they can be stabilized as a phase. Established theoretical examples of a critical phase with T/B scaling include the Tomonaga-Luttinger liquid in half-integer spin chains and the one-dimensional Heisenberg ferromagnet (23, 24). Experimentally, the d-electron metal MnSi is a candidate for a quantum critical phase, with anomalous transport exponents observed over a range of applied pressure (29). Although present work provides a strong indication for existence of such a phase in β-YbAlB4, future studies—in particular, under pressure—are necessary in order to establish it definitively.

Supporting Online Material

Materials and Methods

SOM Text

Figs. S1 to S4


References and Notes

  1. Materials and methods, as well as the details of the scaling analysis and the difference from the previously reported χ(T) and CM(T) (9), are available as supporting material on Science Online.
  2. We thank H. Ishimoto, D. E. MacLaughlin, K. Miyake, T. Senthil, Q. Si, T. Tomita, K. Ueda, and S. Watanabe for useful discussions. This work is partially supported by grants-in-aid (21684019) from the Japan Society for the Promotion of Science; by grants-in-aid for Scientific Research on Priority Areas (19052003) and on Innovative Areas (20102007 and 21102507) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan; by Global Centers of Excellence Program “the Physical Sciences Frontier,” MEXT, Japan; by Toray Science and Technology Grant; and by a grant from NSF DMR-NSF-0907179 (P.C. and A.H.N.). P.C. and A.H.N. acknowledge the hospitality of the Aspen Physics Center.
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