Strange metal without magnetic criticality

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Science  31 Jul 2015:
Vol. 349, Issue 6247, pp. 506-509
DOI: 10.1126/science.1262054

A mysteriously misbehaving phase

Two electrons in a vacuum repel each other because they carry like charges. But in a metal, with a bunch of other electrons, in some ways they behave as if they didn't interact. This phenomenon, known as the Fermi liquid (FL) behavior, comes with a typical quadratic dependence of the solid's electrical resistance on temperature. Cases where the resistance deviates from the FL dependence are commonly associated with magnetic quantum criticality. Tomita et al. measured the resistivity of the heavy-fermion compound β-YbAlB4 over a range of pressures and temperatures to identify a non-FL phase removed from a magnetic phase by an intervening FL. Explaining the non-FL behavior in the absence of magnetism presents a challenge to theorists.

Science, this issue p. 506


A fundamental challenge to our current understanding of metals is the observation of qualitative departures from Fermi liquid behavior. The standard view attributes such non-Fermi liquid phenomena to the scattering of electrons off quantum critical fluctuations of an underlying order parameter. Although the possibility of non-Fermi liquid behavior isolated from the border of magnetism has long been speculated, no experimental confirmation has been made. Here, we report on the observation of a strange metal region away from a magnetic instability in an ultrapure single crystal. In particular, we show that the heavy-fermion superconductor β-YbAlB4 forms a possible phase with strange metallic behavior across an extensive pressure regime, distinctly separated from a high-pressure magnetic quantum phase transition by a Fermi liquid phase.

Qualitative deviations from the standard theory of metals, Landau’s Fermi liquid (FL) theory (1), develop almost routinely in the vicinity of a magnetic quantum phase transition (2, 3). Conventionally, the origin of such non-Fermi liquid (NFL) behavior is attributed to the strong damping of the quasiparticle’s lifetime by quantum critical fluctuations of an underlying order parameter (49).

Physics delineates between the concept of a phase, occupying a finite parameter region of ground state, and quantum critical points, appearing at the transition between phases. Although the possible existence of strange metal phases with NFL behavior, occupying a finite region of the ground-state phase diagram, has long been speculated (1019), the close proximity of such phenomena to magnetic instability and a strong sensitivity to impurities has impeded an experimental confirmation of this idea. One of the most challenging questions is whether a fully paramagnetic strange metal phase is possible without magnetic criticality, retaining full symmetry of the underlying crystal structure.

Many prototypical quantum critical (QC) materials have been found within the class of 4f heavy-fermion compounds. The highly tunable characterstic energy scales and availability of high-purity crystals make them ideal candidates for the study of quantum criticality (2, 3). In these materials, quantum criticality develops from a competition between local moment magnetism and the conduction electron screening of the local moments (the Kondo effect). Most QC heavy-fermion materials are known to have an almost integral valence that stabilizes the local moments considered essential for the criticality.

An exception to this rule is β-YbAlB4, which exhibits quantum criticality despite strong mixed valency (2023). Ultrapure single crystals of this material exhibit intrinsically singular thermodynamic and transport behavior up to an upper limit scale of several K, including a divergent temperature (T) dependence of the magnetic susceptibility Embedded Image and an anomalous Embedded Image dependence of the electrical resistivity, both of which are extremely sensitive to a magnetic field B (20, 22, 23). In particular, T/B scaling of the magnetization has been observed over four decades of T/B, projected to extend down to fields as small as 0.1 mT (22). However, the observation of intrinsic quantum criticality as a function of field does not rule out the possibility that this phenomenon is merely a fine-tuned coincidence of lattice structure. Here, we demonstrate that the intrinsic quantum criticality of β-YbAlB4 is not fine-tuned but instead occupies an extended island of pressure in the phase diagram, indicating a formation of a phase without any symmetry breaking external fields for stabilization. Furthermore, we show that the strange metal region is clearly surrounded and separated from a high-pressure magnetic instability by a finite pressure range of Fermi liquid behavior.

Our main experimental observation is that of an extensive region of the strange metal behavior, which evolves to a Fermi liquid phase (Fig. 1). We employed ultrapure crystals of β-YbAlB4 with residual resistivity ratio (RRR) = 300 (with residual resistivity Embedded Imagecm and mean free path of >1000 Å) and performed high-precision resistivity measurements (with noise levels of <40 pVHzEmbedded Image) using a piston-cylinder pressure cell in a dilution refrigerator (24). The pressure was continuously monitored using tin and aluminum superconducting manometers. X-ray diffraction analyses confirm a continuous reduction of the lattice parameters under pressure with a bulk modulus of 189 GPa (fig. S3) (24). Strikingly, under pressures up to 0.25 GPa, the resistivity exhibits the same anomalous power law behavior Embedded Image, with the same slope as at ambient pressure (Fig. 1A). In contrast, above Embedded Image GPa, as we will discuss below, Embedded Image shows a clear deviation from a Embedded Image dependence at low Ts and transitions to a FL-like Embedded Image dependence. This can be clearly seen in the Embedded Image dependence of the power law exponent Embedded Image (solid circle) in Embedded Image (Fig. 1B). Under Embedded Image GPa, Embedded Image increases gradually on cooling and becomes constant Embedded Image below 0.3 K down to the superconducting (SC) transition temperature Embedded Image. In contrast, at Embedded Image GPa, it saturates to Embedded Image, the value characteristic of a FL state.

Fig. 1 Strange metal, Fermi liquid, and magnetic order in β-YbAlB4 under pressure.

Ultrapure single crystals (RRR = 300) were used (24). (A) Zero-field resistivity Embedded Image versus Embedded Image at various pressures (left and right axes). The anomalous Embedded Image dependence was found robust up to Embedded Image 0.4 GPa. The superconducting (SC) transition was observed up to P = 0.59 GPa. Around Embedded Image, a resistivity spike was observed just above Embedded Image (24). (Inset) Embedded Image versus Embedded Image obtained under an in-plane field Embedded Image T. The solid red line indicates a fit to Embedded Image dependence found at 0.72 GPa. (B) Embedded Image dependence of the power law exponent Embedded Image, corresponding to Embedded Image in (A) under B = 0 (solid symbols) and under Embedded Image T (open symbols). (C) Contour plot of the exponent Embedded Image in the Embedded Image-Embedded Image phase diagram for zero field. (For Embedded Image T, see fig. S8). Red, green, and blue circles indicate the superconducting Embedded Image, Néel point Embedded Image, and Embedded Image where Embedded Image starts showing Embedded Image dependence, respectively. Embedded Image, determined using Embedded Image under Embedded Image T, is also shown as purple circles. Embedded Image becomes strongly suppressed near Embedded Image (fig. S8). The solid line is a guide to the eye. (D) P dependence of the exponent Embedded Image and the coefficient A [of the Embedded Image dependence of Embedded Image] estimated in two T ranges: Embedded Image mK under zero field (Embedded Image, large red circles), and Embedded Image mK under zero field at Embedded Image GPa (Embedded Image, yellow circles; A, closed squares) and under Embedded Image T at Embedded Image GPa (Embedded Image, orange circles; A, open squares). Blue crosses: Embedded Image in the T range Embedded Image mK, Embedded Image T and Embedded Image GPa. Solid line: Fit to Embedded Image, yielding Embedded Image, Embedded Image = 0.40(5) GPa, Embedded Image = 0.05(1) Embedded Imagecm GPa/KEmbedded Image and Embedded Image = 0.43(1) Embedded Imagecm/KEmbedded Image. (E) P dependence of the residual resistivity Embedded Image under zero field (red) and under Embedded Image T (orange). The background color (yellow, white, blue, and green) in (D) and (E) is a guide to the eye.

The superconducting Embedded Image continuously decreases with pressure from Embedded Image mK at ambient pressure and finally vanishes around 0.6 GPa (Fig. 1, A and C). To extend our analysis below Embedded Image, we measured the resistivity by suppressing SC under a weak magnetic field along the ab plane, which should be irrelevant to the NFL critical fluctuations, thanks to the Ising character of the 4f moments. Figure 1A, inset, plots the resistivity versus Embedded Image measured at an in-plane field Embedded Image T under various pressures. The corresponding Embedded Image dependence of the exponent (Fig. 1B) indicates that the strange metallic state with Embedded Image extends down to the lowest Embedded Image mK under Embedded Image, whereas the exponent saturates to Embedded Image at Embedded Image and to Embedded Image for Embedded Image. The Fermi liquid temperature Embedded Image, below which Embedded Image shows a Embedded Image dependence, systematically decreases with decreasing pressure and appears to vanish at Embedded Image (fig. S8).

The contour plots of the exponent Embedded Image (Fig. 1C) obtained using the zero-field Embedded Image data in Fig. 1A and fig. S8A reveal an extended region of anomalous resistivity exponent, indicating the formation of the strange metal phase and its subsequent crossover into the high pressure FL phase. From ambient pressure, a NFL (yellow) region with Embedded Image occupies a finite range up to Embedded Image GPa above the SC dome. In contrast, at pressures beyond Embedded Image up to 2.5 GPa, Embedded Image locks into a constant Embedded Image (blue) below Embedded Image mK, indicating the formation of a FL phase. To carefully examine the phase evolution, in Fig. 1D, we plot the exponents Embedded Image= Embedded Image at the midpoints of two temperature ranges: Embedded Image mK under zero field (large red circles) and Embedded Image mK under an in-plane field of 0.1 T to suppress the SC (orange circles). For the non-SC region at Embedded Image GPa, we plot the zero-field exponent for the Embedded Image range Embedded Image mK (yellow circles). To further evaluate the exponent without the ambiguity associated with the residual resistivity, we have also carried out the analysis of the resistivity exponent (blue crosses) using Embedded Image in the Embedded Image range Embedded Image mK by suppressing the SC under Embedded Image T at Embedded Image GPa. All the data are consistent with the existence of a NFL phase with a constant Embedded Image at Embedded Image and a FL phase with Embedded Image at Embedded Image (Fig. 1D). The apparent crossover between Embedded Image and 2.0 marked by the two points with intermediate exponents is most likely a consequence of experimental resolution and a small inhomogeneity in the pressure. The presence of the superconducting resistivity spike close to Embedded Image (Fig. 1A) supports this interpretation (24).

In the FL phase, the A coefficient for the Embedded Image law is found to be field-independent at Embedded Image T (fig. S8C) (24). The A coefficient has a component that diverges at Embedded Image, following Embedded Image with Embedded Image GPa (Fig. 1D). In addition, changes in P dependence of Embedded Image are observed around Embedded Image for both Embedded Image and 0.1 T (Fig. 1E). Taken together with the change in the exponent Embedded Image, these anomalies suggest a possible quantum phase transition at Embedded Image separating the strange metal phase from the high-pressure FL.

Each of the putative NFL phases reported to date directly adjoin a magnetic phase and are thus linked to magnetic criticality (11, 12, 14, 17, 19). Generally, in Yb-based heavy-fermion compounds, both physical and chemical pressure induce magnetism, stabilizing an “YbEmbedded Image” state with a 4f magnetic moment and a smaller ionic radius than its nonmagnetic “YbEmbedded Image” counterpart (25). To clarify the relation between magnetism and the observed extensive regime of NFL behavior in β-YbAlB4, we have performed a detailed study using high pressure and chemical substitution.

We performed “high-Embedded Image K)” resistivity measurements in a cubic anvil cell, which allowed us to reach much higher pressures, up to 8 GPa (Fig. 2A and fig. S6B) (24). Whereas a systematic change is found in the resistivity Embedded Image at Embedded Image K, no change was found in Embedded Image at Embedded Image GPa below Embedded Image K (24). In the contour plots of the resistivity exponent Embedded Image (Fig. 2B), by far the most prominent feature of the phase diagram is the wide (red) region of anomalous T-linear resistivity (24). This region spans a pressure range from ambient pressure to 3 GPa, extending over a decade of Embedded Image from Embedded Image to 20 K (Fig. 2, A and B). Beyond the critical pressure Embedded Image, GPa (fig. S6) (24), the temperature derivative of resistivity Embedded Image shows an onset of increase on cooling at a pressure-dependent temperature Embedded Image (Fig. 2A, inset) (24). This temperature marks the development of antiferromagnetic (AF) order, as we will discuss below. Embedded Image rises rapidly to 18 K at 8 GPa, which is very high for an Yb-based heavy-fermion system.

Fig. 2 Pressure-induced antiferromagnetism in β-YbAlB4.

(A) T dependence of the in-plane resistivity Embedded Image obtained under various pressures in a cubic anvil cell above Embedded Image K. (Inset) Embedded Image versus T. The arrows indicate the Néel temperatures at different pressures. (B) Contour plots of the power law exponent Embedded Image of Embedded Image in the Embedded Image-Embedded Image phase diagram of an ultrapure single-crystal of β-YbAlB4 (RRR = Embedded Image). Its low T and low P region specified by the blue frame in (B) corresponds to the one in Fig. 1C. For clarity, the values of Embedded Image are multiplied by a factor of 10.

Correspondingly, in the “low-Embedded Image K)” measurements using the dilution refrigerator, application of pressures exceeding Embedded Image GPa in a piston cylinder cell gives rise to a sudden decrease in Embedded Image (Fig. 1E); moreover, a kink develops in the resistivity and its T derivative at a temperature Embedded Image, which rapidly rises from 80 mK at 2.72 GPa to ~4 K at 2.8 GPa (Fig. 1A and fig. S9). Within the pressure uncertainty, this coincides with the onset of antiferromagnetism found in the cubic anvil cell (fig. S6A, inset). This rapid increase of Embedded Image, as well as the jump in Embedded Image across Embedded Image, suggests that the pressure-induced magnetic phase transition is first-order.

Chemical substitution confirms a similar phase evolution to that under pressure. In particular, Fe substitution for Al is found to lead to a crossover from a distinct region with quantum critical behavior to a Fermi liquid. Figure 3, A and B show the T dependence of the resistivity and its power law exponent, respectively. The chemical analysis, as well as the systematic increase in Embedded Image, confirm a homogeneous distribution of Fe ions (Fig. 3A, inset) (24). With 1% doping of Fe, we found that the power law exponent Embedded Image in Embedded Image approaches 1.5 upon cooling below 1 K, the same anomalous exponent as in pure β-YbAlB4, indicating the formation of the strange metal phase. At higher Fe content of x = 2 and 3%, on the other hand, the exponent Embedded Image approaches 1.7 and 2.0, respectively. In addition, both Embedded Image and Embedded Image for x = 3% show no magnetic anomaly but level off on cooling, signaling the formation of a Fermi liquid (fig. S4) (24).

Fig. 3 Chemical substitution effects in β-YbAl1–xFexB4.

(A) Inelastic component Embedded Image and (B) the corresponding power law exponent Embedded Image versus Embedded Image for β-YbAl1–xFexB4 with various x (Fe) at ambient pressure. The inset indicates the Fe doping dependence of the residual resistivity Embedded Image. (C) T dependence of the DC susceptibility M/H (right axis) measured in both zero-field-cooling (ZFC) and field-cooling (FC) sequences under a field of 0.1 T parallel and perpendicular to the ab plane and the magnetic part of the zero-field specific heat Embedded Image (left axis) obtained for β-YbAl1–xFexB4 (Embedded Image) at ambient pressure (26). The in-plane susceptibility for β-LuAl1–xFexB4 (Embedded Image) is also shown.

Moreover, a 6% substitution of Fe contracts the volume by 0.6(2)% and induces antiferromagnetism (Fig. 3C and table S1) (24, 26). The susceptibility Embedded Image shows a kink at 9 K and a weak hysteresis between field-cooled and zero-field-cooled sequences, typically a signature of canted AF (26). The specific heat Embedded Image confirms the bulk nature of the magnetism, showing an anomaly at 8.5 K. By contrast, the Lu nonmagnetic analog, β-LuAl1–xFexB4, exhibits diamagnetism. Thus, the magnetism derives from the Yb rather than the Fe sites.

Application of pressure to the 6% Fe–substituted β-YbAlB4 systematically increases the Néel temperature Embedded Image up to 25 K at Embedded Image GPa (Fig. 4 and fig. S7). For x = 2% Fe substitution, pressure also induces magnetism at a critical pressure Embedded Image GPa, a lower value than in the undoped crystals (2.5 GPa) (24). Figure 4 summarizes the combined data in a single-phase diagram spanned by pressure (P), Fe concentration (x), and temperature (T) axes. The smooth evolution of Embedded Image as a function of pressure and doping suggests that the pressure-induced phase in pure β-YbAlB4 involves the same type of AF order found in the Fe-doped β-YbAlB4.

Fig. 4 Three-dimensional phase diagram of emergent electronic phases versus pressure P, Fe concentration x, and temperature T for β-YbAl1–xFexB4.

Embedded Image and Embedded Image, respectively, denote the superconducting transition temperature and the onset of Fermi liquid Embedded Image dependence of the in-plane resistivity Embedded Image. The P dependence of the Néel point Embedded Image obtained for three different samples with x(Fe) = 0, 0.02, and 0.06 is shown (24). For clarity, the values of Embedded Image and Embedded Image are multiplied by a factor of 10. The regions connecting the (non-)Fermi liquid regions in Embedded Image-Embedded Image and Embedded Image(Fe)-Embedded Image phase diagrams are schematically shown in blue (yellow). Solid and broken lines are guides to the eye.

Conventionally, quantum criticality develops at a zero temperature phase transition into a broken symmetry state. In β-YbAlB4, however, we find an intermediate FL phase nestled between the NFL region and the AF phase, showing that the NFL is not associated with the broken symmetry phase transition. This indicates that the origin of the low-pressure quantum criticality is a different kind of electronic instability. Notable possibilities include topological phase transitions and a quantum valence transition (10, 13, 19, 24, 2732).

Various experiments can be used to differentiate between these scenarios. Thermodynamic measurements such as magnetization and Grüneisen parameter (33) are important to confirm the strange metal phase and its quantum phase transition to the FL phase under pressure. In particular, it would be useful to know if the T/B scaling observed in the thermodynamics of β-YbAlB4 at ambient pressure extends throughout the region of criticality; this would indicate that the observed behavior is associated with a critical line, forming a branch cut in the pressure-field phase diagram. Finally, it would be useful to probe the 4f valence by x-ray and Mössbauer spectroscopy to examine how the average and fluctuating valence of the 4f state evolve in the critical pressure region.

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S9

Table S1

References (3440)

References and Notes

  1. Supporting materials are available on Science Online.
  2. Acknowledgments: We thank Y. Shimura, H. Takahashi, G. G. Lonzarich, Y. Matsumoto, E. C. T. O’Farrell, K. Matsubayashi, N. Horie, and K. Ueda for support and useful discussions. This work is partially supported by grants-in-aid (nos. 25707030 and 24740243) and the Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers (no. R2604) from the Japanese Society for the Promotion of Science; by PRESTO, Japan Science and Technology Agency (JST); by Grant for Basic Science Research Projects from the Sumitomo Foundation; by U.S. National Science Foundation grant DMR-1309929 (P.C.) and U.S. National Science Foundation I2CAM International Materials Institute Award, grant DMR-0844115 (P.C. and S.N.); and by NSF grant no. PHYS-1066293 and the hospitality of the Aspen Center for Physics (P.C. and S.N.). The use of the facilities of the Materials Design and Characterization Laboratory at the Institute for Solid State Physics, The University of Tokyo, is gratefully acknowledged.
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