A challenge for the standard model known as Fermi liquid theory is anomalous (non-Fermi-liquid) properties of metals and still rare but increasing number of unconventional superconductivity, both of which are often found in the vicinity of the quantum critical point, where a magnetic ordering temperature is driven to absolute zero by tuning a physical control parameter. These nontrivial collective phenomena are subtle and highly sensitive to sample purity and homogeneity of the environment, and thus the study of highly clean system under ambient conditions is ideal. For the study of such exotic phenomena, heavy-electron metals of 4f intermetallic compounds have provided prototype materials. An important issue concerning 4f intermetallics is the possibility of the parallelism between the physical properties of electron-like (4f 1 Ce) and hole-like (4f 13 Yb) compounds. The study of Ce and Yb materials indeed finds many similarities deriving from the antiferromagnetic coupling between the 4f magnetic moments and the conduction electrons (the Kondo screening) which leads to highly enhanced electronic effective masses at low temperatures. There has been however, a remarkable difference among their superconducting properties. While a number of Ce-based unconventional superconductors is found in the vicinity of a quantum critical point, in Yb-based heavy fermion compounds superconductivity has never been reported.
Recently, we have succeeded in synthesizing a new structural form of YbAlB4,
β-YbAlB4 [1.1]. Our detailed work on the low temperature transport and thermodynamic
β-YbAlB4 has revealed that this material is a rare example of a highly clean metal that displays quantum critical behavior at ambient pressure and under no applied magnetic fields [1.2]. Furthermore, we have discovered superconductivity in the quantum critical regime characterized by pronounced non-Fermi-liquid response [1.2,1.3].
The superconductivity is observed below Tc =80 mK (Fig. 1.1). Interestingly, the system exhibits pronounced non-Fermi-liquid behavior above Tc, which is characterized by T 1.5 dependence of the resistivity (Fig. 1.1), T -1/3 dependence of the susceptibility and logarithmic divergent electronic specific heat coefficient [1.2]. Furthermore, the magnetic field dependence of the non-Fermi-liquid behavior in Fig. 1-1 indicates that β-YbAlB4 is a rare example of a pure metal that is quantum critical without tuning any physical parameter, i.e. without doping, applied pressure and magnetic field [1.2]. Our study using high-purity single crystals indicates that the superconductivity is in the clean limit and most likely involves heavy quasi-particles [1.3]. Upper critical fields are anisotropic, and strongly suppressed for the field along the c-axis, possibly because of the paramagnetic effect due to the divergent c-axis susceptibility (Fig. 1.2). Strong sensitivity of Tc to sample purity suggests that the superconductivity is of an unconventional, non-s-wave type (Fig. 1.1 inset).
Fig. 1.1: Contour plot of the resistivity exponent α defined by Δ ρ= (ρ(T )-ρ(0)) ∼ T α in the temperature-field phase diagram [1.2]. Inset: Zero-field in-plane resistivity ρ ab vs. T for various single crystals with different qualities, showing high sensitivity of Tc to sample purity. This indicates strong pair-breaking effects due to impurities, probably of nonmagnetic type, and suggests an unconventional character of the superconductivity [1.3].
[1.1] Robin T. Macaluso, Satoru Nakatsuji, Kentaro Kuga, Evan Lyle Thomas, Yo Machida, Yoshiteru Maeno, Zachary Fisk, and Julia Y. Chan, Chem. Mater. 19, 1918 (2007).Fig. 1.2: Temperature dependence of the upper critical field Bc2 along the ab-plane (square, blue) and the c-axis (circle, red)[1.3]. Different single crystals with similar Tc were used for each field direction measurement. Broken lines represent the curves obtained by fitting to the experimental results near Tc using the WHH model. Inset, the crystallographic unit cell of b-YbAlB4 [1.1].
[1.2] S. Nakatsuji, K.Kuga, Y. Machida, T. Tayama, T. Sakakibara, Y. Karaki, H. Ishimoto, S. Yonezawa, Y. Maeno, E. Pearson, G. G. Lonzarich, L.Balicas, H. Lee, and Z. Fisk, Nature Phys. 4, 603-607 (2008).
[1.3] K. Kuga, Y. Karaki, Y. Matsumoto, Y. Machida, and S. Nakatsuji, Phys. Rev. Lett. 101, 137004 (2008)
Singularities where the smoothness of physical laws breaks down and is replaced by mathematical infinities are studied extensively in physics as sources of fascinating new forms of behavior. For instance, lightning bolts, tornadoes and black holes are show-cases of singularities in physics. In the past few decades, it has been recognized that similar singularities also develop inside "ordinary" materials at low temperatures, and when they do, a new kind of emergent phenomena appears. In particular, the electron fluid that carries electricity in metals becomes unstable. For the past 80 years, Fermi liquid theory, which is the idealized picture of electron liquid, has provided a mainstay of our understanding of metals. However, in β-YbAlB4, a new ytterbium-based material, this picture breaks down, revealing an underlying singularity at zero temperature i.e. quantum critical point (QCP) [2.1,2.3].
In order to study a singularity in a material (QCP), a tuning of a physical parameter such as temperature, magnetic field, pressure, and doping is necessary to make the material approaches to QCP in the phase space. This can be compared to traveling around space and time searching black holes. However, like the black hole forms around a singularity masking the mathematical singularity at its center in the fabric of space and time, the breakdown of the Fermi liquid picture in β-YbAlB4 is masked by superconductivity which prohibits any direct measurement of the underlying metallic state [2.2,2.3]. Moreover, the singularities have often been obscured by disorder. For all these difficulties, we have succeeded in probing deep inside the singularity using the ultrapure form of the crystals and high fidelity magnetization measurements [2.3]. As a result, we have found that the magnetization M satisfies a scaling equation -dM/dT = B-1/2f(T/B) over a wide range of temperature and field at T ≲ 3 K and B ≲ 2 T. In other words, this means that the singularity lets its presence be known by affecting the material properties at elevated temperatures and magnetic fields, several orders of magnitude larger than the boundaries of the superconducting domain. It bears resemblance to a black hole, which cannot be probed directly, but whose gravitational pull is felt by the surrounding stars, long distances away. The T/B scaling proves not only unconventional quantum criticality but, furthermore, that the singularity in β-YbAlB4 occurs just at zero magnetic field with an experimental error comparable to Earth’s magnetic field at ambient pressure.
This is quite surprising compared to canonical QCP materials which require a tuning of a control parameter to approach QCP. This raises an intriguing possibility that this metal may be part of a new, quantum critical, phase of matter i.e. the spontaneous quantum criticality in β-YbAlB4 would persist in a finite region of the phase-space, as a function of pressure or doping [2.3]. This strange state, which displays an intriguing correspondence of self-similarity under applied magnetic field and temperature, can be viewed as the third state of the matter in addition to the Fermi liquid and a superconductor. It is expected that this observation opens new horizons in our understanding of quantum criticality and provides us with important information which can help shed light on the strange metal state and superconductivity not only in this material, but also in other families of unconventional superconductors.
Fig. 2.1: A) Crystal structures of β-YbAlB4. B) A picture of single crystals of β-YbAlB4.
[2.1] S. Nakatsuji, K.Kuga, Y. Machida, T. Tayama, T. Sakakibara, Y. Karaki,H. Ishimoto, S. Yonezawa, Y. Maeno, E. Pearson, G. G. Lonzarich, L.Balicas,H. Lee, and Z. Fisk, Nature Phys. 4, 603-607 (2008).Fig. 2.2: Scaling observed for the magnetization at T ≲ 3 K and B ≲ 2 T. Here, the magnetization M satisfies a scaling equation -dM/dT = B-1/2f(T/B) over a wide range of temperature and field shown in the inset. This means that the physical properties around QCP are determined only by the ratio T/B. In addition this proves that the QCP is located just at zero-magnetic field. Inset shows 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 ~ gmBB line (broken line). The open circles are the temperature scale TFL, below which the T2 dependence of the resistivity is observed [2.1].
[2.2] K. Kuga, Y. Karaki, Y. Matsumoto, Y. Machida, and S. Nakatsuji, Phys.Rev. Lett. 101, 137004 (2008)
[2.3] Yosuke Matsumoto, Satoru Nakatsuji, Kentaro Kuga, Yoshitomo Karaki, Naoki Horie, Yasuyuki Shimura, Toshiro Sakakibara, Andriy H. Nevidomskyy, Piers Coleman, Science 331, 316 (2011).
4f -based heavy-fermion (HF) systems have attracted much attention with interesting phenomena such as unconventional superconductivity and non-Fermi-liquid (NFL) behavior found in the vicinity of quantum critical points. Our recent studies have found the first Yb- (4f 13) based HF superconductivity with the transition temperature Tc = 80 mK in the compound β-YbAlB4 [3.1, 3.2]. Pronounced NFL behavior above Tc and its magnetic field dependence indicate that the system is a rare example of a pure metal that displays quantum criticality at ambient pressure and close to zero magnetic field [3.1]. Furthermore, the T/B scaling found in our recent high- precision magnetization measurements clarifies its unconventional zero-field quantum criticality without tuning [3.3], which cannot be explained by the standard theory based on spin-density-wave fluctuations. In contrast to the canonical quantum critical materials, hard x-ray photoemission spectroscopy (HXPES) measurements have revealed a strongly intermediate valence of Yb+2.75 [3.4], providing an example of quantum criticality in a mixed- valence system. Whether the valence fluctuation is relevant for the mechanism of quantum criticality and superconductivity is an interesting open question.
Here, we measured the specific heat, magnetization, and resistivity of α-YbAlB4 down to very low temperature [3.5]. This compound is the locally isostructural polymorph of β-YbAlB4 with a different arrangement of distorted hexagons made of Yb atoms [space groups Pbam(α-YbAlB4) and Cmmm(β-YbAlB4), see Fig. 1.2]. According to the HXPES measurement [3.4], α-YbAlB4 also has an intermediate valence of Yb+2.73. The results indicate a Fermi-liquid (FL) ground state for α-YbAlB4 in contrast to the unconventional quantum criticality observed in β-YbAlB4. Interestingly, both systems exhibit Kondo lattice behavior with a small renormalized temperature scale of T* 8 K, although both of them have a large valence-fluctuation scale of 200 K. Below T*, α-YbAlB4 forms a heavy-Fermi-liquid state with an electronic specific heat coefficient γ 130 mJ/mol K2 and a large Wilson ratio greater than 7, which indicates a ferromagnetic correlation between Yb moments. A Kadowaki- Woods ratio is found that is similar to those found in the normal Kondo lattice systems and considerably larger than mixed valence systems. Furthermore, the resistivity of α-YbAlB4 exhibits one of the strongest anisotropies in heavy fermions (Fig. 3.1). The ratio between in plane and c-axis resistivity ρab and ρc, ρab/ ρc, reaches 11 at low temperatures below T*. This strongly suggests anisotropic hybridization between 4f and conduction electrons which is stronger in the ab-plane. This is the key to understanding the mechanism of heavy- fermion formation as well as the Kondo lattice behavior found in the intermediate-valence system. Thus the system should be one of the best systems to study for elucidating the effects of anisotropic hybridization.
[3.1] Robin T. Macaluso, Satoru Nakatsuji, Kentaro Kuga, Evan Lyle Thomas,Yo Machida, Yoshiteru Maeno, Zachary Fisk, and Julia Y. Chan, Chem. Mater.19, 1918 (2007).Fig 3.1. Temperature dependence of the in-plane and c-axis resistivity ρab and ρc of α-YbAlB4 and ρab of β- YbAlB4. The magnetic part of the resistivity ρm is obtained by subtracting the nonmagnetic contribution estimated by ρab of α- and β-LuAlB4 (solid and dash-dotted lines, respectively) or ρc of α-LuAlB4 (dashed line). The inset shows the temperature dependence of the ratios ρab/ρc for α-YbAlB4 and α-LuAlB4.
[3.2] S. Nakatsuji, K.Kuga, Y. Machida, T. Tayama, T. Sakakibara, Y. Karaki,H. Ishimoto, S. Yonezawa, Y. Maeno, E. Pearson, G. G. Lonzarich, L.Balicas,H. Lee, and Z. Fisk, Nature Phys. 4, 603-607 (2008).
[3.3] K. Kuga, Y. Karaki, Y. Matsumoto, Y. Machida, and S. Nakatsuji, Phys.Rev. Lett. 101, 137004 (2008
[3.4] M. Okawa et al. Strong Valence Fluctuation in the Quantum Critical Heavy Fermion Superconductor β-YbAlB4: A Hard X-Ray Photoemission Study. Phys. Rev. Lett. 104, 247201, (2010).
[3.5]Yosuke Matsumoto, Kentaro Kuga, Takahiro Tomita, Yoshitomo Karaki, and Satoru Nakatsuji, Physical Review B 84, 125126 (2011).
Several recent studies of the two polymorphs α and β-YbAlB4 by the Nakatsuji Group and collaborators have revealed the remarkable properties of these two closely related materials. However, within a conventional understanding of f-electron intermetallics some of these experimental observations are seemingly contradictory to one another; but now, a study of the Hall effect of β-YbAlB4 by O’Farrell et al. [4.1], published in Physical Review Letters, sheds new light on the evolution of the electronic structure of β-YbAlB4 and suggests how these behaviors can coexist.
Following the preparation of high quality crystals of β-YbAlB4 in the Nakatsuji group several remarkable behaviors have been reported, as we now describe. β-YbAlB4 is the first Yb-based material in which highly renormalized electronic quasiparticles (with effective masses ＞ 100 times larger than the bare electronic mass) superconduct [4.2]. β-YbAlB4 was also found to have a quantum critical point at exactly zero magnetic field and zero pressure [4.3], suggesting the vanishing of an energy scale associated with an ordered electronic state, such as magnetism. In apparent contradiction to these low temperature properties, the Yb f-moment in β-YbAlB4 shows large valence fluctuations [4.4] (the Yb valence is +2.75) and has a high Kondo temperature (TK = 200 K) [4.2]; behaviors that usually lead to a Fermi liquid ground state with weakly renormalized quasiparticles.
The Hall effect measurements by O’Farrell, Matsumoto and Nakatsuji found that the Hall coefficient has strong temperature dependence and a minimum at T = 40 K (see Fig. 4.1), that bears a close similarity to Kondo rather than mixed valence systems. The usual interpretation of this result would put the Kondo temperature at 40 K rather than 200 K, as was measured by the longitudinal resistivity. To answer the question of how β-YbAlB4 can appear to have two Kondo temperatures separated by almost an order of magnitude the authors suggested a two component Hall effect, which was supported by the magnetic field dependence of the Hall resistivity.
Fig 4.1. The Hall coefficient (RH) of β-YbAlB4 vs temperature (T) for several different quality samples compared to a mixed valent compound YbAl3. β-YbAlB4 shows a strong temperature dependence characterisitic for materials with incoherent skew-scattering from localized moments as expected when the Kondo interaction dominates.
As the temperature is lowered below 100 K the Hall resistivity becomes non-linear; a careful analysis showed that the field dependence can be explained by the material having two field independent Hall coefficients, the combination of which leads to a non-linear Hall resistivity. Furthermore, this analysis allowed the authors to show that the mobility of the material strongly increases approaching the minimum at 40 K; thus demonstrating that the minimum in the Hall coefficient is indeed due to the onset of coherent transport in a second component of the electronic transport that has just 10% of the total carrier density.
The reason that these two components have such different Kondo temperatures is suggested by the Fermi surface of β-YbAlB4 together with a recent theoretical analysis showing that the hybridization between conduction electrons and f-moments may vanish at certain points in momentum space [4.5] (See Fig. 4.2). These nodal points lead to a large difference in hybridization strength and thereby a different Kondo temperature between the two Fermi surfaces, one of which passes close to the nodal region and the other of which is well separated.
Fig 4-2. The tight binding Fermi Surface of β-YbAlB4 calculated by Ramires et al. [4.5]. The red line indicates points in momentum space where the hybridization vanishes. The experimental Fermi surface of β-YbAlB4 has a second sheet that is well separated from the region of vanishing hybridization and therefore weakly affected by this node.
These results therefore suggest that the emergent second component arises from the Fermi surface that lies close to the region of vanishing hybridization that may be responsible for the quantum critical and superconducting behavior observed at low temperature. This scenario is consistent with a recent theoretical work by Ramires et al. that provides a phenomenological model for quantum criticality in β-YbAlB4 [4.5].[4.1] E. C. T. O’Farrell, Y. Matsumoto, and S. Nakatsuji. Evolution of c-f¬ hybridization and two component Hall effect in β-YbAlB4. Phys. Rev. Lett. 109, 176405 (2012).
[4.2] S. Nakatsuji, K.Kuga, Y. Machida, T. Tayama, T. Sakakibara, Y. Karaki,H. Ishimoto, S. Yonezawa, Y. Maeno, E. Pearson, G. G. Lonzarich, L.Balicas,H. Lee, and Z. Fisk, Nature Phys. 4, 603-607 (2008).
[4.3]Yosuke Matsumoto, Kentaro Kuga, Takahiro Tomita, Yoshitomo Karaki, and Satoru Nakatsuji, Physical Review B 84, 125126 (2011).
[4.4] M. Okawa et al. Strong Valence Fluctuation in the Quantum CriticalHeavy Fermion Superconductor β-YbAlB4: A Hard X-Ray Photoemission Study.Phys. Rev. Lett. 104, 247201, (2010).
[4.5] A Ramires et al. β-YbAlB4: A Critical Nodal Metal. Phys. Rev. Lett. 109, 176404, (2012).
In condensed matter physics, it is highly important subject to find a novel state of matter. Through many extensive work performed on strongly correlated electron systems including heavy electron metals, high-Tc superconductors and organic conductors, it has been one of the most intriguing and significant possibility that there should be another class of a metallic phase that cannot be described by the standard frame work of metal, namely, Fermi liquid theory. It has been often called as strange metal phase. However, its strong sensitivity to impurity has made it hard to distinguish it from quantum criticality associated with a singular zero temperature point, namely, quantum critical point.
Nakatsuji and Uwatoko groups at ISSP University of Tokyo and a theoretical group at Rutgers University, USA, have studied the pressure effect on heavy fermion system β-YbAlB4 and found the first evidence of a strange metal phase over an extensive region of pressure by suppressing superconductivity. Just as the melting of ice involves a transition from solid to liquid state, strongly correlated materials exhibit transition between magnetic ordered state and a Fermi liquid state. Among the strongly correlated electron systems, the heavy fermion metals often offer the convenient cases where the transition temperature can be easily tuned, for example, by varying a physical parameter, such as magnetic field, external pressure and chemical composition. In particular, the parameter where the transition temperature is suppressed to absolute zero temperature (minus 273.15 °C) is called quantum critical point (QCP). Nearby the quantum critical point, at low temperatures, a strange metal state is believed to emerge due to quantum critical fluctuations associated with the instability between the magnetic ordered state and Fermi liquid state (Figure 5.1(a)). As the quantum critical point is very unstable, the quantum critical phenomena such as the non-Fermi liquid state and superconductivity appear only nearby quantum critical point (conventional QCP in Fig. 5.1).
Here, we show our experimental study on an ultrapure single crystal of the Yb-based heavy fermion compound β-YbAlB4. β-YbAlB4 is known to have a zero magnetic field and zero pressure quantum critical point and exhibits pronounced strange metal state and superconductivity at low temperatures [5.1-5.3]. We applied high pressures and performed high resolution measurements of electric resistivity at low temperatures to investigate its electrical properties. A small magnetic field was used to suppress the superconductivity to reveal the strange metal state. Eventually, we succeeded in elucidating a number of surprising phenomena (Figure 5.1 (b)) [5.4]:
1) In the extensive pressure regime (0 < P < 0.4 GPa), a strange-metal region with non-Fermi liquid properties exists beneath the superconducting dome.
2) Both the strange metal phase as well as the superconducting (SC) phase (0 < P < 1 GPa) are isolated from the border of magnetism ( P > 2.5 GPa).
3) The strange metal phase can not be described by the standard theory of metals based on spin fluctuations. A new type of mechanism is expected, and is most likely associated with valence fluctuations.
These discoveries imply the existence of a strange metallic phase (Figure 5.1 (b)), which should be well distinguished from the quantum critical states associated with a magnetic quantum critical point (conventional QCP in Figure 5.1 (a)). This insight may be useful for clarifying the mechanism for the strange metal phase. It may also contribute to understanding the superconducting mechanisms associated with other classes of strongly correlated materials such as Cu-based and Fe-based high-temperature superconductors.
Fig. 5.1: Conventional quantum critical point and new types of strange metal phase realized in β-YbAlB4
[5.1] S. Nakatsuji, K. Kuga, Y. Machida, T. Tayama, T. Sakakibara, Y. Karaki, H. Ishimoto, S. Yonezawa, Y. Maeno, E. Pearson, G. G. Lonzarich, L. Balicas, H. Lee, and Z. Fisk, Nature Physics 4, 603 (2008).
[5.2] K. Kuga, Y. Karaki, Y. Matsumoto, Y. Machida, and S. Nakatsuji, Physical Review Letters 101, 137004 (2008).
[5.3] Yosuke Matsumoto, Satoru Nakatsuji, Kentaro Kuga, Yoshitomo Karaki, Naoki Horie, Yasuyuki Shimura, Toshiro Sakakibara, Andriy H. Nevidomskyy, Piers Coleman, Science 331, 316 (2011).
[5.4] Takahiro Tomita, Kentaro Kuga, Yoshiya Uwatoko, Piers Coleman, and Satoru Nakatsuji, Science 349, 506–509 (2015)