Quantum critical behaviour and Lifshitz transition in intermediate valence -YbAlB
Intermetallic compounds containing -electron elements have been prototypical materials for investigating strong electron correlations and quantum criticality (QC). Their heavy fermion ground state evoked by the magnetic -electrons is susceptible to the onset of quantum phases, such as magnetism or superconductivity, due to the enhanced effective mass () and a corresponding decrease of the Fermi temperature. However, the presence of -electron valence fluctuations to a non-magnetic state is regarded an anathema to QC, as it usually generates a paramagnetic Fermi-liquid state with quasiparticles of moderate . Such systems are typically isotropic, with a characteristic energy scale of the order of hundreds of kelvins that require large magnetic fields or pressures to promote a valence or magnetic instability. Here we show that the intermediate valence compound -YbAlB surprisingly exhibits both quantum critical behaviour and a Lifshitz transition under low magnetic field, which is attributed to the anisotropy of the hybridization between the conduction and localized -electrons. These findings suggest a new route to bypass the large valence energy scale in developing the QC.
A quantum critical point (QCP) occurs when the ground state of a system is continuously tuned between two states1, 2. The strong incipient quantum fluctuations modify the system’s electronic state over large regions of its phase diagram. This has led to the notion that understanding quantum criticality (QC) is the key to understanding emergent phases in materials such as spin liquids and the high temperature superconductors.
Heavy fermion (HF) materials, often -electron based intermetallics containing Ce or Yb, have been prototypical for the investigation of QC1, 2, 3, 4: the enhanced entropy of the Fermi sea makes them susceptible to low temperature instabilities such as magnetism and superconductivity. The energy differences between these possible ground states are small, and therefore can typically be tuned by applying small magnetic fields or pressures.
The conventional paradigm of QC in HF materials, the Doniach phase diagram and its recent extensions5, requires a stable valence of the magnetic ion, which for Yb is the state. Yb intermetallics with fluctuating valence, such as YbAl or YbAgCu, have a large valence fluctuation energy scale making new phases difficult to achieve; YbAl is not known to order magnetically at all 6, while YbAgCu requires extremely high pressure7. The recently synthesized YbAlB is therefore remarkable because QC coexists with intermediate valence, and for and polymorphs 8, respectively. In this article we focus on -YbAlB and show the presence of QC that is tuned with small magnetic fields.
The two polymorphs of YbAlB are locally isostructural, with a highly anisotropic structure where sheets of boron separate the layers containing Yb and Al. The local structure is shown in Fig. 1a) illustrating the anisotropy, atypical for intermediate valence (IV) systems, and the slightly lowered symmetry of -YbAlB. Both materials have a valence fluctuation scale K 8, 9 and only a minor change of valence 10 in magnetic fields up to 40 T.
Ii Summary of main results
We use a complete set of thermodynamic, magnetotransport and microscopic experimental techniques to probe the electronic anisotropy as -YbAlB is driven toward two electronic instabilities with a magnetic field that is small compared to K. Most prominently, we find that the sign of the thermal expansion, which directly probes the pressure dependence of the entropy 11, changes at T due to a change in the nature of the relevant fluctuation scale from magnetic correlations at lower fields, to the Kondo or valence correlations at higher fields; a signature of the proximity to a QCP. At a slightly lower field T, the Shubnikov–de Haas measurements show the appearance of a new, strikingly anisotropic Fermi surface (FS), indicating a Lifshitz transition. We refer to these fields as = 2.1 T and = 3.6 T in what follows. At both of these fields nuclear magnetic resonance (NMR) measurements find a diverging spin lattice relaxation rate of B nuclei down to 50 mK. The magnetostriction, i.e. the rate of change of the lattice constant with magnetic field, is highly anisotropic: for the -axis the magnetostriction is maximum at , while for the -axis it is maximum at . The resistivity shows non-FL (NFL) behaviour at and but only for current applied parallel to the -axis. The striking two dimensionality (2D) of the FS that appears at is consistent with it being a remnant of the Lifshitz transition proposed to drive criticality in -YbAlB12, while the change in energy scales at indicates the proximity to a QCP. We provide a tentative theoretical explanation for this anisotropy in terms of the momentum-dependent nature of the Kondo hybridization, and resolve the mismatch between the large intermediate valence energy scale and the much smaller energy associated with fields and .
By combining these complementary experimental techniques together with theoretical arguments we establish a complete picture of the system and find that the hybridization anisotropy provides a means to overcome the large valence energy scale in the YbAlB system.
We first describe the magnetic field dependence of the thermal expansion , where is the sample length along the -axis, shown in Fig. 1b. At and low temperatures ( ) the linear thermal expansion coefficient is positive for both the and axes, and thus the volume coefficient .This is surprising since in Yb-based Kondo-lattice (KL) or IV systems usually, as is magnetostriction (see Supplementary Note 1 for details). However, measures the pressure dependence of the entropy, which in -YbAlB implies that the dominant contribution does not arise from KL or IV type fluctuations, but from an energy scale that increases with pressure. In magnetic Yb-based systems this is usually (anti-)ferromagnetic (A)FM order mediated by the RKKY interaction 13, 14. The high Wilson ratio () and indicates that FM correlations dominate the ground state. In fact, it has been found that pressure induces an AFM state in -YbAlB through a first-order phase transition at 2.5 GPa 15, 16, and an AFM state emerges in -YbAlB by Fe doping17, 18 at 1.5%. Although the AFM critical field of the Fe-doped system 18 is close to , the features observed at in pure -YbAlB have a different origin and cannot be related to the AFM-to-PM field-induced QCP. We argue this by further discussing our results.
Under magnetic field the magnetic correlations are suppressed and the sign of changes between 3.3 T and 4 T, indicating a change of the relevant energy scale as expected at a crossover or a phase transition between an ordered and a disordered phase. At a QCP it is expected that becomes divergent and changes sign 11. However, in -YbAlB this change is rather smooth and asymmetric, similar to the one found in YbAgGe 19, and the system remains paramagnetic (PM) on both sides. This behaviour is typical of metamagnetic materials, like SrRuO 20 or CeRuSi 21 where the entropy is dominated by magnetic fluctuations (see Supplementary Note 1 for details). The relatively smooth change in suggests that -YbAlB is located either close to a quantum critical endpoint, or in the proximity of a field-induced QCP.
Within both scenarios a clear anomaly in magnetic susceptibility and specific heat coefficient is expected. The magnetization and measured with are shown in Fig. 1 c, with displaying a clear enhancement at before decreasing rapidly at higher magnetic field. Compared with true metamagnetic materials the enhancement is weak and is not symmetric around . The same features are seen in the specific heat coefficient and in the NMR Knight shift measured at the nucleus, as shown in Fig. 1d. Since all three quantities are governed by the Fermi surface properties: , with the density of states at the Fermi level , it is clear that the anomaly at involves a continuous reduction of as a consequence of the suppression of the correlations by the field. This is different from a possible suppression of the HF state by magnetic field through a metamagnetic transition. In fact, when compared to HF metamagnetic compounds 22 the value of is not large enough. In -YbAlB the relevant magnetic field scale, estimated from mJ/molK, should exceed 20 T, and hence metamagnetism cannot account for the observed phenomena.
A rigorous test of the ground state properties is the character of underlying excitations. To probe them we have measured the NMR relaxation rate (see Methods) of the nuclei (plotted in Figs. 2a,b and c) as a function of temperature at various magnetic fields for . The data are shown as so that they can be easily contrasted to = const. that is typically observed for simple metals at low temperature. Below K and below 1.5 T, where the coherent FL state is formed, is Ks. This is two orders of magnitude lower than what is typical with dominant magnetic correlations 23, 22, 24, which shows that the system is not close to magnetism. As the magnetic field approaches , we surprisingly find a pronounced divergence at low temperatures with exponent . This power-law behaviour persists down to 50 mK, two orders of magnitude lower than , and reveals the presence of quantum critical fluctuations that destabilize the FL and lead to a NFL ground state. The exponent show that this is not a magnetic QCP - it is too low for the divergence to appear in alone, which would be usual for a magnetic QCP. Also, across the NMR spectrum remains unchanged (Supplementary Fig. 1) which would split if magnetic or charge order appeared at , due to symmetry breaking.
We have also analysed the Knight shift behaviour , where is the hyperfine magnetic field at the nucleus site. While is sensitive to spin excitations at , the Knight shift is related to the static spin susceptibility , and in a FL it is expected to be constant against temperature and magnetic field. As seen in Fig. 1d, for the -axis Knight shifts of B () and Al () nuclei (Supplementary Fig. 2) are constant, and at show a weak feature (similar to ) confirming the absence of magnetic mode at and . Across 4 T, both and show only a slow change arising from , which is also supported by the measurements of the quadupolar coupling of Al and B (Supplementary Figs. 3 and 4).
To summarize, from the presented data we can deduce that at the underlying ground state is NFL, no magnetism emerges, the critical spin fluctuations originate at and there is a change in the static charge density distribution.
So far, we have focused on the QC at T visible in thermodynamic and NMR measurements. In addition, the NMR measurements show the existence of another unusual behaviour that suddenly appears at a lower field T. Indeed, the magnetic field dependence of the NMR relaxation rate at 142 mK shows two pronounced peaks (Fig. 2a) at and . The temperature dependence reveals another NFL power-law divergence at , , with similar to the behaviour at . Between and the system behaves as a standard metal, down to lowest temperatures, which indicates are separate phenomena. Their qualitatively different nature is evidenced by the critical dynamics at that, unlike the crossover at , shows no distinct signature in susceptibility or specific heat despite clear evidence of divergent spin fluctuations from the NMR.
Further indications about the nature of the transition at are given by the magnetostriction coefficient measured along both the -axis () and -axis () but with magnetic field , shown in Fig. 2d. , which is sensitive to structural, magnetic and electronic structure transitions, here shows a strong anisotropy: shows a maximum at as expected from , while shows a maximum at the smaller field implying that it is related to a 2D effect within the -plane.
Anisotropy in the critical behaviour of -YbAlB is also evidenced by resistivity measurements along the and axes, and , respectively. Low temperature NFL behaviour due to QC is quantified by observing the exponent and the coefficient as , (see Methods). At K, shows an enhancement (Fig. 2c) by at and before decreasing at higher fields, and the exponent shows deviations (Fig. 2c) from the FL value of 2 in the vicinity of and – at the lowest measured temperature K. This is not observed18 in the Fe-doped -YbAlB at 3.5 T, where the exponent retains the FL value of 2, and is therefore different from the pristine sample. The value in the pristine -YbAlB is close to that expected at a FM or AFM QCP at which and 3/2, respectively 25. For instance, -YbAlB shows at its critical point consistent with the AFM QCP. Importantly, the NFL behaviour in -YbAlB occurs only for , whereas remains for (Supplementary Fig. 5). This shows that quantum critical fluctuations do not originate from the suppression of the AFM phase in Fe-doped case, and that they manifest themselves in specific regions of momentum space, which confirms the conclusion derived from anisotropic magnetostriction and the NMR data. Intrinsic electronic anisotropy is therefore an important factor for understanding this system.
We now consider the changes in the electronic structure in the vicinity of and , by measuring quantum oscillations (QO). Figure 3(a) shows as a function of the magnetic field, from which we separate the slow-varying () and the oscillatory () component (see Methods). A clear kink in (Fig. 3(b)) is observed at below which oscillations appear. The logarithmic scale is used to show that the amplitude decays linearly as expected from the Dingle relation. The amplitude is also shown against temperature in Fig. 3(c) together with a fit to the Lifshitz-Kosevich (LK) relation, that describes the decay of QOs with and gives . The excellent agreement to the LK relation confirms these are indeed QOs with frequency T, while their sudden appearance at suggests they emerge as the result of a Lifshitz transition at . This remarkable result makes -YbAlB the first IV compound with a Lifshitz transition induced by such a low magnetic field.
By rotating the sample relative to we probe the extremal cross-section of this FS perpendicular to . Figure 3(d) shows against the -axis component of , i.e. where the is rotated in the plane of  and . The is observed to be constant with indicating the FS is cylindrical and arises from 2D carriers, the volume of this cylinder is very small, corresponding to carriers per Brillouin zone. The QOs of different spin components of a 2D FS are normally split by the Zeeman interaction and interfere to produce spin zeros where the QO amplitude vanishes and the phase shifts by . Using a factor of 2.3 obtained by electron spin resonance 26 and the determined value of , we would expect a strong angular dependence of the amplitude and a spin zero at (see Supplementary Note 3 for details). However, no spin zeros are found, suggesting that this pocket is spin polarized, consistent with a Zeeman-driven Lifshitz transition at . If this pocket is fully spin polarized and taking27 , the magnetic moment of this pocket is , which is consistent with the extremely small polarization increase observed in in Fig. 1c).
The 2D nature of the FS, the strong divergence in at the B site and the pronounced maximum in indicate that due to anisotropy the electronic properties within boron layers are most affected at . However, the small size of the FS pocket relative to the total number of carriers makes it difficult to account for the strong quantum critical fluctuations and suggests that larger sheets of the FS are affected. While large QO frequencies are observed at higher magnetic fields (see Supplementary Note 3 for details), these were difficult to observe at low magnetic fields of and , and we therefore turn to the Hall coefficient () shown in Fig. 3e. decreases smoothly above , which would within a single-band model indicate a change in the carrier density by a factor of 2. However, this is not supported by the small changes in , and therefore it could be interpreted as changes in the FS velocity on larger sheets of the FS. The data and the gradual change in confirm that -YbAlB is close to a QCP at the field .
Intermediate valence compounds have been mainly outside the main focus of research on QC since, due to large characteristic energy scales of the valence fluctuations ( K in YbAl), application of large magnetic fields or pressures is required to notably modify their properties or induce new phases 28. In contrast, for both YbAlB polymorphs magnetic moments survive well below K, and become strongly correlated below K, showing QC at low magnetic fields. Although the quantum critical behaviour in the two polymorphs has different phenomenology, the presence of highly anisotropic Kondo hybridization in both compounds appears crucial for establishing it. The conventional picture of IV materials contains the hybridization energy scale , where is Anderson’s coupling strength between conduction band and localized -electrons. In the strong coupling limit approaches . The NFL behaviour and Kondo hybridization can be connected 27 by appealing to the tensorial, momentum-dependent nature of the hybridization stemming from the dominant nature of Yb ground state doublet expressed in terms of the creation/annihilation operators of conduction () and localized () electrons. Subsequent theoretical work on -YbAlB showed 12 that the tensor is indeed highly anisotropic and vanishes upon approaching the line in the Brillouin zone as , which is also confirmed by recent ARPES 29 measurement. In the lower symmetry polymorph -YbAlB, this dispersion is not expected to persist, however, the observation of a Lifshitz transition of the strikingly 2D FS pocket and the deviation from FL behaviour suggest that is remnant of the QCP in -YbAlB. In this case is detuned from zero field by a non-zero value of the renormalized chemical potential : , whereas is believed to be zero 12 in -YbAlB. This is consistent with the experimental data in -YbAlB if is meV, such that the signature of QC appears at T, when the chemical potential reaches the bottom of the majority-spin band.
Unlike the transition at , the NFL behaviour at is much more a 3D phenomenon that drastically changes the global properties of the system. The FM fluctuations present in the ground state at zero field (shown by and high Wilson ratio) are suppressed as the magnetic field is increased to . Near , although FM () fluctuations are suppressed, finite-q fluctuations get enhanced, causing power-law divergence of and anomaly in resistivity, susceptibility and specific heat, which imply the proximity to a QCP unrelated to the Fe-doped case. At the same time, magnetostriction, QO and thermal expansion show that the transition is followed by a FS change. The QCP also differs from the one in -YbAlB, where the well understood FS of -YbAlB 30, 31, 12, 29 has no small pocket like the one observed to appear at in -YbAlB, and signatures of spin zeros were observed on larger FS sheets 32 which are absent in the -YbAlB.
Although in IV compounds magnetic field can induce a valence QCP 33, our measured data and the band structure analysis exclude this for the case of -YbAlB (see Supplementary Note 2 for details). Recently 34, Mössbauer spectroscopy found a change in the quadrupolar moment of Yb between 1 and 5 T. This implies the QC at originates from a multipole-type QCP in -YbAlB. However, due to the low symmetry of the Yb site, its understanding will require further measurements which are beyond the scope of the current work.
It is remarkable that critical fluctuations survive in - and -YbAlB, which are both IV materials – particularly since there has been only one reported quantum critical IV compound so far: CeRhSn 35. Although it is also anisotropic, there the 2D frustrations drive the QC. This is indicative of the fact that in YbAlB family a robust physical mechanism emerges in the presence of strong electron anisotropy, that is unaffected by the change of local symmetry. What makes YbAlB family special are the Yb chains interpenetrating 36 the sheets of B, which cause the anisotropic hybridization and allow for QC to emerge. The tensorial nature of the Kondo hybridization and its vanishing at certain high-symmetry points in the Brillouin zone is also a central thesis of the theory of topological Kondo insulators 37, and has shown promising results in a recent theoretical treatment of the quasi-1D Kondo lattice 38. However, a more detailed analysis is required to encompass the richness of the phase diagram of YbAlB family.
From all the data shown above we can conclude that anisotropy, along with magnetic frustration, offer a new route of overcoming the high energy scale of IV compounds resulting in quantum criticality and new phases of matter.
Materials and Methods
Single crystals of -YbAlB were grown from Al flux. The stoichiometric ratio of Yb:4B was heated in excess Al in an alumina crucible under an Ar atmosphere as described elsewhere 39. Chemical compositions of single crystals were determined by a inductively coupled plasma - atomic emission spectrometry (ICP-AES, HORIBA JY138KH ULTRACE) at ISSP, and the analysis of both polymorphs are in good agreement with the ideal compositions of YbAlB within the error bars. We analysed diffraction patterns to determine the crystal structure and lattice constant using the Rietveld
analysis program PDXL (Rigaku) and found no impurity phase. The thermal expansion and magnetostriction were measured with a high-resolution capacitive CuBe dilatometer in a dilution refrigerator 40. Specific heat was measured using relaxation calorimetry 41. Further detail and subtraction of nuclear contributions will be described elsewhere. Magnetization was measured using capacitance Faraday method 42.
Resistivity and Shubnikov-de Haas measurements were performed using conventional lock-in amplifier techniques. For measurements with pristine crystals were used, while for and for Hall effect measurements larger crystals were polished to form thin plates perpendicular to the -axis. Low temperature deviations from Fermi liquid (FL) behaviour due to quantum critical fluctuations are quantified by expressing . Taking we extracted . Similarly, we extracted the temperature exponent by assuming is constant in as .
For quantum oscillations the small number of oscillation periods made the extraction of the oscillatory component of the resistance challenging; we applied two methods that gave consistent results. We assume that where is assumed to be slowly varying. In the first method we apply a low pass filter in that subtracts a locally quadratic polynomial, this is described in detail elsewhere 43. In the second we first subtract a linear component from the entire field range and then take a derivative. A comparison between the Fourier spectrum of these methods is shown in the supporting information (Supplementary Note 3).
The NMR measurements of B were performed using a pulsed spectrometer. The spectra were collected by Hahn echo sequence with typical value of s, and a pulse of 6 s. The measurements were performed using a saturation-recovery technique on a satellite NMR line determined in previous work44. Data were fitted to a magnetic relaxation function of the form: . The sample was oriented in situ by observing the quadrupolar splitting of using a two-axis goniometer for measurement temperatures 1.5 - 300 K, and with a single axis goniometer for measurements at lower temperatures. In all cases the magnetic field orientation was within 2 of the crystal c axis.
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Acknowledgments We acknowledge the help of Naoki Horie. M.S.G. acknowledges the support of Croatian Science Foundation (HRZZ) under the project IP-2018-01-2970, the Unity Through Knowledge Fund (UKF Grant No. 20/15) and the support of project CeNIKS co-financed by the Croatian Government and the European Union through the European Regional Development Fund - Competitiveness and Cohesion Operational Programme (Grant No. KK.01.1.1.02.0013). The work at Rice University (A.H.N.) was supported by the U.S. National Science Foundation CAREER grant no. DMR-1350237 and the Welch Foundation grant C-1818. R.K. is supported by the German Science Foundation through Project No. KU 3287/1-1. M. L. S. acknowledges the support of the EPSRC and the Royal Society.
Author contributions E.C.T.O., M.S.G. and Y.M. contributed equally to this work. E.C.T.O. performed the resistivity, Hall effect and quantum oscillations measurements, M.S.G., M.Y. and M.T. performed NMR experiments, Y.M. performed specific heat measurements. Y.M., K.K., Y.K., Y.S. and T.S. performed magnetization measurements and K.K. synthesized the samples. R.K. and Y.M. performed the thermal expansion and magnetostriction measurements. M.L.S., M.B. and A.H.N. contributed to data interpretation. M.T. and S.N. conceived the project, planed the research and contributed to data interpretation. E.C.T.O., M.S.G. and Y.M. wrote the paper. All authors took part in discussing results and editing the manuscript.
Correspondence and requests for materials should be addressed to E. C. T. O., M.S.G, Y. M., M.T. and S.N.
Competing financial interests The authors declare no competing financial interests.