Heavy fermion materials

Steffen Wirth,¹ Manuel Brando,¹ Zachary Fisk,² Sven Friedemann,³ Christoph Geibel,¹ Stefan Kirchner,⁴ Cornelius Krellner,⁵ Heike Pfau,¹ Naren H. Ranganath,⁶ S. Seiro,¹ Qimiao Si,⁷ F. Steglich,¹ Joe D. Thompson,⁸ and Gertrud Zwicknagl⁹

¹Max-Planck-Institut für Chemische Physik fester Stoffe, 01187 Dresden, Germany ²University of California, Irvine, California 92697, USA ³HH Wills Laboratory, University of Bristol, Bristol BS8 1TH, UK ⁴Center for Correlated Matter, Zhejiang University, Hangzhou, Zhejiang 310058, China ⁵Institute of Physics, Goethe-Universität Frankfurt, 60438 Frankfurt/Main, Germany ⁶Dept. of Condensed Matter Physics, Weizmann Institute of Science, 76100 Rehovot, Israel ⁷Dept. of Physics and Astronomy, Rice University, Houston, TX 77005, USA ⁸Los Alamos National Laboratory, Los Alamos, NM 87545, USA ⁹Institut für Mathematische Physik, TU Braunschweig, 38106 Braunschweig, Germany

Electronic correlations give rise to a plethora of interesting phenomena and phases. For example, hybridization between 4f and conduction electrons in heavy fermion metals may result in the generation of low-energy scales that can induce quantum criticality and unconventional superconductivity (see Special Issue, Ref. 1). These complex interactions often operate, and compete, on different lengths scales. One of the most important techniques that helped shaping our understanding of nonlocal correlations, both magnetic and superconducting, has been scanning tunneling spectroscopy (STS) with its unique ability to give local, microscopic information that directly relates to the one-particle Green’s function. We complement these local measurements with global probes, most importantly magnetotransport (see review on Hall effect, Ref. 2), magnetization and x-ray diffraction measurements, in order to unambiguously assign the local, surface-sensitive findings by STS to bulk properties of the single crystals under scrutiny.

As a pivotal prerequisite for a possible disentanglement of the involved interactions, to rule out inadvertent influences of impurities as well as to successfully relate local with global properties, samples of highest possible quality need to be investigated. Here, the so-called 115 family of compounds (CeMIn₅ with M = Co, Ir, Rh) exhibiting superconductivity, magnetism and Fermi surface changes, are a clear-cut example for which we could identify nanometer-scale defects, Wirth et al. 2014 [3]. A substitution series of these compounds was recently instrumental in identifying the Ce 4f wave function as a significant parameter for the realization of either a superconducting or an antiferromagnetic ground state, Willers et al. 2015 [4]. For CeCoIn₅ a field-induced quantum critical point (QCP) of spin-density-wave (SDW) type is extrapolated to exist inside the superconducting phase. Single crystals of 115 compounds were provided through our long-term collaboration with groups led by Dr. Zachary Fisk (UC Irvine, USA) and Dr. Joe D. Thompson (Los Alamos National Laboratory, USA). Similarly, in CeCu₂Si₂ a SDW type of QCP was established in the vicinity of which superconductivity was demonstrated to be mediated by spin fluctuations, Steglich et al. 2013 [5].

Very strong correlation effects can be realized through the above-mentioned hybridization between 4f and conduction electrons via the Kondo effect. In case of the prototypical heavy fermion metal YbRh₂Si₂ this can result in quasiparticle masses beyond a thousand times the free electron mass. Our low-temperature electronic transport measurements at high magnetic field (μ₀H ≤ 15 T) in concert with thermal transport measurements (conducted in the group Novel states of matter on the border of magnetism) and renormalized band structure calculations by Prof. Gertrud Zwicknagl indicate that the de-renormalization of the quasiparticles, i.e. the destruction of the local Kondo singlets, occurs smoothly and likely requires magnetic fields of the order of 30 T to complete, Naren et al. 2013 [6], Pfau et al. 2013 [7]. On the other hand, the spin splitting of the very flat, hybridized bands results in three successive Lifshitz transitions which are confined to relatively narrow magnetic field intervals and clearly visible in magnetotransport [6] (see Fig. 1, lower panel) and thermopower measurement [7].

FIG. 1: (upper panel) Calculated density of states with van Hove singularity. (middle) Isoenergy surfaces for the two most prominent bands. Four different topologies can be distinguished (marked by different colors), with Lifshitz transitions between them. (lower) Example of the field derivative of the magnetoresistivity (MR) showing clear signatures of the Lifshitz transitions. Note that the Lifshitz transition at small fields corresponds to the one closest to the Fermi energy EF, dashed line in upper panel. — FIG. 1: (upper panel) Calculated density of states with van Hove singularity. (middle) Isoenergy surfaces for the two most prominent bands. Four different topologies can be distinguished (marked by different colors), with Lifshitz transitions between them. (lower) Example of the field derivative of the magnetoresistivity (MR) showing clear signatures of the Lifshitz transitions. Note that the Lifshitz transition at small fields corresponds to the one closest to the Fermi energy E_F, dashed line in upper panel.

FIG. 1: (upper panel) Calculated density of states with van Hove singularity. (middle) Isoenergy surfaces for the two most prominent bands. Four different topologies can be distinguished (marked by different colors), with Lifshitz transitions between them. (lower) Example of the field derivative of the magnetoresistivity (MR) showing clear signatures of the Lifshitz transitions. Note that the Lifshitz transition at small fields corresponds to the one closest to the Fermi energy E_F, dashed line in upper panel.

FIG. 2: Topography of a Si-terminated surface of YbRh2Si2, area shown is 40×40 nm2. The more pronounced defects can be related to a site exchange between Si and Rh: Rh on a Si site causes a single protrusion whereas Si on a Rh site (which has to take place in the second-to-topmost layer) results in dumbbell-shaped dents. The overall small number of defects evidences the excellent sample quality of our single crystals. — FIG. 2: Topography of a Si-terminated surface of YbRh₂Si₂, area shown is 40×40 nm². The more pronounced defects can be related to a site exchange between Si and Rh: Rh on a Si site causes a single protrusion whereas Si on a Rh site (which has to take place in the second-to-topmost layer) results in dumbbell-shaped dents. The overall small number of defects evidences the excellent sample quality of our single crystals.

FIG. 2: Topography of a Si-terminated surface of YbRh₂Si₂, area shown is 40×40 nm². The more pronounced defects can be related to a site exchange between Si and Rh: Rh on a Si site causes a single protrusion whereas Si on a Rh site (which has to take place in the second-to-topmost layer) results in dumbbell-shaped dents. The overall small number of defects evidences the excellent sample quality of our single crystals.

The renormalized band structure calculations indicate a neck-forming Lifshitz transitions in the hybridized band 37 at around 3 T (from region I to II) along the Γ→X direction, a neck disruption in band 37 as well as the formation of a single surface in the f-derived band 35 at around 9 T, and finally the disappearance of a pocket along X→P→u in band 37 around 11 T (from region III to VI). As can be seen, the calculations are in excellent agreement with our measurements. Neither our measurements, nor the renormalized band structure calculations provided any hint of a metamagnetic transition, Friedemann et al. 2013 [8]. Again, all these measurements require high-quality single crystalline YbRh₂Si₂ which are provided by the Materials design and synthesis group.

Numerous electronic transport and thermodynamic measurements indicate an unconventional nature of the QCP in YbRh₂Si₂ related to the break-up of the quasiparticles right at the QCP, Steglich et al. 2014 [9]. However, the corresponding large and small Fermi surfaces are sharply separated at zero temperature, only. At finite temperature, the quantum critical fluctuations broaden this cross-over.

In consequence, the single-particle spectral function at T ≳ 0.5 K and zero magnetic field has significant weight at both the small and large Fermi surfaces, Paschen et al. 2015 [10]. Therefore, for unambiguous measurements of the Fermi surface volume sufficiently low temperatures are required.

YbRh₂Si₂ single crystals were studied by low-temperature scanning tunneling microscopy (STM) and spectroscopy (STS). The topography on samples cleaved in situ at low temperature and perpendicular to the crystallographic c-direction confirms the excellent quality of the latest generation of samples, see Fig. 2. Two types of defects can be distinguished, which are likely caused by site exchange: The single protrusions may originates from a larger Rh ion occupying a Si site at the surface while the dumbbell-shaped dents result from a smaller Si occupying a Rh site within the second-to-topmost layer and affecting the two adjacent Si in the topmost layer. Our studies support two conclusions (see Wirth et al. 2012 [11]): i) The exact stoichiometry of the single crystals can be determined down to an atomic level. ii) More importantly, such surfaces provide a first indication that they are Si terminated. This assignment is could be supported further by the first-ever observation of crystalline electric field (CEF) excitations by STS. Moreover, our STS spectra are dominated by a gap-like feature around zero-bias resulting from the hybridization of conduction and 4f electrons. Clearly, the local Kondo entanglement modifies the local density of states (DOS) govern the measured tunneling spectra g(V,T). The temperature dependence of the reduction in g(V,T) can nicely be reproduced by calculations within the non-crossing approximation (NCA) whereas renormalized band structure calculation (which are restricted to zero temperature) capture 4f-derived intersite correlations. An additional hump in our tunneling spectra at around V = -6 mV strongly evolves at low temperatures and is likely caused by a lattice Kondo resonance.

The interpretation of the results of our studies requires in many cases profound theoretical support. Here we rely primarily on collaborations in house and with Dr. Qimiao Si (Rice University, USA).

[1] Special Issue: “Quantum Criticality and Novel Phases”, edited by S. Kirchner, O. Stockert and S. Wirth, Phys. Stat. Sol. 250(3), pp. 417-659 (2013) http://onlinelibrary.wiley.com/doi/10.1002/pssb.v250.3/issuetoc.

[2] S. Nair, S. Wirth, S. Friedemann, F. Steglich, Q. Si and A. J. Schofield Hall effect in heavy fermion metals. Adv. Phys. 61, 583-664 (2012) http://dx.doi.org/10.1080/00018732.2012.730223.

[3] S. Wirth, Y. Prots, M. Wedel, S. Ernst, S. Kirchner, Z. Fisk, J. D. Thompson, F. Steglich and Y. Grin Structural investigations of CeIrIn₅ and CeCoIn₅ on macroscopic and atomic length scales. J. Phys. Soc. Jpn. 83, 061009 (2014) http://dx.doi.org/10.7566/JPSJ.83.061009.

[4] T. Willers, F. Strigari, Z. Hu, V. Sessi, N. B. Brookes, E. D. Bauer, J. L. Sarrao, J. D. Thompson, A. Tanaka, S. Wirth, L. H. Tjeng and A. Severing Correlation between ground state and orbital anisotropy in heavy fermion materials. Proc. Natl. Acad. Sci. USA 112, (2015) http://dx.doi.org/10.1073/pnas.1415657112.

[5] F. Steglich, O. Stockert, S. Wirth, C. Geibel, H. Q. Yuan, S. Kirchner and Q. Si Routes to heavy-fermion superconductivity. J. Phys.: Conf. Series 449, 012028 (2013) http://iopscience.iop.org/article/10.1088/1742-6596/449/1/012028.

[6] H. R. Naren, S. Friedemann, G. Zwicknagl, C. Krellner, C. Geibel, F. Steglich and S. Wirth Lifshitz transitions and quasiparticle de-renormalization in YbRh₂Si₂. New J. Phys. 15, 093032 (2013). http://dx.doi.org/10.1088/1367-2630/15/9/093032.

[7] H. Pfau, R. Daou, S. Lausberg, H. R. Naren, M. Brando, S. Friedemann, S. Wirth, T. Westerkamp, U. Stockert, P. Gegenwart, C. Krellner, C. Geibel, G. Zwicknagl and F. Steglich Interplay between Kondo suppression and Lifshitz transitions in YbRh₂Si₂ at high magnetic fields. Phys. Rev. Lett. 110, 256403 (2013) http://dx.doi.org/10.1103/PhysRevLett.110.256403.

[8] S. Friedemann, S. Paschen, C. Geibel, S. Wirth, F. Steglich, S. Kirchner, E. Abrahams and Q. Si Comment on “Zeeman-Driven Lifshitz Transition: A Model for the Experimentally Observed Fermi-Surface Reconstruction in YbRh₂Si₂”. Phys. Rev. Lett. 111, 139701 (2013) http://dx.doi.org/10.1103/PhysRevLett.111.139701.

[9] F. Steglich, H. Pfau, S. Lausberg, Peijie Sun, U. Stockert, M. Brando, S. Friedemann, C. Krellner, C. Geibel, S. Wirth, S. Kirchner, E. Abrahams and Qimiao Si Evidence of a Kondo destroying quantum critical point in YbRh₂Si₂. J. Phys. Soc. Jpn. 83, 061001 (2014) http://dx.doi.org/10.7566/JPSJ.83.061001.

[10] S. Paschen, S. Friedemann, S. Wirth, F. Steglich, S. Kirchner and Q. Si Kondo destruction in heavy fermion quantum criticality and the photoemission spectrum of YbRh₂Si₂. J. Magn. Magn. Mater. (2015) accepted http://dx.doi.org/10.1016/j.jmmm.2015.09.008.

[11] S. Wirth, S. Ernst, R. Cardoso-Gil, H. Borrmann, S. Seiro, C. Krellner, C. Geibel, S. Kirchner, U. Burkhardt, Y. Grin and F. Steglich Structural investigations on YbRh₂Si₂: from the atomic to the macroscopic length scale. J. Phys.: Condens. Matter 24 294203 (2013) http://dx.doi.org/10.1088/0953-8984/24/29/294203.