No abstract

The concept of mass-generation via the Higgs mechanism was strongly inspired by earlier works on the Meissner-Ochsenfeld effect in superconductors. In quantum field theory, the excitations of longitudinal components of the Higgs field manifest as massive Higgs bosons. The analogous Higgs mode in superconductors has not yet been observed due to its rapid decay into particle-hole pairs. Following recent theories, however, the Higgs mode should decrease below the pairing gap 2∆ and become visible in two-dimensional systems close to the superconductor-insulator transition (SIT). For experimental verification, we measured the complex terahertz transmission and tunneling density of states (DOS) of various thin films of superconducting NbN and InO close to criticality. Comparing both techniques reveals a growing discrepancy between the finite 2∆ and the threshold energy for electromagnetic absorption which vanishes critically towards the SIT. We identify the excess absorption below 2∆ as a strong evidence of the Higgs mode in two dimensional quantum critical superconductors.The Higgs mechanism, which has great implications to recent developments in particle physics [1], originates in Anderson's pioneering work on symmetry breaking with gauge fields in superconductors [2]. A superconductor spontaneously breaks continuous U (1) symmetry and acquires the well-known Mexican hat potential with a degenerate circle of minima described by the order parameter Ψ = Ae iϕ , see Fig. 1a. Excitations from the ground state can be classified as transverse Nambu-Goldstone (phase) modes and massive longitudinal Higgs (amplitude) modes (see blue and red lines in Fig. 1a). In particle physics, the latter manifest themselves as the Higgs boson which was recently discovered at CERN [3]. Indications of a Higgs mode in correlated many-body systems have been found in one-dimensional charge-densitywave systems [4], quantum antiferromagnets [5] and twodimensional superfluid to Mott transition in cold atoms [6]. An amplitude mode, also named Higgs mode, was theoretically predicted for superconductors [7] and recently reported to be measured by pump-probe spectroscopy [8]. This amplitude mode describes pairing fluctuations, which are qualitatively distinct from the purely bosonic mode expected from the O(2) field theory. The Higgs-amplitude mode analogous to the highenergy Higgs Boson has not yet been observed in superconductors. A partial reason is that in homogeneous, BCS superconductors the Higgs mode is short-lived and decays to particle hole (Bogoliubov) pairs [9,10]. Nevertheless, collective modes were recently predicted to be significant in strongly disordered superconductors [11], and, in particular it was shown [12][13][14] that the Higgs mode softens but remains sufficiently sharp near a quantum critical point (QCP) in two dimensions since it is found to be a critical energy scale of the quantum phase transition. Hence, the Higgs mass can be reduced below twice the pairing gap, 2∆, making this mode experimentally visible. Such a critical...

We present a broadband microwave spectrometer covering the range from 45 MHz up to 20 GHz (in some cases up to 40 GHz) which employs the Corbino geometry, meaning that the flat sample terminates the end of a coaxial transmission line. This setup is optimized for low-temperature performance (temperature range 1.7–300 K) and for the study of highly conductive samples. The actual sensitivity in reflection coefficient can be as low as 0.001, leading to a resolution of 10% in absolute values of the impedance or complex conductivity. For optimum accuracy a full low-temperature calibration is necessary; therefore up to three calibration measurements (open, short, and load) are performed at the same temperature as the sample measurement. This procedure requires excellent reproducibility of the cryogenic conditions. We compare further calibration schemes based on just a single low-temperature calibration measurement or employing a superconducting sample as a calibration standard for its normal state, and we document the capability of the instrument with test measurements on metallic thin films. Finally we apply the spectrometer to thin films of a heavy-fermion compound as an example for a strongly correlated electron system.

Deterministic enhancement of the superconducting (SC) critical temperature T c is a longstanding goal in material science. One strategy is engineering a material at the nanometer scale such that quantum confinement strengthens the electron pairing, thus increasing the superconducting energy gap ∆ [1-6], as was observed for individual nanoparticles [7]. A true phasecoherent SC condensate, however, can exist only on larger scales and requires a finite phase stiffness J [13]. In the case of coupled aluminium (Al) nanograins [8][9][10], T c can exceed that of bulk Al by a factor of three, but despite several proposals the relevant mechanism at play is not yet understood. Here we use optical spectroscopy on granular Al to disentangle the evolution of the fundamental SC energy scales, ∆ and J, as a function of grain coupling. Starting from wellcoupled arrays, ∆ grows with progressive grain decoupling, causing the increasing of T c . As the grain-coupling is further suppressed, ∆ saturates while T c decreases, concomitantly with a sharp decline of J. This crossover to a phase-driven SC transition is accompanied by an optical gap persisting above T c . These findings identify granular Al as an ideal playground to test the basic mechanisms that enhance superconductivity by nanoinhomogeneity.Bulk samples of pure Al represent a prototypical BCS superconductor (SC) with relatively low T c0 ≈ 1.2 K. Several studies since the late 1960s [8][9][10] have shown a quite different situation for granular Al, i.e. thin films composed of 2 nm grains separated by thin insulating barriers, where a superconducting condensate is established via Josephson-coupling across the grain array. The coupling between the grains can be controlled during film growth, leading to samples with strong coupling and low resistivity (LR) in electrical transport compared to high resistivity (HR) samples with weak intergrain coupling. In LR samples T c can be enhanced up to several times T c0 , whereas it is suppressed to zero in HR samples, shaping a superconducting dome in the phase diagram, see Fig. 1(a).To understand the behavior of T c it is crucial to access the underlying SC energy scales associated with the amplitude and phase of the complex order parameter ψ = ∆e iφ . Indeed, while the SC energy gap ∆ measures the pairing strength between the electrons, the true superfluid behavior can only be established if the Cooper pairs acquire the same macroscopic SC phase φ. The energy scale controlling the rigidity of the condensate with respect to a deformation of this collective phase-coherent state is the so-called superfluid stiffness J. In ordinary BCS superconductors J exceeds ∆ by orders of magnitudes, and the SC transition at T c is amplitude-driven. However, in the unconventional situation where ∆ exceeds J the transition is expected to be phase-driven, due to the loss of phase coherence at a temperature scale of order of J. Consequently, even though several finite-size effects have been proposed to explain the enhancement of ∆ in isolated nano-grai...

The electrical conduction of metals is governed by how freely mobile electrons can move throughout the material. This movement is hampered by scattering with other electrons, as well as with impurities or thermal excitations (phonons). Experimentally, the scattering processes of single electrons are not observed, but rather the overall response of all mobile charge carriers within a sample. The ensemble dynamics can be described by the relaxation rates, which express how fast the system approaches equilibrium after an external perturbation. Here we measure the frequency-dependent microwave conductivity of the heavy-fermion metal UPd2Al3 (ref. 4), finding that it is accurately described by the prediction for a single relaxation rate (the so-called Drude response). This is notable, as UPd2Al3 has strong interactions among the electrons that might be expected to lead to more complex behaviour. Furthermore, the relaxation rate of just a few gigahertz is extremely low--this is several orders of magnitude below those of conventional metals (which are typically around 10 THz), and at least one order of magnitude lower than previous estimates for comparable metals. These observations are directly related to the high effective mass of the charge carriers in this material and reveal the dynamics of interacting electrons.

Electrical conductance through InAs nanowires is relevant for electronic applications as well as for fundamental quantum experiments. Here we employ nominally undoped, slightly tapered InAs nanowires to study the diameter dependence of their conductance. Contacting multiple sections of each wire, we can study the diameter dependence within individual wires without the need to compare different nanowire batches. At room temperature we find a diameter-independent conductivity for diameters larger than 40 nm, indicative of three-dimensional diffusive transport. For smaller diameters, the resistance increases considerably, in coincidence with a strong suppression of the mobility. From an analysis of the effective charge carrier density, we find indications for a surface accumulation layer.

Abstract-We report on terahertz frequency-domain spectroscopy (THz-FDS) experiments in which we measure charge carrier dynamics and excitations of thin-film superconducting systems at low temperatures in the THz spectral range. The characteristics of the set-up and the experimental procedures are described comprehensively. We discuss the single-particle density of states and a theory of electrodynamic absorption and optical conductivity of conventional superconductors. We present the experimental performance of the setup at low temperatures for a broad spectral range from 3 to 38 cm -1 (0.1 -1.1 THz) by the example of ultra-thin films of weakly disordered superconductors niobium nitride (NbN) and tantalum nitride (TaN) with different values of critical temperatures Tc. Furthermore, we analyze and interpret our experimental data within the framework of conventional Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity. By and large, we find the properties of our NbN and TaN thin films to be well described by the theory. Our results on NbN resemble tendencies towards anomalous behavior of the ratio 2∆(0)/kBTc as a function of Tc.Index Terms-Frequency-domain THz spectroscopy, superconducting thin films, BCS theory, density of states of a superconductor, optical conductivity of superconductors, superconductorinsulator transition, TaN, NbN.

Investigating solids with light gives direct access to charge dynamics, electronic and magnetic excitations. For heavy fermions, one has to adjust the frequency of the probing light to the small characteristic energy scales, leading to spectroscopy with microwaves. We review general concepts of the frequency-dependent conductivity of heavy fermions, including the slow Drude relaxation and the transition to a superconducting state, which we also demonstrate with experimental data taken on UPd 2 Al 3 . We discuss the optical response of a Fermi liquid and how it might be observed in heavy fermions. Microwave studies with focus on quantum criticality in heavy fermions concern the charge response, but also the magnetic moments can be addressed via electron spin resonance (ESR). We discuss the case of YbRh 2 Si 2 , the open questions concerning ESR of heavy fermions, and how these might be addressed in the future. This includes an overview of the presently available experimental techniques for microwave studies on heavy fermions, with a focus on broadband studies using the Corbino approach and on planar superconducting resonators.