Gap Inhomogeneity Research Articles

Visualization of spatial inhomogeneity in the superconducting gap using micro-ARPES

ABSTRACT Electronic inhomogeneity arises ubiquitously as a consequence of adjacent and/or competing multiple phases or orders in strongly correlated electron systems. Gap inhomogeneity in high- T c cuprate superconductors has been widely observed using scanning tunneling microscopy/spectroscopy. However, it has yet to be evaluated by angle-resolved photoemission spectroscopy (ARPES) due to the difficulty in achieving both high energy and spatial resolutions. Here, we employ high-resolution spatially-resolved ARPES with a micrometric beam (micro-ARPES) to reveal the spatial dependence of the antinodal electronic states in optimally-doped Bi 2 Sr 2 CaCu 2 O 8 + δ . Detailed spectral lineshape analysis was extended to the spatial mapping dataset, enabling the identification of the spatial inhomogeneity of the superconducting gap and single-particle scattering rate at the micro-scale. Moreover, these physical parameters and their correlations were statistically evaluated. Our results suggest that high-resolution spatially-resolved ARPES holds promise for facilitating a data-driven approach to unraveling complexity and uncovering key parameters for the formulation of various physical properties of materials.

Simulating superconducting properties of overdoped cuprates: The role of inhomogeneity

Theoretical studies of disordered $d$-wave superconductors have focused, with a few exceptions, on optimally doped models with strong scatterers. Addressing recent controversies about the nature of the overdoped cuprates, however, requires studies of the weaker scattering associated with dopant atoms. Here we study simple models of such systems in the self-consistent Bogoliubov-de Gennes (BdG) framework, and compare to disorder-averaged results using the self-consistent-T-matrix-approximation (SCTMA). Despite surprisingly linear in energy behavior of the low-energy density of states even for quite disordered systems, the superfluid density in such cases retains a quadratic low-temperature variation of the penetration depth, unlike other BdG results reported recently. We trace the discrepancy to smaller effective system size employed in that work. Overall, the SCTMA performs remarkably well, with the exception of highly disordered systems with strongly suppressed superfluid density. We explore this interesting region where gap inhomogeneity dominates measured superconducting properties, and compare with overdoped cuprates.

Carrier Dynamics in Thin Germanium–Tin Epilayers

The Si-based mid-infrared photonics is an emerging technology in which group-IV germanium–tin (Ge1–xSnx) binary alloys can play a fundamental role in the development of a Si-compatible photonic components including monolithically integrated coherent light sources and detectors, on the same Si or SOI substrate. Although the Ge1–xSnx-on-Si lasers, at low temperatures, have already been demonstrated, the knowledge of the material properties necessary for such device optimization and real-life usage is very limited. In particular, carrier relaxation kinetics, relaxation pathways, and accompanied physical mechanisms, important for the laser’s dynamics, have not been subjected to in-depth research and understanding. In this work, we present detailed spectroscopic studies on photoinjected carrier dynamics in Ge1–xSnx epilayers, as a function of Sn content (6–12%) and temperature (20–300 K), by utilizing time-resolved differential reflectivity and photoluminescence. The latter technique allowed us to track separated electron and hole dynamics with a femtosecond time resolution, while the former experiment exploited a joined electron–hole recombination. This experimental approach allowed us to identify (i) two initial electron relaxation processes after photoexcitation; (ii) radiative electron–hole recombination on below-band gap states; (iii) nonradiative carrier recombination involving the Shockley–Read–Hall mechanism; and (iv) nonradiative recombination through the surface states. The research results significantly expand the knowledge on the initial carrier relaxation dynamics in the Ge1–xSnx epitaxial material. It provides unknown up-to-date kinetic parameters of the initial stage of electron relaxation and further carrier recombination dynamics, unveils the critical role of band gap inhomogeneity for the relaxation dynamics, and highlights the role of below-band gap states that can participate in the light generation process in Ge1–xSnx epilayers.

Resistivity phase diagram of cuprates revisited

The phase diagram of the cuprate superconductors has posed a formidable scientific challenge for more than three decades. This challenge is perhaps best exemplified by the need to understand the normal-state charge transport as the system evolves from Mott insulator to Fermi-liquid metal with doping. Here we report a detailed analysis of the temperature (T) and doping (p) dependence of the planar resistivity of simple-tetragonal HgBa$_2$CuO$_{4+\delta}$ (Hg1201), the single-CuO$_2$-layer cuprate with the highest optimal $T_c$. The data allow us to test a recently proposed phenomenological model for the cuprate phase diagram that combines a universal transport scattering rate with spatially inhomogeneous (de)localization of the Mott-localized hole. We find that the model provides an excellent description of the data. We then extend this analysis to prior transport results for several other cuprates, including the Hall number in the overdoped part of the phase diagram, and find little compound-to-compound variation in (de)localization gap scale. The results point to a robust, universal structural origin of the inherent gap inhomogeneity that is unrelated to doping-related disorder. They are inconsistent with the notion that much of the phase diagram is controlled by a quantum critical point, and instead indicate that the unusual electronic properties exhibited by the cuprates are fundamentally related to strong nonlinearities associated with subtle nanoscale inhomogeneity.

Spectroscopic evidence for strong correlations between local superconducting gap and local Altshuler-Aronov density of states suppression in ultrathin NbN films

Disorder has different profound effects on superconducting thin films. For a large variety of materials, increasing disorder reduces electronic screening which enhances electron-electron repulsion. These fermionic effects lead to a mechanism described by Finkelstein: when disorder combined to electron-electron interactions increases, there is a global decrease of the superconducting energy gap $\Delta$ and of the critical temperature $T_c$, the ratio $\Delta$/$k_BT_c$ remaining roughly constant. In addition, in most films an emergent granularity develops with increasing disorder and results in the formation of inhomogeneous superconducting puddles. These gap inhomogeneities are usually accompanied by the development of bosonic features: a pseudogap develops above the critical temperature $T_c$ and the energy gap $\Delta$ starts decoupling from $T_c$. Thus the mechanism(s) driving the appearance of these gap inhomogeneities could result from a complicated interplay between fermionic and bosonic effects. By studying the local electronic properties of a NbN film with scanning tunneling spectroscopy (STS) we show that the inhomogeneous spatial distribution of $\Delta$ is locally strongly correlated to a large depletion in the local density of states (LDOS) around the Fermi level, associated to the Altshuler-Aronov effect induced by strong electronic interactions. By modelling quantitatively the measured LDOS suppression, we show that the latter can be interpreted as local variations of the film resistivity. This local change in resistivity leads to a local variation of $\Delta$ through a local Finkelstein mechanism. Our analysis furnishes a purely fermionic scenario explaining quantitatively the emergent superconducting inhomogeneities, while the precise origin of the latter remained unclear up to now.

Criticality across the energy spectrum from random artificial gravitational lensing in two-dimensional Dirac superconductors

We numerically study weak, random, spatial velocity modulation ["quenched gravitational disorder" (QGD)] in two-dimensional massless Dirac materials. QGD couples to the spatial components of the stress tensor; the gauge-invariant disorder strength is encoded in the quenched curvature. Although expected to produce negligible effects, wave interference due to QGD transforms all but the lowest-energy states into a quantum-critical "stack" with universal, energy-independent spatial fluctuations. We study five variants of velocity disorder, incorporating three different local deformations of the Dirac cone: flattening or steepening of the cone, pseudospin rotations, and nematic deformation (squishing) of the cone. QGD should arise for nodal excitations in the $d$-wave cuprate superconductors (SCs), due to gap inhomogeneity. Our results may explain the division between low-energy "coherent" (plane-wave-like) and finite-energy "incoherent" (spatially inhomogeneous) excitations in the SC and pseudogap regimes. The model variant that best matches the cuprate phenomenology possesses quenched random pseudospin rotations and nematic fluctuations. This model variant and another with pure nematic randomness exhibit a robust energy swath of stacked critical states, the width of which increases with increasing disorder strength. By contrast, quenched fluctuations that isotropically flatten or steepen the Dirac cone tend to produce strong disorder effects, with more rarified wave functions at low- and high-energies. Our models also describe the surface states of class DIII topological SCs.

Influence of silicon doping on internal quantum efficiency and threshold of optically pumped deep UV AlGaN quantum well lasers

The influence of silicon doping on optically pumped AlGaN quantum well (QW) lasers has been analyzed. Either the QWs, quantum barriers, or the whole heterostructure were doped with Si concentrations between 6 × 1016 and 1 × 1019 cm−3. When the QWs are doped up to 5 × 1017 cm−3, laser threshold is reduced and internal quantum efficiency is increased most probably due to a reduction of group-III-vacancies during growth. Higher Si concentrations in the wells can increase the density of group-III-vacancies causing an increased laser threshold. High concentrations can also enhance band gap inhomogeneities and the free carrier density, which improves the internal quantum efficiency measured by photoluminescence. Doping of all layers with a Si concentration of 1 × 1019 cm−3 deteriorates the QW quality, most probably because the silicon acts as an anti-surfactant reducing and delaying Ga incorporation.

Percolative nature of the direct-current paraconductivity in cuprate superconductors

Despite extraordinary scientific efforts over the past three decades, the cuprate high-temperature superconductors continue to pose formidable challenges. A pivotal problem, essential for understanding both the normal and superconducting states, is to clarify the nature of the superconducting pre-pairing above the bulk transition temperature Tc. Different experimental probes have given conflicting results, in part due to difficulties in discerning the superconducting response from the complex normal-state behavior. Moreover, it has proven challenging to separate common properties of the cuprates from compound-specific idiosyncrasies. Here we investigate the paraconductivity—the superconducting contribution to the direct-current (dc) conductivity—of the simple-tetragonal model cuprate material HgBa2CuO4+δ. We are able to separate the superconducting and normal-state responses by taking advantage of the Fermi-liquid nature of the normal state in underdoped HgBa2CuO4+δ; the robust and simple quadratic temperature-dependence of the normal-state resistivity enables us to extract the paraconductivity above the macroscopic Tc with great accuracy. We find that the paraconductivity exhibits unusual exponential temperature dependence, and that it can be quantitatively explained by a simple superconducting percolation model. Consequently, the emergence of superconductivity in this model system is dominated by the underlying intrinsic gap inhomogeneity. Motivated by these insights, we reanalyze published results for two other cuprates and find exponential behavior as well, with nearly the same characteristic temperature scale. The universal intrinsic gap inhomogeneity is not only essential for understanding the supercoducting precursor, but will also have to be taken into account in the analysis of other bulk measurements of the cuprates.

Spatial variation of the two-fold anisotropic superconducting gap in a monolayer of FeSe0.5Te0.5 on a topological insulator

We present our investigations on the superconducting properties of monolayers of FeSe$_{0.5}$Te$_{0.5}$ grown on the 3D topological insulator Bi$_{2}$Se$_{1.2}$Te$_{1.8}$ using low temperature scanning tunneling spectroscopy (STS). While the morphology and the overall transition temperature resemble those of similarly doped bulk crystals, the spatially resolved spectroscopic data at 1.1K shows a much larger spatial inhomogeneity in the superconducting energy gaps. Despite the gap inhomogeneity all the spectra can be fitted with a two-fold anisotropic s-wave gap function. The two-fold nature of the gap symmetry is evident from the Bogoliubov quasiparticle interference (QPI) pattern which shows distinct C$_{2}$ symmetric scattering intensities. We argue that the gap inhomogeneity emerges as a result of intrinsic disorder in our system similar to disordered conventional superconductors. Even though most of our findings clearly differ from the current understanding of the corresponding bulk system, it provides an ideal platform to study unconventional superconductivity in Fe chalcogenides thinned down to a single layer and in close proximity to a topological insulator.

Modeling study of peak-dip-hump structure in tunneling spectra of high-temperature superconducting cuprates

We propose new specific model for quasiparticle (QP) tunneling across the superconductor-insulator-normal metal (SIN) junction based on two mechanisms. Origin of the many features of the tunneling spectra, such as peak-dip-hump (PDH) structure, U- and V-shapes, temperature dependence of differential tunneling conductance, asymmetric conductance peaks, zero-bias conductance, subgap feature and gap inhomogeneity have been explained by the proposed model. We show that the energy scales of the binding energies of large polarons and polaronic Cooper pairs are identified by pseudogap (PG) crossover temperature on the cuprate phase diagram.

Pseudogap formation and unusual quasiparticle tunneling in cuprate superconductors: Polaronic and multiple-gap effects on the tunneling spectra

We propose new simple and generalized multiple-gap models of quasiparticle tunneling across the high-Tc cuprate superconductor (HTSC)/insulator/normal metal (SIN) junction based on the two different mechanisms for tunneling at positive and negative bias voltages, and the gap inhomogeneity (i.e., multiple-gap) picture. The tunneling of electrons from the normal metal into the quasiparticle states in HTSC with the BCS-type density of states (DOSs) takes place at V>0, while the tunneling of Cooper pairs and large polarons from the HTSC with the BCS DOS and quasi-free state DOS (which appears only in the dissociation of polarons) into the normal metal occurs at V<0. We show that most of the unusual features of tunneling spectra such as nearly U- and V-shaped subgap features, peak-dip-hump structure (appearing systematically at V<0) and asymmetry of the conductance peaks and their temperature and doping dependences, and shoulder-like features inside the main conductance peaks arise naturally in our specific models of SIN tunneling. The experimental tunneling spectra of Bi2Sr2CaCu2O8+δ are adequately reproduced by using the specific multiple-gap models and taking into account the distribution of BCS and polaronic gap values.

Strong charge carrier localization interacting with extensive nonradiative recombination in heteroepitaxially grown m-plane GaInN quantum wells

The development of GaInN quantum well structures with nonpolar crystal orientation for light-emitting diodes and semiconductor lasers is currently one of the main foci of III-nitride-based optoelectronics research. One of the advantages of nonpolar orientations is the absence of polarization fields perpendicular to the quantum well plane. As a consequence, radiative recombination rates are higher compared to quantum wells on polar surfaces. However, due to high densities of threading dislocations and basal plane stacking faults in the case of heteroepitaxially grown nonpolar layers, and due to band gap inhomogeneities in the GaInN quantum wells, characterization of radiative and nonradiative recombination mechanisms is a complex challenge. So far, most published data about band gap fluctuations, charge carrier localization and internal quantum efficiency in nonpolar quantum wells are ambiguous. Here, we present temperature and excitation power density-dependent photoluminescence data featuring multiple characteristics related to strong charge carrier localization in m-plane (1–100) GaInN quantum wells. Thermally activated redistribution of charge carriers between localization sites in these quantum wells is weaker than in polar c-plane ones. The localization strength increases with higher indium concentration in the quantum wells. In the heteroepitaxially grown quantum well structures, the internal quantum efficiency is reduced even at low temperatures (T = 10 K) and especially for m-plane quantum wells with high indium mole fractions.

Disappearance of zinc impurity resonance in large-gap regions ofBi2Sr2CaCu2O8+δprobed by scanning tunneling spectroscopy

Using Scanning tunneling spectroscopy (STS), we report the correlation between spatial gap inhomogeneity and the zinc (Zn) impurity resonance in single crystals of Bi$_{\mathrm{2}}$Sr$_{\mathrm{2}}$Ca(Cu$_{\mathrm{1-}x}$Zn$_{x}$)$_{\mathrm{2}}$O$_{\mathrm{8+}\delta}$ with different carrier (hole) concentrations ($p$) at a fixed Zn concentration ($x$ $\sim$ 0.5 % per Cu atom). In all the samples, the impurity resonance lies only in the region where the gap value is less than $\sim$ 60 meV. Also the number of Zn resonance sites drastically decreases with decreasing $p$, in spite of the fixed $x$. These experimental results lead us to a conclusion that the Zn impurity resonance does not appear in the large gap region although the Zn impurity evidently resides in this region.

Electronic inhomogeneity and optical response of Bi2212

We measured the in-plane optical reflectivity spectra of Bi2Sr2CaCu2O8+δ single crystals from under-to overdoped regime. Appreciable residual conductivity is observed in the low energy region even in the superconducting state well below Tc. Comparing with the results of the Scanning Tunneling Microscope/Scanning Tunneling Spectroscopy (STM/STS) measurement made on the same set of crystals, we discuss how the observed gap inhomogeneity in the CuO2 plane affects the optical spectrum.

Impurity effect on superconducting gap and pseudogap

Co substituted Bi2212 samples of Bi 2.1Sr 1.9Ca(Cu 1−xCo x)O 8+ δ ( x = 0, 0.01, 0.02, 0.04, 0.06, and Tc = 85,71,52,35 K) were prepared, and their electronic density states were characterized by a low temperature STM. The experimental results revealed that superconducting gap-like features were rapidly destroyed, while pseudogap-like features were enhanced with increasing impurities, suggesting that the origin of the pseudogap is different from that of a superconducting gap. We suggested a two phase-like model to explain the behavior of a gap map and a gap distribution function. On the other hand, spatial gap and pseudogap inhomogeneity persisted in all Co concentrations. This indicates that the origin of inhomogeneity is not related to in-plane disorder or impurities.

Comment on “Thermodynamic transitions in inhomogeneousd-wave superconductors”

Scanning-tunneling spectroscopy studies on suggest the presence of electronic inhomogeneity with a large spatial variation in gap size. have modeled this variation by assuming a spatially varying pairing interaction. We show that their calculated specific heat is incompatible with the experimental data which exhibit narrow transitions. This calls into question the now-common assumption of gap and pairing inhomogeneity.

Signatures of superconducting gap inhomogeneities in optical properties

Scanning tunneling spectroscopy applied to the high-${T}_{c}$ cuprates has revealed significant spatial inhomogeneity on the nanoscale. Regions on the order of a coherence length in size show variations of the magnitude of the superconducting gap of order $\ifmmode\pm\else\textpm\fi{}20%$ or more. An important unresolved question is whether or not these variations are also present in the bulk, and how they influence superconducting properties. As many theories and data analyses for high-${T}_{c}$ superconductivity assume spatial homogeneity of the gap magnitude, this is a pressing question. We consider the far-infrared optical conductivity and evaluate, within an effective-medium approximation, what signatures of spatial variations in gap magnitude are present in various optical quantities. In addition to the case of $d$-wave superconductivity, relevant to the high-${T}_{c}$ cuprates, we have also considered $s$-wave gap symmetry in order to provide expected signatures of inhomogeneities for superconductors in general. While signatures of gap inhomogeneities can be strongly manifested in $s$-wave superconductors, we find that the far-infrared optical conductivity in $d$-wave is robust against such inhomogeneity, such that any changes maintain the character of a curve for a single $d$-wave gap.

Fermi-arc superconductivity, pseudogap and charge order in Bi 2Sr 2CaCu 2O 8+δ

We report STM/STS observations on the 4 a×4 a charge order in the pseudogap (PG) and superconducting (SC) states of Bi 2Sr 2CaCu 2O 8+ δ , and suggest that the charge order is associated with incoherent quasiparticle or pair states around the antinodal fermi surface (FS), which are also responsible for the PG, and it can coexist with the superconductivity caused by the pairing of coherent quasiparticles on the nodal fermi arc. We also suggest that the nanometer-scale gap inhomogeneity in the SC state, reported previously [Pan, et al., Nature 413 (2001) 282; Mommo, et al., J. Phys. Soc. Jpn. 74 (2005) 2400; Hashimoto, et al., Phys. Rev. B74 (2006) 064508] arises from that in the PG state, which occurs around the antinodal FS.

Mesoscopic gap fluctuations in an unconventional superconductor

We study mesoscopic disorder fluctuations in an anisotropic gap superconductor, which lead to the spatial variations of the local pairing temperature and formation of superconducting islands above the mean-field transition. We derive the probability distribution function of the pairing temperatures and superconducting gaps. It is shown that above the mean-field transition, a disordered BCS superconductor with an unusual pairing symmetry is described by a network of superconducting islands and metallic regions with a strongly suppressed density of states due to superconducting fluctuations. We argue that the phenomena associated with mesoscopic disorder fluctuations may also be relevant to the high-temperature superconductors, in particular, to recent scanning tunnel microscope (STM) experiments, where gap inhomogeneities have been explicitly observed. It is suggested that the gap fluctuations in the pseudogap phase should be directly related to the corresponding fluctuations of the pairing temperature.

Light absorption and emission in semiconductors with band gap fluctuations—A study on Cu(In,Ga)Se2 thin films

This work investigates the influence of lateral fluctuations of the fundamental band gap on the macroscopic light absorptance and emission spectra of spatially inhomogeneous semiconductors. A model assuming a Gaussian distribution for the local band gaps yields closed-form expressions for the spectral absorptance and emission. Band gap fluctuations broaden the absorption edge of the fundamental band gap, as well as the associated emission peak. The spectral position of the photoluminescence emission peak depends on the length scale of the fluctuations in relation to the characteristic charge carrier transport length. We apply the model to experimental results from Cu(In1−x,Gax)Se2 thin films routinely used as photovoltaic absorbers in thin-film ZnO/CdS/Cu(In,Ga)Se2 heterojunction solar cells. The films feature band gap fluctuations with standard deviations between 15 and 65 meV which would lead to losses in the range of 5–80 mV for the open circuit voltage of solar cells made from these films. The pure ternary compounds CuInSe2 and CuGaSe2 exhibit smaller standard deviations than their quaternary alloys. This experimental finding indicates alloy disorder as one possible source of band gap inhomogeneities. The length scale of the observed fluctuations turns out to be much smaller than the minority carrier diffusion length. Hence, the band gap fluctuations occur on a length scale below 100 nm.