Classifying Charge Carrier Interaction in Highly Compressed Elements and Silane

Since the pivotal experimental discovery of near-room-temperature superconductivity (NRTS) in highly compressed sulphur hydride by Drozdov et al. (Nature 2015, 525, 73–76), more than a dozen binary and ternary hydrogen-rich phases exhibiting superconducting transitions above 100 K have been discovered to date. There is a widely accepted theoretical point of view that the primary mechanism governing the emergence of superconductivity in hydrogen-rich phases is the electron–phonon pairing. However, the recent analysis of experimental temperature-dependent resistance, R(T), in H₃S, LaH_x, PrH₉ and BaH₁₂ (Talantsev, Supercond. Sci. Technol. 2021, 34, accepted) showed that these compounds exhibit the dominance of non-electron–phonon charge carrier interactions and, thus, it is unlikely that the electron–phonon pairing is the primary mechanism for the emergence of superconductivity in these materials. Here, we use the same approach to reveal the charge carrier interaction in highly compressed lithium, black phosphorous, sulfur, and silane. We found that all these superconductors exhibit the dominance of non-electron–phonon charge carrier interaction. This explains the failure to demonstrate the high-T_c values that are predicted for these materials by first-principles calculations which utilize the electron–phonon pairing as the mechanism for the emergence of their superconductivity. Our result implies that alternative pairing mechanisms (primarily the electron–electron retraction) should be tested within the first-principles calculations approach as possible mechanisms for the emergence of superconductivity in highly compressed lithium, black phosphorous, sulfur, and silane.     

Dynamical Casimir effect in a Josephson metamaterial

The zero-point energy stored in the modes of an electromagnetic cavity has experimentally detectable effects, giving rise to an attractive interaction between the opposite walls, the static Casimir effect. A dynamical version of this effect was predicted to occur when the vacuum energy is changed either by moving the walls of the cavity or by changing the index of refraction, resulting in the conversion of vacuum fluctuations into real photons. Here, we demonstrate the dynamical Casimir effect using a Josephson metamaterial embedded in a microwave cavity at 5.4 GHz. We modulate the effective length of the cavity by flux-biasing the metamaterial based on superconducting quantum interference devices (SQUIDs), which results in variation of a few percentage points in the speed of light. We extract the full 4 × 4 covariance matrix of the emitted microwave radiation, demonstrating that photons at frequencies symmetrical with respect to half of the modulation frequency are generated in pairs. At large detunings of the cavity from half of the modulation frequency, we find power spectra that clearly show the theoretically predicted hallmark of the Casimir effect: a bimodal, “sparrow-tail” structure. The observed substantial photon flux cannot be assigned to parametric amplification of thermal fluctuations; its creation is a direct consequence of the noncommutativity structure of quantum field theory.     

Tuning interactions between spins in a superconductor

Novel many-body and topological electronic phases can be created in assemblies of interacting spins coupled to a superconductor, such as one-dimensional topological superconductors with Majorana zero modes (MZMs) at their ends. Understanding and controlling interactions between spins and the emergent band structure of the in-gap Yu–Shiba–Rusinov (YSR) states they induce in a superconductor are fundamental for engineering such phases. Here, by precisely positioning magnetic adatoms with a scanning tunneling microscope (STM), we demonstrate both the tunability of exchange interaction between spins and precise control of the hybridization of YSR states they induce on the surface of a bismuth (Bi) thin film that is made superconducting with the proximity effect. In this platform, depending on the separation of spins, the interplay among Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, spin–orbit coupling, and surface magnetic anisotropy stabilizes different types of spin alignments. Using high-resolution STM spectroscopy at millikelvin temperatures, we probe these spin alignments through monitoring the spin-induced YSR states and their energy splitting. Such measurements also reveal a quantum phase transition between the ground states with different electron number parity for a pair of spins in a superconductor tuned by their separation. Experiments on larger assemblies show that spin–spin interactions can be mediated in a superconductor over long distances. Our results show that controlling hybridization of the YSR states in this platform provides the possibility of engineering the band structure of such states for creating topological phases.

The goal of realizing topological electronic phases using combinations of superconductivity, magnetism, and spin–orbit interaction has motivated efforts in creating spin chains and other magnetic assemblies on the surfaces of superconductors (1–10). There is now considerable evidence that a topological superconducting phase forms in closely packed one-dimensional ferromagnetic chains made of magnetic atoms on the surface of a superconductor with strong spin–orbit coupling, where at the ends of the chains Majorana zero modes (MZMs) have been detected in various scanning tunneling microscope (STM) experiments (11–15). The next step in advancing the study of MZMs in atomic chains is to build chains using STM atomic manipulation techniques starting from single magnetic atoms to be able to systematically probe the topological phase diagram (1, 3, 5–8). These experiments could also make it possible to test the MZMs’ non-Abelian properties, such as their fusion rules, and perhaps ultimately to braid them (16). The key parameters of the topological phase diagram of atomic chains are their spin texture and the bandwidth of their overlapping in-gap Yu–Shiba–Rusinov (YSR) states. Demonstration of atomic manipulation experiments that can control these parameters would open up a wide range of future experiments. While there have been efforts to successfully create closely packed iron chains with nearest neighbor exchange interaction on the surface of superconducting rhenium using STM atomic manipulation (17), a platform that meets the above tunability requirements has not been realized.Here, we show that magnetic atoms on the surface of bismuth (Bi) thin films made superconducting by the proximity effect provides such a platform. The relatively large Fermi wavelength of this surface as compared to its atomic lattice spacing, and its strong spin–orbit interaction make it possible to use atomic manipulation with STM to create spin assemblies with different spin alignments. By fine-tuning the distance between pairs of interacting spins, we have precisely measured the splitting of their YSR states and observed a quantum phase transition (QPT) between phases with different electron number parity tuned by their separation. In larger spin assemblies, we find evidence that noncollinear alignment of spins controls the splitting of YSR states, thereby illustrating the potential for YSR band structure engineering with topological properties in this platform.     

Avoided valence transition in a plutonium superconductor

The d and f electrons in correlated metals are often neither fully localized around their host nuclei nor fully itinerant. This localized/itinerant duality underlies the correlated electronic states of the high-T_c cuprate superconductors and the heavy-fermion intermetallics and is nowhere more apparent than in the 5f valence electrons of plutonium. Here, we report the full set of symmetry-resolved elastic moduli of PuCoGa₅—the highest T_c superconductor of the heavy fermions (T_c = 18.5 K)—and find that the bulk modulus softens anomalously over a wide range in temperature above T_c. The elastic symmetry channel in which this softening occurs is characteristic of a valence instability—therefore, we identify the elastic softening with fluctuations of the plutonium 5f mixed-valence state. These valence fluctuations disappear when the superconducting gap opens at T_c, suggesting that electrons near the Fermi surface play an essential role in the mixed-valence physics of this system and that PuCoGa₅ avoids a valence transition by entering the superconducting state. The lack of magnetism in PuCoGa₅ has made it difficult to reconcile with most other heavy-fermion superconductors, where superconductivity is generally believed to be mediated by magnetic fluctuations. Our observations suggest that valence fluctuations play a critical role in the unusually high T_c of PuCoGa₅.PuCoGa₅ enters the superconducting state below T_c = 18.5 K (1)—an order of magnitude higher than all Ce- or U-based superconductors. This contrast raises the question of whether PuCoGa₅ simply benefits from a higher superconducting-pairing energy scale than its U- and Ce-based relatives or instead, whether PuCoGa₅ is host to a completely different pairing mechanism. In many lanthanide and actinide compounds, the f electrons are nearly degenerate with the conduction band. In addition, the outer f-shell states are close in energy and may support two (or more) nearly degenerate valence configurations (2). In some cases, this valence degeneracy becomes unstable, leading to valence fluctuations and ultimately, a transition to a different valence state as a function of temperature, pressure, and/or doping (3). X-ray and photoemission spectroscopy (4, 5), neutron form factor measurements (6), and theoretical calculations (7) all indicate that PuCoGa₅ is in an intermediate valence state, with the 5f⁶ (Pu²⁺), 5f⁵ (Pu³⁺), and 5f⁴ (Pu⁴⁺) orbitals all residing near the chemical potential and all partially occupied. PuCoGa₅ exhibits no localized magnetic moments (6), and like other plutonium systems, its 5f electrons cannot be treated as fully localized or fully itinerant (4). [Strong Curie–Weiss-like magnetic susceptibility was initially reported for PuCoGa⁵, consistent with a local moment. However, additional susceptibility measurements (including polarized neutron scattering) have not reproduced this result (6).]In contrast, the analogous CeMIn₅ (M = Co, Rh, and Ir) series of superconductors has localized Ce 4f magnetic moments, with a tendency toward antiferromagnetic order (8). These systems reside close to an antiferromagnetic quantum critical point (9), where antiferromagnetic fluctuations are believed to mediate unconventional superconductivity. Because there is no evidence for PuCoGa₅ being in proximity to a magnetically ordered state (10), it is unlikely that magnetic fluctuations are the primary driver of its anomalously high T_c of 18.5 K (11). Valence fluctuations—where the total number of f electrons per ion fluctuates with the conduction band or the configuration of a fixed number of f electrons fluctuates between two or more nearly degenerate f states (sometimes known as orbital fluctuations)—have been proposed as a possible alternative mechanism for superconductivity in several heavy-fermion systems (12–15). Here, we report that PuCoGa₅’s elastic moduli soften over a large temperature range above T_c. Analysis of the observed temperature dependence of the softening, the symmetry channel in which it occurs, and its interplay with the superconducting transition suggests that valence fluctuations are critical for superconductivity in this system.     

High-harmonic spectroscopy of quantum phase transitions in a high-Tc superconductor

We report on the nonlinear optical signatures of quantum phase transitions in the high-temperature superconductor YBCO, observed through high harmonic generation. While the linear optical response of the material is largely unchanged when cooling across the phase transitions, the nonlinear optical response sensitively imprints two critical points, one at the critical temperature of the cuprate with the exponential growth of the surface harmonic yield in the superconducting phase and another critical point, which marks the transition from strange metal to pseudogap phase. To reveal the underlying microscopic quantum dynamics, a strong-field quasi-Hubbard model was developed, which describes the measured optical response dependent on the formation of Cooper pairs. Further, the theory provides insight into the carrier scattering dynamics and allows us to differentiate between the superconducting, pseudogap, and strange metal phases. The direct connection between nonlinear optical response and microscopic dynamics provides a powerful methodology to study quantum phase transitions in correlated materials. Further implications are light wave control over intricate quantum phases, light–matter hybrids, and application for optical quantum computing.

Attosecond technology (1), specifically the process of high harmonic generation (HHG) (2–4), provides an all-optical probe of the microscopic dynamics of atoms, molecules, and solids. Shortly after the first observation of high harmonics in atoms, their generation was understood (4–6) as arising from electron recollision after strong field photoionization and excursion in the continuum. Since the harmonic signal strongly depends on the electron recollision angle and time, high-harmonic spectroscopy (HHS) is a sensitive nonlinear probe of microscopic electronic structure with atomic spatial and suboptical cycle temporal resolution. HHS of solids (7, 8), two-dimensional materials (9, 10), or nanostructured media (11, 12) differs from the gas phase since the optical field–driven electronic wave packet is delocalized over many lattice sites, the wave function depends on the lattice momentum, and a hole has to match the electron’s momentum for recombination to occur (13, 14). Recent experimental efforts extended HHS as nonperturbative probe to quantum materials (9, 10, 15, 16) and to topological insulators (17–19). There have also been several theoretical advances, which suggest using strong fields to probe the physics of Mott insulators (20, 21), alongside the possibility of optically modifying strongly correlated matter (22) and tracking optically induced phase transitions (23), with a recent experiment reported in ref. 24.The sensitivity of HHS to the intricate microscopic details of carriers and lattice predestines HHS to investigate strong interactions and quantum correlations which lead to fascinating new states of matter such as superconductivity. The phase transition into a strongly correlated superconductive state is described by the spontaneous symmetry breaking of the U(1) redundancy when cooling below the critical temperature Tc of the material. As we will show, HHS is a sensitive probe of the dynamic evolution of the superconducting phase transition since the formation of composite bosons by pairing two fermionic spin-1/2 particles (Cooper pairs) changes the distribution of charge carriers, and this sensitively registers in the high harmonic amplitudes and spectral distribution. Pictorially, this is described in SI Appendix, Fig. S1, by a three-step model, consisting of 1) interband excitation process, 2) intraband acceleration, and 3) interband recombination. Pairing below T_c splits the bands by opening a superconducting gap Δ, and in the strongly correlated phase, the three processes of harmonic generation occur within the effective band structure for the Cooper pairs. We will also show that HHS can identify additional phase transitions between quantum phases in the strongly correlated material which are not accessible through the linear optical response, and they are difficult to detect with established methods such as superconducting quantum interference device (SQUID) magnetometry or four-probe transport measurements.A conventional superconductor can be described by the Anderson–Higgs mechanism, which explains that an optical nonlinear response is due to a gapless phase mode (Nambu–Goldstein) and a gapped amplitude mode (Higgs) of the ordering parameter. In the simplest case, and depending on the strength and type of excitation, Boltzmann and Ginzburg–Landau theories (25, 26) predict a second-order response, which mixes with the excitation mode (27, 28), thus the generation of the third harmonic (29). Unconventional high-Tc superconductors are of tremendous interest for a wide range of applications ranging from electronic devices and information processing devices to optical quantum computers and quantum simulators. However, due to their rich landscape of quantum phases and the difficulties of experimental methods to probe the microscopic dynamics, our understanding is still very limited.Among the well-established methods, e.g., transport measurements (30) or magnetic torque measurements (31, 32), photoemission measurements such as angle-resolved photoemission spectroscopy (ARPES) (33, 34) provide direct access to a material’s microscopic carrier distribution and dynamics. The interpretation of such ARPES measurement is, however, complicated by the interpretation of the bulk spectral function and the assumption of independent electron emission despite measuring a strongly correlated electronic state of matter. These are central questions to access the nature of the multibody state, which call for further developments and powerful new tools to aid in the interpretation of the physical mechanism.Therefore, the development of all-optical and ultrafast probes of the macroscopic dynamics inside such materials, which is compatible with existing methods, is highly desirable. To this aim, we apply HHS to investigate the transition between the different phases of the unconventional high-Tc superconductor YBa₂Cu₃O7−d (YBCO). We elucidate the connection between the measured optical spectra, the transition between strange metal and pseudogap phases, and the superconducting phase transition with a strong-field Hubbard model. The HHS measurement clearly shows a departure from the normal conducting phase with an increased formation of Cooper pairs upon cooling. The variation in harmonic orders is linked to phenomenological energy and phase relaxation times, which identify the transition to the fluctuation regime (35, 36), i.e., between the strange metal and pseudogap phases, and the sudden transition at Tc into the superconducting phase. Unconventional superconductors, like YBCO, are material systems in which the formation of composite bosons out of paired fermions is mediated not by phonon exchange but by some other kind of energy exchange (37), for instance, due to spin fluctuations. Such systems present many standing fascinating questions. It is thus important to have new powerful experimental techniques like HHS that provide a fresh and alternate view of the problem.     

Coherent emission from surface Josephson plasmons in striped cuprates

The interplay between charge order and superconductivity remains one of the central themes of research in quantum materials. In the case of cuprates, the coupling between striped charge fluctuations and local electromagnetic fields is especially important, as it affects transport properties, coherence, and dimensionality of superconducting correlations. Here, we study the emission of coherent terahertz radiation in single-layer cuprates of the La_2-xBa_xCuO₄ family, for which this effect is expected to be forbidden by symmetry. We find that emission vanishes for compounds in which the stripes are quasi-static but is activated when c-axis inversion symmetry is broken by incommensurate or fluctuating charge stripes, such as in La_1.905Ba_0.095CuO₄ and in La_1.845Ba_0.155CuO₄. In this case, terahertz radiation is emitted by surface Josephson plasmons, which are generally dark modes, but couple to free space electromagnetic radiation because of the stripe modulation.

Nonlinear terahertz (THz) spectroscopy has recently emerged as a new tool to study the microscopic properties of quantum materials, being susceptible to the symmetry of low-energy degrees of freedom and complementing already-existing nonlinear optical probes (1). For example, THz third-harmonic generation was shown to be a sensitive probe of superfluid stripes, which do not couple to light at linear order but participate in higher-order responses (2, 3). As such, the study of THz nonlinear optics in presence of frustrated couplings provides new opportunities to explore the symmetry of quantum materials. Here, we focus on THz emission from high-TC cuprates and demonstrate how this method is highly sensitive to the spatial arrangement of the superconducting state and its interaction with charge-stripe order.The emission of THz radiation from materials illuminated with femtosecond optical pulses (4–7) is generally enabled by two classes of mechanisms. The first mechanism, active in transparent noncentrosymmetric materials such as ZnTe or LiNbO₃, is based on optical rectification, where the second-order nonlinear optical susceptibility causes a time-dependent electrical polarization (8). The second mechanism relies on the excitation of time-dependent charge currents and is well documented for biased high-mobility semiconductors (8). A number of additional reports of coherent THz radiation have been made for complex quantum materials, typically related to the perturbation of electronic and magnetic interactions. THz emission in colossal magnetoresistance manganites (7, 9, 10), magnetic and multiferroic compounds (11–20), are some of the best-known examples.In the case of high-T_C superconductors, coherent THz emission has been reported only for situations in which time-dependent supercurrents, J˙s(t), are set in (8). These situations range from near-single-cycle THz pulses in biased antennae fabricated from YBa₂Cu₃O_7-δ or Bi₂Sr₂CaCu₂O_8+δ films (5, 21, 22), to multicycle narrowband emissions governed by the Josephson effect in the case of applied out-of-plane magnetic fields (23). It has also been shown that the use of Josephson junction stacks in mesa-type resonant structures allows orders of magnitude increase in THz emission efficiency, also providing narrow bandwidths and tuneable frequency (24–26).Here, we report anomalous THz emission in high-TC cuprates, observed for photoexcitation with femtosecond near-infrared pulses, in absence of external magnetic fields and current biases. The effect is detected only when superconductivity coexists with charge-stripe order in the Cu-O planes (27–30) and when these stripes are either incommensurate with the lattice or fluctuating.We studied cuprates belonging to the “214” family, with one Cu-O layer per unit cell. As a prototypical “homogeneous” cuprate, we considered optimally doped La_2-xSr_xCuO₄ (LSCO), with a critical temperature of 38 K (phase diagram in Fig. 1A). Although in the LSCO family fluctuating striped charge and spin orders have been reported in the underdoped region of the phase diagram (31), there is no evidence for stripes at optimal 0.16 doping (32). This sample was compared with the response of La_2-xBa_xCuO₄ (LBCO), for which superconductivity coexists with charge stripes (27). We focused on three LBCO compounds: La_1.885Ba_0.115CuO₄ (LBCO 11.5%, TC = 13 K), where the superconducting transition is highly depleted by a robust stripe phase below the charge ordering temperature TCO = 53 K, La_1.845Ba_0.155CuO₄ (LBCO 15.5%, TC = 30 K, TCO = 40 K), placed at the nominal optimal doping and characterized by weak, highly fluctuating stripes (27), and La_1.905Ba_0.095CuO₄ (LBCO 9.5%, TC = TCO = 33 K), for which the stripes have an intermediate intensity and correlation length compared with the other two compounds (27), but in contrast to them are here highly incommensurate (33, 34). The location of the three samples in the LBCO phase diagram is shown in Fig. 1 B–D.Open in a separate windowFig. 1.(A–D) Temperature-doping phase diagrams of the four compounds investigated in the present study. TCO, TSO, and TC stand for the charge ordering, the spin ordering, and the superconducting critical temperature, respectively. (E–H) Time-dependent THz emission traces taken for a pump fluence of 2.5 mJ/cm² at the temperatures indicated by full circles in (A–D). Solid lines represent multicomponent fits to the data (SI Appendix). The vertical scales in the three panels are mutually calibrated. (I–L) Fourier transforms (circles) of selected time-domain traces in (E–H). Solid lines are multi-Gaussian fits. Insets: Experimental geometry. Near-infrared (NIR) pump pulses are shone at normal incidence onto an ac-oriented sample surface, with polarization parallel to the c axis. As a result of photoexcitation, c-polarized THz radiation is emitted. Ampl., amplitude.We note that LBCO is the same cuprate in which signatures of optically enhanced superconductivity have been measured (35–38) and attributed to the ultrafast perturbation of the stripe order (39, 40). In addition, a number of nonlinear optical effects, such as THz parametric amplification (41) and third-harmonic generation (2), related to the resonant driving of Josephson plasma waves, have also been measured.The main result of our experiment is summarized in Fig. 1 E–L, where the measured THz emission traces are reported for the four investigated compounds for selected temperatures, at a constant pump fluence of 2.5 mJ/cm². The experimental geometry is shown in the insets of the lower panels. We used the output of an amplified Ti:Sa laser as pump pulses, with a duration of 100 fs and photon energy of 1.55 eV (800 nm wavelength). These were focused at normal incidence onto an ac-oriented sample surface on an ∼ 500 μm spot. The emitted THz pulses were collimated with a parabolic mirror and refocused on a 1-mm-thick ZnTe crystal to perform electro-optic sampling directly yielding THz electric field traces in time domain.In optimally doped LSCO (Fig. 1E), the THz emission signal was measurable only in the superconducting state below TC and displayed a very small amplitude, just above the noise level. This effect consisted of a single-cycle trace, with a flat and featureless spectrum (Fig. 1I). A similar response was also found in LBCO 11.5% (Fig. 1 F and J), where charge stripes are robust, quasi-static, and quasi-commensurate. Here, a barely detectable emission signal was also found for T>TC.On the other hand, in LBCO 15.5% [weak, highly fluctuating, but quasi-commensurate stripes (33, 34) (Fig. 1 G and K)], the THz emission in the superconducting state acquired an appreciable amplitude, with oscillations at a frequency of ∼600 GHz (depending on temperature).In the compound with incommensurate, relatively strong stripes, i.e., LBCO 9.5%, the THz emission amplitude was even higher than LBCO 15.5% and greater by a factor of ∼5–10 compared with LSCO and LBCO 11.5%. Coherent multicycle oscillations were observed (Fig. 1H), corresponding to a narrow spectral peak (Fig. 1L). The frequency of these oscillations shifted to the red with increasing temperature, while also reducing in amplitude and disappearing at TC.The rest of the analysis in this paper is focused on LBCO 9.5%, which yielded the largest signal and highest coherence. We verified that the emission was entirely polarized along the out-of-plane crystallographic axis and could be induced only for a pump polarization aligned along the same direction (SI Appendix).Fig. 2A displays the pump fluence dependence measured at a constant temperature of 7 K. These experimental traces were modeled using fits in time domain (solid lines), for which we report the single components in the SI Appendix. These include a “single-cycle” pulse at early times, which was absent at the lowest fluences and grew quadratically with irradiation, and a quasi-monochromatic, long-lived oscillation, which grew linearly up to about 1 mJ/cm² and tended to saturate for higher excitation fluence (Fig. 2B). This linear trend of the main oscillation is compatible with the impulsive excitation of a coherent mode. In the fluence-dependent behavior of lifetime and oscillation frequency (Fig. 2 C and D), we identify a linear excitation regime where these quantities are weakly dependent on fluence and seem to stabilize at constant values of ∼4 ps and ∼0.45 THz, respectively. In this weak excitation regime, the driven mode parameters are well determined.Open in a separate windowFig. 2.Pump fluence-dependent THz emission in La_1.905Ba_0.095CuO₄ at T = 7 K. (A) Experimental traces taken for different pump fluences (full circles). Solid lines are multicomponents fits to the data, which include a quasi-monochromatic, long-lived oscillation and a “single-cycle” component around time 0 (SI Appendix). (B–D) Fluence-dependent parameters of the quasi-monochromatic oscillation extracted from the fits in (A). Ampl., amplitude; Freq., frequency.In Fig. 3, we report the temperature dependence of this effect. We show a comparison between the oscillation frequency in the THz emission signal in LBCO 9.5%, and the bulk Josephson plasma resonance measured at equilibrium with time-resolved THz spectroscopy in the same sample. In the inset of Fig. 3A, we show the experimental geometry in which we illuminated the sample with weak broadband THz pulses (generated in a 200-μm-thick GaP), polarized along the out-of-plane direction, that were then detected in another 200-μm-thick GaP crystal via electro-optic sampling after being reflected from the sample surface.Open in a separate windowFig. 3.Comparison with the equilibrium Josephson plasma resonance (JPR) in La_1.905Ba_0.095CuO₄. (A, Inset) Experimental geometry for the equilibrium THz time-domain characterization. A weak broadband THz pulse was shone at normal incidence onto the sample surface with polarization along the c direction. The electric field profile of the same THz pulse was then detected after reflection. (A) Reflectivity taken at two different temperatures in the superconducting state, normalized by the same quantity measured at T = 35 K > TC (full circles). The solid lines are fits to the data performed with a JPR model. (B) Temperature dependence of the equilibrium Josephson plasma frequency (gray circles), as determined from the fits in (A). The oscillation frequencies in the THz emission signal measured in the same sample are also reported for two different excitation fluences (legend). Error bars indicate uncertainties extracted from fits such as those in Fig. 2 (see also SI Appendix). Solid lines are guides to the eye.Fig. 3A displays examples of reflectivity ratios at two temperatures below TC, normalized by the same quantity measured in the normal state. These curves evidence a Josephson plasma resonance, the exact frequency of which was determined by fitting the experimental data with a Josephson plasma model (solid lines) (35, 38). The key result of this analysis is displayed in Fig. 3B, in which we show a comparison of the temperature dependence of the Josephson plasma frequency at equilibrium (gray) with the frequency of the emitted oscillations for two pump fluences. Notably, the emitted mode frequency hardens with decreasing fluence and approaches the equilibrium plasma frequency measured at the corresponding base temperature.In interpreting our results, we first note that in a centrosymmetric cuprate, impulsive excitation of Josephson plasmons is forbidden by symmetry. Josephson plasma modes are in fact symmetry-odd (infrared-active), while impulsive photoexcitation couples only to totally symmetric modes (42). As discussed in a related theory work (43), a prerequisite for the excitation of these modes is that charge order breaks inversion symmetry. However, this does not happen for commensurate quasi-static stripes as those expected for dopings x ≳1/8 (33, 34), which exhibit a twofold screw axis along the out-of-plane direction (see Fig. 4A) (44). A symmetry breaking is expected instead for incommensurate or highly fluctuating stripes, as in the case of LBCO 9.5% and LBCO 15.5%. Here, the charge order correlation length along the out-of-plane direction is of the order of one unit cell (27), resulting in a loss of the phase relation between stripes in next-nearest-neighboring planes (Fig. 4B).Open in a separate windowFig. 4.(A) Charge density pattern (gradual scale of blue) in three neighboring planes of a cuprate with commensurate stripes (red dashed lines are spaced by 4a, where a is the lattice constant). Stripes in next-nearest layers are off-phased by π (30). Here, inversion symmetry is preserved (black dot and vertical dashed line are an inversion center and a screw axis, respectively). (B) Once commensurability is lost, stripes are fluctuating, or there is no phase relation between next-nearest layers, inversion symmetry can be broken, and THz emission is enabled (43). (C, Left) In-plane dispersion of bulk Josephson plasma polaritons (red line). Emission from these modes (Right) is expected to be very broad, as it encompasses a wide range of in-plane momenta, qx (gray shading) (43). (D, Left) Out-of-plane dispersion of surface Josephson plasmons (solid blue line). These modes are localized at the surface and propagate along z (out-of-plane direction). As their dispersion lies below the light cone (red shading), they are not expected to radiate into vacuum. However, Bragg scattering off the stripe order induces a backfolding, defined by the stripe wave vector, Qstripes, into a reduced Brillouin zone (dashed horizontal line). Hence, these surface modes are redirected into the light cone and can radiate out at frequencies just below ωJPR(q=0). (Right) Calculated emission spectrum from surface Josephson plasmons in a striped superconductor (43). JPR, Josephson plasma resonance.Once inversion symmetry is broken, electromagnetic emission at a frequency ω≪ωpump can result from rectification of the optical pulse. We associate the optically rectified drive for plasma oscillations with the excitation of a shift current (43, 44) at the sample surface. This is expected to interact with modes at ω≃ωJPR, of which one finds at least two: 1) a bulk Josephson plasma polariton, sustained by tunneling supercurrents oriented in the z (out-of-plane) direction and propagating along the x (in-plane) direction; and 2) a surface Josephson plasmon, also sustained by plasma oscillations in the z direction, but localized at the surface of the material and propagating along z. The dispersion relations for these two modes are shown in Fig. 4 C and D, respectively (45).Radiation from bulk plasma polaritons (Fig. 4C), excited over a depth between ∼200 nm (skin depth of the pump) and ∼1 μm (46), would be expected to be broad in frequency and overdamped. This is because excitation by the near-infrared pump covers a wide range of in-plane momenta, qx, which in the first instance, is limited only by the envelope bandwidth of the pump pulse (gray shading in Fig. 4C). The spectrum of Josephson plasmons would, in this case, also be independent of the details of the stripe order and of its correlation lengths, as is instead observed. Moreover, one would expect radiation at frequencies ω≳ωJPR, in contrast to the experimental observation of a slightly redshifted emission with respect to the plasma frequency (Fig. 3B).Coherent narrowband emission by surface Josephson plasmons is instead more likely. Although the dispersion of these modes lies below the light cone and, hence, they are not expected to radiate into vacuum (Fig. 4D), we argue here that Bragg scattering off the stripe order induces a backfolding, defined by the stripe wave vector, into a reduced Brillouin zone (dashed horizontal line in Fig. 4D). For this reason, these surface modes can radiate, much like a situation in which a fabricated corrugation would be used to achieve the coupling (47–50).In the right panel of Fig. 4D, we report the emission spectrum calculated for a striped superconductor through the excitation of surface Josephson plasmons. As extensively discussed in our related theory work (43, 44), in the presence of stripes, the pump pulse is expected to give origin to an Umklapp shift current, JUcos(Qstripesz), that is modulated in space by the stripe wave vector, Qstripes. This naturally drives high-momenta surface plasmons, which can radiate out due to the aforementioned backfolding mechanism.In summary, we have reported the observation of coherent THz emission just below the Josephson plasma frequency in cuprates for which the superconducting state coexists with stripes. We assigned this effect to the excitation of surface Josephson plasmons, which become Raman active due to the breaking of inversion symmetry induced by the stripes and can radiate out thanks to the backfolding of their dispersion curve onto the light cone. Based on these findings, the characterization of coherent THz emission emerges as a sensitive method to unveil broken symmetry states, which may not be detectable with other conventional techniques. Moreover, the absence of THz emission in LBCO 11.5%, where the stripes are more robust and quasi-static, may suggest a qualitative difference in the nature of charge and spin order between compounds that are in the vicinity of the commensurate 1/8 doping and those that are far from it.     

Phonon mechanism in the most dilute superconductor n-type SrTiO3

Superconductivity of n-doped SrTiO₃, which remained enigmatic for half a century, is treated as a particular case of nonadiabatic phonon pairing. Motivated by experiment, we suggest the existence of the mobility edge at some dopant concentration. The itinerant part of the spectrum consists of three conduction bands filling by electrons successively. Each subband contributes to the superconducting instability and exhibits a gap in its energy spectrum at low temperatures. We argue that superconductivity of n-doped SrTiO₃ results from the interaction of electrons with several longitudinal (LO) optical phonons with frequencies much larger than the Fermi energy. Immobile charges under the mobility edge threshold increase the “optical” dielectric constant far above that in clean SrTiO₃ placing control on the electron–LO phonon interaction. T_C initially grows as density of states at the Fermi surface increases with doping, but the accumulating charges reduce the electrons–polar-phonon interaction by screening the longitudinal electric fields. The theory predicts maxima in the T_C-concentration dependence indeed observed experimentally. Having reached a maximum in the third band, the transition temperature finally decreases, rounding out the T_C (n) dome, the three maxima with accompanying superconducting gaps emerging consecutively as electrons fill successive bands. This arises from attributes of the LO optical phonon pairing of electrons. The mechanism of LO phonons opens the path to increasing superconducting transition temperature in bulk transition-metal oxides and other polar crystals, and in charged 2D layers at the LaAaO₃/SrTiO₃ interfaces and on the SrTiO₃ substrates.In the rapidly growing field of transition-metal-oxide electronics (1) SrTiO₃ is a key material with the unique physical properties. A broadband insulator, SrTiO₃ is the rare case of a paraelectric with the ferroelectric transition aborted by the quantum fluctuations and is remarkable, in particular, for a large dielectric constant that at helium temperatures reaches the enormous value of 25,000 (2).A number of features in the low-temperature behavior of SrTiO₃ are still poorly understood, primarily its electronic properties. Most challenging, perhaps, is the nature of superconductivity in doped SrTiO₃. Upon doping (via a chemical substitution or reducing the oxygen content), SrTiO₃ displays low-temperature superconductivity already at concentrations of doped electrons as low as n_s ≈ 5.5 × 10¹⁷cm^?3; as such, doped SrTiO₃ was dubbed the “most dilute superconductor” (3). At LaAlO₃/SrTiO₃ interfaces the electrons form a metallic layer on the SrTiO₃ side that displays 2D ferromagnetism and superconductivity (4, 5). Further, a 1-unit-cell-thick layer of FeSe on the surface of SrTiO₃ becomes superconducting at the record temperatures T_C ≥ 100?K (6). These and other examples suggest that the mechanism of superconductivity in SrTiO₃ might differ from that in the weak-coupling BCS (Bardeen, Cooper, Schrieffer) phonon model. Below we focus on superconductivity in bulk doped SrTiO₃ and argue that it owes its origin to interaction between electrons and longitudinal (LO) polar phonons with frequencies far exceeding the Fermi energy. Implications for other polar crystals and 2D superconductivity of the atomic-thick FeSe layer on substrates of SrTiO₃ are briefly mentioned.Superconductivity in SrTiO₃, first discovered in 1964 (7), was supposed initially to be another case of the phonon-mediated Cooper pairing generalized to the case of doped multivalley semiconductors (8). The limit of a small number of carriers is known to present a special challenge for theory as at low doping the Fermi energy E_F is small.The dimensionless coupling parameter of the BCS theory λ_BCS is proportional to the product of a matrix element of electron–phonon scattering V_e?ph and density of states v(E_F) at the Fermi surface: λ_BCS ∝ V_e?ph × ν(E_F). Whereas the former is of atomic scale, as in ordinary metals, ν(E_F) is small. It was therefore suggested in ref. 9 that in polar semiconductors the long-range interaction with an LO optical mode can compensate the smallness of density of states. After the analysis the authors, however, came to the conclusion that this mechanism may be effective only if the frequency ω_LO of such phonons is smaller than the Fermi energy.Understanding of superconductivity in the doped insulator SrTiO₃ has met with additional difficulties (for a brief review see refs. 10, 11). Thus, the popular multivalley model (8) turned out not to be relevant as in the energy spectrum of the cubic SrTiO₃ there is only one minimum at the Γ-point built on the titanium 3dt_2g levels (11–13).That the Debye temperature for strontium titanate is comparable with or, at low enough doping, even larger than E_F, was probably first stressed in ref. 10.Recent experiments (3, 14, 15) demonstrated that upon doping SrTiO₃ passes several stages before at last reaching at high enough concentration a metallic ground state. We show below that taking this evolution into consideration is necessary for understanding the low-temperature physics in doped SrTiO₃. In what follows, we explore interactions between the electrons doped into the conduction band and LO phonons. One vital difference from ref. 9 is that there are four LO phonons at the Γ-point in SrTiO₃ [in the cubic phase (16)]. The other one comes about from the fact that the presence of immobile charges embedded via the doping process significantly affects the dielectric characteristics of the material and, hence, the matrix element for the interactions of electrons with LO phonons. Most of the results below are obtained analytically. Among other things, the theory predicts the appearance of maxima in the dependence T_C(n_s) on the number of carriers n_s, providing the explanation for one of the most intriguing experimental findings (14).As is well known, in metals the weak-coupling BCS theory is generalized to the case of arbitrary strong electron–phonon interactions in the Migdal–Eliashberg equations (17, 18).As the frequencies of the LO-polar phonon modes in bulk pure SrTiO₃ are known and are high (19), electron–phonon interactions in doped SrTiO₃ present the extreme case of the “antiadiabatic” limit ω₀ ? E_F. We argue that apart from the self-evident significant interest in the realization of such extreme conditions specifically in the doped strontium titanate, the notion of superconductivity mediated by phonons with a frequency higher than or of the same order as the Fermi energy calls for more general exploration of peculiarities inherent in nonadiabatic pairing mechanisms.One part of the problem concerns competition between the Coulomb repulsion and the phonon-mediated attraction. From the condition ω₀/E_F ? 1 in the adiabatic regime follows the “retardation” effect of the BCS theory: Two electrons of the pair evade each other, being a distance d apart, on the order of v_F/ω₀ larger by a factor E_F/ω₀ ? 1 than the atomic scale a ≈ 1/p_F. Effectively, this decreases the role of the Coulomb repulsion because the latter is screened on the atomic distances.In the opposite limit of ω₀/E_F ? 1 electrons of the pair feel both the Coulomb repulsion and the potential of the local lattice distortion instantaneously. For the Cooper pair to be formed the strength of the phonon attraction must exceed the direct Coulomb repulsion. For the general theory of electron–phonon interaction in the nonadiabatic case the immediate difficulty is that the customary mathematical apparatus of the Migdal–Eliashberg equations is not applicable at ω₀/E_F ≥ 1; the case of doped SrTiO₃ is of particular value where the most significant peculiarities proper to a nonadiabatic superconductivity can be deduced in the logarithmic approximation.     

Signatures of two-dimensional superconductivity emerging within a three-dimensional host superconductor

Spatial disorder has been shown to drive two-dimensional (2D) superconductors to an insulating phase through a superconductor–insulator transition (SIT). Numerical calculations predict that with increasing disorder, emergent electronic granularity is expected in these materials—a phenomenon where superconducting (SC) domains on the scale of the material’s coherence length are embedded in an insulating matrix and coherently coupled by Josephson tunneling. Here, we present spatially resolved scanning tunneling spectroscopy (STS) measurements of the three-dimensional (3D) superconductor BaPb_1−xBi_xO₃ (BPBO), which surprisingly demonstrate three key signatures of emergent electronic granularity, having only been previously conjectured and observed in 2D thin-film systems. These signatures include the observation of emergent SC domains on the scale of the coherence length, finite energy gap over all space, and strong enhancement of spatial anticorrelation between pairing amplitude and gap magnitude as the SIT is approached. These observations are suggestive of 2D SC behavior embedded within a conventional 3D s-wave host, an intriguing but still unexplained interdimensional phenomenon, which has been hinted at by previous experiments in which critical scaling exponents in the vicinity of a putative 3D quantum phase transition are consistent only with dimensionality d = 2.

In the two-dimensional (2D) limit of the superconductor–insulator transition (SIT) (1–3), there is evidence of emergent electronic granularity in numerical simulations (4–6) and thin-film experiments (7–10). It has been recently discovered (11, 12) that bulk three-dimensional (3D) BaPb_1−xBi_xO₃ (BPBO) (Fig. 1A) exhibits a magnetic field–tuned SIT with unexpected 2D scaling which hints at a “hidden” two dimensionality in the superconductivity. Lacking further experiments, the origin of this interdimensional order was not clear. For example, the phase could be emergent due to increased interactions and localization near the SIT. Such unusual scaling behavior has also been connected to a partially disordered stripe-like nanoscale structural phase separation due to the fact that the stripe width matches the superconducting coherence length at the Bi composition which maximizes T_c (the optimal doping point) (13, 14). Additionally, the role played by the high-temperature dimorphic nature of BPBO (two coexisting crystal structures with different electronic properties) in the composition range x = [0.18−0.30] (Fig. 1B) on the low-temperature electronic structure is not known. This possible competition between emergence, macroscopic electronic properties, and microscopic structure is fascinating, but a detailed study of the local electronic superconducting and localization properties and their connection to disorder potential was necessary to provide further insight.Open in a separate windowFig. 1.BPBO structure and electronic phase diagram. (A) Perovskite-like crystalline structure of three-dimensional BPBO single crystal. Orthorhombic and tetragonal phases result from rotational instabilities of the O₆ units. (B) Electronic phase diagram of BPBO as a function of temperature and Bi composition (x), exhibiting a superconducting dome peaked at x = 0.25 (optimal doping). Critical temperature (T_c) values were obtained from measurements of resistivity (full and open squares) and magnetic susceptibility (full circles) (11). Compositions studied in this work are highlighted in green (x = 0.19), blue (x = 0.25), and red (x = 0.28). (C) Diagram summarizing coherence peak height (h_cp, a measure of SC pairing amplitude), energy gap size (Δ), zero-bias conductance (g0, a measure of disorder potential), and topography (z, a measure of physical disorder) from our spatially resolved observations for BPBO near the SIT (x = 0.28). The inset shows anticorrelation between measured features as a function of Bi composition. Anticorrelation between Δ and h_cp (squares) increases near the SIT, while anticorrelation between Δ and g0 (circles) and between h_cp and g0 (triangles) remain low.Scanning tunneling microscopy and spectroscopy (STM/STS) enable a local probe of the electronic properties through measurements of local density of states (LDOS) along with topographic information. This technique has been used to study the LDOS homegeneity in quasi-2D superconductors (SC), including high-T_c superconductors (HTSC) (15–17) and 2D thin-film SC systems (10, 18–20). STS reports that examine the influence of disorder on the SC state in 3D systems are limited (21, 22).Motivated by these challenges, we performed topographic and spatially resolved LDOS measurements of BPBO for three Bi compositions, x = 0.19, 0.25, and 0.28, which we refer to as x_low, x_opt, and x_high, respectively. All measurements were performed at 4.4 K, well below T_c for all dopings measured; hence, we are exploring the component of the phase diagram consisting of the superconducting dome as we approach the SIT from under-doping through optimal doping to over-doping (Fig. 1A). Our results are summarized in Fig. 1C, which show spatially resolved properties for x = 0.28, the Bi composition closest to the SIT. The electronic properties extracted from LDOS measurements (coherence peak height h_cp and gap size Δ) show a distinct spatial anticorrelation and have a characteristic length scale ξ, which we show is very close to the calculated BPBO coherence length. These properties are uncorrelated with measurements of disorder potential g0 and topography z. These observations are all consistent with an emergent electronic granularity at x = 0.28. The spatial correlations between properties for all Bi compositions are summarized in the anticorrelation plot in Fig. 1C.dI/dV measured along a line for the three Bi compositions is shown in Fig. 2, providing an overview of the inhomogeneity analyzed in the rest of the paper. The spectra contain SC gaps of varying type, ranging from those with strong coherence peaks (yellow) to those with muted coherence peaks (red). The gap size Δ, defined as half the peak-to-peak distance, is measured utilizing derivative analysis to determine maxima. For optimally doped samples (Fig. 2B), the spectra are mostly symmetric with strong coherence peaks. In contrast, spectra measured for samples away from x_opt tend to be asymmetric with coherence peaks less well defined (Fig. 2 A and C). The gap size and coherence peaks are strongly inhomogeneous for x_high. These spatial variations in the magnitude of Δ have been explored before in STS studies of HTSCs (15, 23, 24) and SC thin films (8–10, 18–20, 25) but thus far not reported in a conventional 3D SC system.Open in a separate windowFig. 2.Spatial variation snapshots of the local density of states. Measurements of dI/dV along a line of length 15 nm for x = 0.19 (A), x = 0.25 (B), and 40 nm for x = 0.28 (C) (I = 200 pA). Upper insets show corresponding position in the x−T_c phase diagram and the spatially-averaged dI/dV curve. STS spectra show symmetric gap with (yellow spectra) and without (red spectra) coherence peaks at the gap edge. Each lower color snapshot plots all dI/dV spectra acquired along the line, but here spectra are stacked up along the vertical axis and arranged by increasing gap magnitude. For each snapshot, the horizontal axis matches the sample bias scale for the corresponding linecut (A–C), and dI/dV magnitude is represented by a color scale ranging from black through orange up to yellow (regions outside the gap are identified in light blue). In this visualization, gap edges with strong coherence peaks are highlighted in yellow, gap edges with weak coherence peaks appear orange, and the gap region appears in dark colors.The gap (Δ) and normalized coherence peak height (h∼cp) variations are quantified by the histograms plotted in Fig. 3 A and B. The coherence peak height h_cp is determined by measuring the average of the coherence peak height at positive and negative bias relative to a curve passing through the high bias background (see Fig. 5A for schematic). That value is then normalized by the highest h_cp value to determine h∼cp. The most frequent Δ and h∼cp values were found by fitting the histograms to Gaussians and are shown with corresponding bars depicting the SDs in Fig. 3 C and D, respectively. Here we observe that both Δ and h∼cp as a function of Bi composition follow a trend similar to the dome-like behavior of the critical temperature T_c (see inset between Fig. 3 C and D). This is in stark contrast to other SC systems such as hole-doped cuprates where measured gap with doping does not appear to follow T_c (15). Since the coherence peaks are directly associated with the phase coherence in the superconductor (5, 6, 8, 26), the height of the coherence peaks provides a direct measure of the local SC order parameter phase stiffness. The obtained h∼cp dome-like behavior (Fig. 3D) confirms an increase of SC disorder in BPBO (loss of global superconductivity) for Bi compositions away from x_opt.Open in a separate windowFig. 3.Statistics of the gap size and coherence peak inhomogeneity. (A and B) Histograms of Δ and h∼cp respectively obtained from STS for samples with x = 0.19, 0.25, and 0.28. A Gaussian has been fit to each histogram to quantify the mean gap value and the gap variation. Since Δ for x = 0.28 and h∼cpfor x = 0.25 are so asymmetric, two separate Gaussians were fit for each side of the data (only one shown in the case of gap). Values of h_cp were normalized to the highest measured value. (C and D) Gap size and coherence peak height as a function of Bi concentration obtained from histogram in A and B, with corresponding bars denoting SD. Δ and h∼cp values versus x follow a dome-like shape matching the x-versus-T_c phase diagram (inset). However, strong asymmetry for Δ variations for x = 0.28 is observed (red bar), biased toward higher gap values (dashed black line).Open in a separate windowFig. 5.Correlation between superconducting gap and pairing amplitude. (A) Representative spectra taken at points α, β, and γ in B (or C equivalently). Spatial variation of SC gap (B) and coherence peak height (C) for Bi composition x = 0.28 (9 × 9 nm²). An anticorrelation between Δ and h∼cp is clearly observed for this Bi composition closest to the SIT. (D) Two-parameter histogram of Δ and h∼cp for x = 0.19 (green color scale), x = 0.25 (blue color scale) and x = 0.28 (red color scale). Cross-correlation values r for each composition are displayed in green (+0.1), blue (−0.14), and red (−0.42).According to the observed spatial gap variation, which is widely understood as a measure of SC disorder (25, 27–29), SC disorder in BPBO increases considerably for samples with Bi composition away from x_opt. From x_opt to x_high, variations in gap size range from 35 to 243%. At x_high, there is an important asymmetry in the variation of Δ (red histogram in Fig. 3A), weighted toward higher values. Observed asymmetry in Δ variations for x_high suggests that disorder displaces Δ toward higher values (instead of toward lower values as a SC dome would indicate). This leads to a SC phase half-dome shape similar to the one observed for Ba_1−xK_xBiO₃ (30–33).The normalized SC gap, 2Δ/kBTc, where kB is the Boltzmann constant, is the best indicator of the electron–phonon coupling strength within mean field theory (34). For a superconductor with strong coupling, 2Δ/kBTc is larger than the value expected for BCS theory in the weak coupling limit, 3.52 (35). The value of the electron–phonon coupling strength for each doping, according to our tunneling spectroscopy data combined with critical temperatures obtained from resistivity and magnetic susceptibility measurements (Fig. 1B), is 2Δ/kBTc ∼ 9, 9.7, and 11.6 for x = 0.19, 0.25, and 0.28, respectively. This trend might be due to a progressive increase in electron–phonon coupling strength as function of Bi composition. A similar increase has been seen in BPBO using macroscopic point-contact tunneling measurements (36, 37). However, those reports showed a 2Δ/kBTc value of 2.3 for x = 0.18 that saturates to the BCS value above x = 0.22.These results can be understood in terms of disorder. For strongly disordered superconductors, Δ is expected to be much larger than the pairing field which leads (when moving toward the SIT) to a lower macroscopic T_c value that is no longer determined by Δ (19). For BPBO samples near the SIT, macroscopic T_c values (11, 36, 38, 39) would be then suppressed due to phase incoherence effects originated in the spatial disorder. Hence controlling disorder would be a key to increasing T_c in this material, which in turn would decrease the 2Δ/kBTc ratio, approaching the BCS estimate. Similar interpretation of an anomalously large 2Δ/kBTc has been explored in NbN (8) and InO (19) thin films.To visualize spatial variations, we map the SC gap values for the three Bi compositions over nanometer-scale areas (Fig. 4). The color bar below each map shows the range of Δ measured for that particular composition, which varies from 1.1 to 12 mV across all compositions. Percentage variation with respect to the mean gap value as previously seen in Fig. 3A is represented by the right color scale bar. Large spatial variations of Δ were found for x_high (Fig. 4C), in contrast to the more homogeneous gap values measured for x_opt doping (Fig. 4B). The characteristic length scale of the gap inhomogeneity for x = 0.28 is measured to be 4.5 ± 0.5 nm according to spatial correlation analysis. This is comparable to the Ginzburg–Landau coherence length ξ (∼5 nm, obtained from upper critical field Hc2(T) measurements through the standard Werthamer–Helfand–Hohenberg approximation) for the same doping, in addition to being similar to the length scale of structural phase separation, ∼8 nm (11, 13).Open in a separate windowFig. 4.Spatial gap inhomogeneity in the superconducting dome. Map of spatial variations of the SC gap of BPBO, deduced from STS experiments for different Bi concentrations: A, x_low (15 × 15 nm²), B, x_opt (15 × 15 nm²), and C, x_high (9 × 9 nm²) (I = 100 to 200 pA). Overall gap size variations range from 1.6 to 6.6 mV. Corresponding gap values are represented by the color scale bar below each map. The right color scale bar is defined as percentage variation of gap size with respect to the mean gap value (0% or white). Areas with colors ranging from green to yellow correspond to points where gap size exceeds the mean Δ value, while colors ranging from light to dark blue are associated with gaps below the mean Δ value.The observed spatial inhomogeneity in the SC gap is unrelated to topographic disorder as measured by STM. Due to the 3D nature of the crystal, topographic images show a highly disordered surface for all Bi compositions (SI Appendix, Fig. S1) with characteristic length scales ranging from atomic level up to ∼3 nm. For x_high, both the topography (z) and zero-bias conductance (g0) map are uncorrelated with the observed SC electronic disorder (SI Appendix, Fig. S2).Recent zero-field-cooled magnetic susceptibility measurements of the SC volume fraction (12) show a maximum value of 45% for x = 0.25, which was attributed to structural dimorphism. A higher SC volume fraction of around 70% was measured in a different study (40). Our local measurements show that even though the local gap size is highly inhomogeneous for Bi compositions close to the SIT, there exists a finite gap over all space.It was previously mentioned that Δ and h∼cp values follow the same dome-like behavior of T_c with varying Bi composition, but we now seek further information connecting these two parameters from a spatially resolved point of view. This analysis uncovers a striking behavior: Δ and h∼cp become strongly spatially anticorrelated for Bi compositions near the SIT (Fig. 5). Comparing three representative dI/dV spectra (Fig. 5A, acquired from positions marked α, β, and γ in Fig. 5B from an x_high sample), strong variations in h_cp and in Δ can be seen. In fact, the magnitudes of these two values are visibly anticorrelated. To highlight this anticorrelation, Δ and h∼cp maps for x = 0.28 are plotted in Fig. 5 B and C. The anticorrelation for x_high is quantified in a two-parameter histogram (Fig. 5D) that shows a count distribution with negative slope and a cross-correlation value of r = −0.42, indicating a robust anticorrelation. In contrast, very weak correlation coefficients are found for x = 0.19 and x = 0.25, with r = +0.1 and r = −0.14, respectively (See SI Appendix, Fig. S3 for all maps). In addition, zero-bias conductance measurements, a measure of potential disorder (41), show an even weaker correlation with Δ and h∼cp (r = −0.1 and r = 0.01, respectively) (SI Appendix, Figs. S3 and S4).The signatures presented here suggest that an emergent electronic granularity is present in BPBO at x_high. This has been theoretically predicted to occur in 2D superconductors (4–6, 42) and experimentally observed in 2D systems, such as InO (7, 19), NbN (8, 25, 43), and TiN (2, 18) thin films, where temperature is the parameter used to tune disorder in analogy to how Bi composition acts in our system. In this picture, SC domains in BPBO close to the SIT, which are separated by insulating areas with size comparable to the coherence length, become correlated through coherent Josephson tunneling of Cooper pairs between the SC domains. This would lead to the apparent absence of insulating domains in our local spectroscopic measurements. One possible interpretation of SC gap and coherence peak height maps for x = 0.28 is that areas with wider gap values and smaller coherence peak height (red areas in Fig. 5B and blue areas in Fig. 5C) can be associated with insulating domains (which show a gap due to proximity effects) that are separated by SC domains (smaller gap values and larger coherence peak height in Fig. 5 B and C), although the one-to-one identification might not be as simple as suggested by the anticorrelation relation between gap size and coherence peak height. The h∼cp map (Fig. 5C) shows a granular structure with a characteristic length scale of 4.4 ± 0.5 nm, in close agreement with the length scale for the Δ spectral map quoted above. Both of these length scales are comparable to the calculated coherence length, a feature shared by systems with emergent electronic granularity (44). Additionally, our observation of a finite gap over all space supports this picture. Finally, the anticorrelation between gap size and coherence peak height (associated with pairing amplitude) that occurs most strongly at x = 0.28 provides a third signature of emergent electronic granularity.We contextualize this interpretation within the framework of other intertwined orders such as a relationship between the superconductivity and structural polymorphism [studied previously using high resolution TEM (13)] and a proximal charge density wave (CDW) phase. The similarity between electronic length scales, coherence length, and structural polymorph segregation length scale (13) is enigmatic. However, no strong correlation between the gap size and the two measures of spatial disorder (topography z, potential disorder g0) are observed (SI Appendix, Figs. S3 and S4), suggesting that the electronic length scales are emergent. Notably, the structural polymorphs are not identified via STM topography, although the crystal is viewed through the disordered surface. Interestingly, our BPBO samples are in close proximity to a competing CDW phase. Such a CDW phase—if coexistent with superconductivity—would manifest itself with an energy gap, which creates a possible ambiguity in our data: Are the gaps (especially those with small coherence peaks) due to superconductivity or a charge density wave phase? Evidence favors the superconductivity interpretation. First, CDW order has only been observed in the normal state and not simultaneously in the superconducting state, although coexistence was conjectured due to overlapping crystal dopings (45). However, and most significantly, typical CDW gaps in bismuthates are almost an order of magnitude larger than the gaps observed in this work (45–47).Although BPBO is expected to be an s-wave superconductor, it apparently shares some phenomenology of d-wave layered superconductors such as anticorrelation between Δ and h_cp and the “V”-shape of dI/dV spectra seen in Figs. 5A and ​and22 (typical of underdoped cuprates) (48). This d-wave resemblance has been discussed before for s-wave SC films (4) in terms of gap persistence across phase transitions. However, the role of disorder over this phenomenology, mainly driven by the proximity to a Mott insulator, is expected to be secondary. Additionally, our data show how the pairing amplitude h_cp for x = 0.28 covers a very wide range of values including being completely suppressed for the largest gap values, in contrast to previously reported data from cuprates (48) where such behavior is not as extreme. Gap variations are also extreme at x = 0.28, significantly larger than the reported values for cuprates (15).In conclusion we found in BPBO—a bulk 3D SC material—three signatures expected for 2D SC systems near a SIT: evidence for an emergent electronic granularity on the scale of the SC coherence length ξ, finite energy gap over all space, and strong spatial anticorrelation between energy gap Δ and pairing amplitude h_cp. The length scale of electronic disorder is uncorrelated with local structural disorder as measured by STM, supporting the emergent nature of the observed granularity. Our observations suggest a complex interplay between the low-temperature superconductivity, the electronic disorder landscape, high-temperature microstructure, and localization.The observed crossover in length scales for electronic granularity and ξ could possibly explain the previously reported but unexplained 2D critical scaling at a quantum phase transition in BPBO. There has been recent work in one-dimensional (1D) disordered superconducting nanowires showing that 2D scaling anomalously provides superior “scaling collapse,” which may point to a reverse microscopic phenomenon (49). On the other hand, it was explicitly noted that for BPBO, the scaling data for all of the compositions studied could not be fit to a 3D quantum phase transition and ruled out any temperature-dependent prefactor T−(d−2)/z in the resistivity; hence, d=2 provided the best scaling for more than three orders of magnitude in the scaling variable (11). We note that recent theoretical studies of duality between superconducting and superinsulating phases enable a 3D generalization of 2D emergent granularity (50, 51).Local pairing amplitude measurements for x_high samples are larger than expected based on the macroscopic T_c values, which implies that this material has a higher intrinsic T_c that is limited by disorder. Statistical analysis of spatially resolved measurements suggests that the SIT is determined by phase rather than amplitude fluctuations. Our results also raise the question of whether the origin of V-shape conductance spectra is due to d-wave pairing as in the case of cuprates, or simply due to electronic disorder. Notably, the 3D superinsulator phase has been conjectured to be associated with the pseudogap phase of high-temperature superconductors (52). Very recently, numerical calculations have suggested that a disordered superconductor driven through the 3D Anderson transition shares some of the 2D phenomenology, including strong spatial fluctuations and enhancements of the order parameter, although only in the weak-coupling limit, and specific connections to local variables accessible via STM/STS are out of reach (53). Our work presents a detailed study of local effects of disorder in a 3D SC system, and we expect that it will be a first step in understanding the interdimensional interconnection in local SC behavior between 2D and 3D quantum materials.     

Direct Observation of the Spin Exciton in Andreev Spectroscopy of Iron-Based Superconductors

Quasiparticle excitations provide viable information on the physics of unconventional superconductors. Higgs and Leggett modes are some of the classic examples. Another important bosonic excitation is the spin exciton originating from the sign-changing superconducting gap structure. Here we report a direct observation of the temperature-dependent spin exciton in the Andreev spectra of iron-based superconductors. Combined with the other experimental evidence, our observation confirms the extended s-wave (s±) order parameter symmetry and indirectly proves the spin-fluctuation mechanism of Cooper pairing.     

On Molecular Dynamics and Charge Transport in a Flexible Epoxy Resin Network

An epoxy based on diglycidyl ether of bisphenol A was reacted with a long-chain poly(oxypropylene diamine) hardener in the presence of an accelerator, resulting in a flexible epoxy network. Tensile properties were tested as a function of accelerator concentration. All systems exhibited high levels of extensibility, with strain at failure values in excess of 65%. Molecular dynamics in a formulation containing 10 phr of accelerator were then examined using dielectric spectroscopy over the temperature range of 103–433 K. At low temperatures, a molecular relaxation process (γ relaxation) was observed and shown to conform well to both the Arrhenius equation and activated tunnelling. A stronger relaxation appeared (203–303 K) just before the onset of charge transport, which dominated the behaviour at higher temperatures. The former takes an unusual bimodal form, which we consider a result of overlapping β and α relaxations, consequently termed αβ mode. Analysis of this mechanism revealed a Vogel–Fulcher–Tammann (VFT) behaviour. The temperature-dependent DC conductivity, σ_DC (deduced from the low-frequency charge transport contribution to ε_r″), also revealed VFT behaviour with an onset statistically equivalent to that of the αβ mode, therefore suggesting that charge transport, at this temperature regime, is strongly affiliated with cooperative molecular motion.     

Anomalous superfluid density in quantum critical superconductors

When a second-order magnetic phase transition is tuned to zero temperature by a nonthermal parameter, quantum fluctuations are critically enhanced, often leading to the emergence of unconventional superconductivity. In these “quantum critical” superconductors it has been widely reported that the normal-state properties above the superconducting transition temperature T_c often exhibit anomalous non-Fermi liquid behaviors and enhanced electron correlations. However, the effect of these strong critical fluctuations on the superconducting condensate below T_c is less well established. Here we report measurements of the magnetic penetration depth in heavy-fermion, iron-pnictide, and organic superconductors located close to antiferromagnetic quantum critical points, showing that the superfluid density in these nodal superconductors universally exhibits, unlike the expected T-linear dependence, an anomalous 3/2 power-law temperature dependence over a wide temperature range. We propose that this noninteger power law can be explained if a strong renormalization of effective Fermi velocity due to quantum fluctuations occurs only for momenta k close to the nodes in the superconducting energy gap Δ(k). We suggest that such “nodal criticality” may have an impact on low-energy properties of quantum critical superconductors.     

Charge Injection and Dielectric Characteristics of Polyethylene Terephthalate Based on Semiconductor Electrodes

Employing a novel semiconductor electrode in comparison with the traditional semiconductor electrode made of polyethylene/ethylene-vinyl-acetate copolymer/carbon-black (PE/EVA/CB) composite, characteristic charge carriers are injected into polyethylene terephthalate (PET) as a polymer dielectric paradigm, which will be captured by specific deep traps of electrons and holes. Combined with thermal stimulation current (TSC) experiments and first-principles electronic-state calculations, the injected charges from the novel electrode are characterized, and the corresponding dielectric behavior is elucidated through DC conductance, electrical breakdown and dielectric spectrum tests. TSC experiments with novel and traditional semiconductor electrodes can distinguish the trapping characteristics between hole and electron traps in polymer dielectrics. The observable discrepancy in space charge-limited conductance and the stable dielectric breakdown strength demonstrate that the electron injection into PET film specimen is restricted by using the novel semiconductor electrode. Attributed to the favorable suppression on the inevitable electron injections from metal electrodes, adopting novel i-electrode can avoid the evident abatement of dipole orientation polarization caused by space charge clamp, but will engender the accessional high-frequency dielectric loss from dielectric relaxations of interface charges at i-electrodes.     

Charge Accumulation in the Homo-Crosslinked-Polyethylene Bilayer

The homo-crosslinked-polyethylene (H-XLPE) bilayer simplifies the returned insulation structure of the factory joint in submarine cables, and its dielectric property is key to the reliability of the power transmission system. In this paper, we investigated the charge accumulation phenomenon in a secondary thermocompression H-XLPE bilayer using the pulse electro-acoustic method. The charge accumulation reduces its overall breakdown strength when compared with XLPE. According to X-ray diffraction measurement and thermal analysis results, the specimen forms a homo-junction region between the bilayers, which has overlapping spherulites with a thick lamella, high crystallinity, and high surface free energy. The charge accumulation can be ascribed to fused lamellas and the crystal imperfection of the homo-junction region, which restricts the charge transport process and exhibits a higher number of deep traps. This study emphasizes the importance of the homo-junction region in the H-XLPE bilayer, which should be considered in the design and operation of factory joint insulation.     

Surface Charge of Erythrocytes in Paroxysmal Nocturnal Haemoglobinuria

The electrophoretic mobility of erythrocytes from 3 patients with PNH was measured by standard techniques. Erythrocyte mobility is a measure of the net negative surface charge of these cells. In each of the 3 patients studied the mobility was significantly increased over normal cells indicating that there is an increase in negative surface charge on PNH erythrocytes. Following the Ham's and sucrose lysis tests, surviving cells showed mean mobilities that were not statistically different from normal cells. This suggests that these cells with higher surface charge are vulnerable to complement lysis.     

Experimental and Numerical Study on the Penetration Performance of a Shaped Charge

In guided ammunition, because a shaped energy jet warhead is located behind the control cabin (including the guidance cabin, the steering gear cabin, and the flight control cabin), the penetration order of a shaped energy jet is the control cabin and the target plate. In order to obtain maximum penetration depth by a shaped energy jet into a Q235 steel plate, the penetration performance of shaped energy jets was studied by numerical simulation and experimental verification. Firstly, the penetration performance of a warhead under different conditions at a certain explosion height is studied, which is the penetration performance of a Q235 steel plate with and without the control cabin. Secondly, the numerical simulation results are verified by experimental method. The numerical simulation and experimental results showed that, after penetration of the shaped energy jet warhead into the control cabin, it continued to penetrate the 20 mm-thick Q235 steel plate. At a certain explosion height, the maximum penetration depth of the shaped energy jet warhead into the Q235 steel plate was about 80 mm. Alongside the numerical simulation and experiment, the armor-breaking process of the shaped charge jet was analyzed theoretically. The results show that when the shaped energy jet warhead is located behind the control cabin, although the control cabin will have a certain impact on the penetration ability of shaped energy jet, the penetration performance of the residual jet still has the ability to penetrate light armor.     

From the Cover: Biological role of noise encoded in a genetic network motif

Genetic circuits that regulate distinct cellular processes can differ in their wiring pattern of interactions (architecture) and susceptibility to stochastic fluctuations (noise). Whether the link between circuit architecture and noise is of biological importance remains, however, poorly understood. To investigate this problem, we performed a computational study of gene expression noise for all possible circuit architectures of feed-forward loop (FFL) motifs. Results revealed that FFL architectures fall into two categories depending on whether their ON (stimulated) or OFF (unstimulated) steady states exhibit noise. To explore the biological importance of this difference in noise behavior, we analyzed 858 documented FFLs in Escherichia coli that were divided into 39 functional categories. The majority of FFLs were found to regulate two subsets of functional categories. Interestingly, these two functional categories associated with FFLs of opposite noise behaviors. This opposite noise preference revealed two noise-based strategies to cope with environmental constraints where cellular responses are either initiated or terminated stochastically to allow probabilistic sampling of alternative states. FFLs may thus be selected for their architecture-dependent noise behavior, revealing a biological role for noise that is encoded in gene circuit architectures.     

Charge transfer as a mechanism for chlorophyll fluorescence concentration quenching

Highly concentrated solutions of chlorophyll display rapid fluorescence quenching. The same devastating energy loss is not seen in photosynthetic light-harvesting antenna complexes, despite the need for chromophores to be in close proximity to facilitate energy transfer. A promising, though unconfirmed mechanism for the observed quenching is energy transfer from an excited chlorophyll monomer to a closely associated chlorophyll pair that subsequently undergoes rapid nonradiative decay to the ground state via a short-lived intermediate charge-transfer state. In this work, we make use of newly emerging fast methods in quantum chemistry to assess the feasibility of this proposed mechanism. We calculate rate constants for the initial charge separation, based on Marcus free-energy surfaces extracted from molecular dynamics simulations of solvated chlorophyll pairs, demonstrating that this pathway will compete with fluorescence (i.e., drive quenching) at experimentally measured quenching concentrations. We show that the rate of charge separation is highly sensitive to interchlorophyll distance and the relative orientations of chromophores within a quenching pair. We discuss possible solvent effects on the rate of charge separation (and consequently the degree of quenching), using the light-harvesting complex II (LH2) protein from rps. acidophila as a specific example of how this process might be controlled in a protein environment. Crucially, we reveal that the LH2 antenna protein prevents quenching, even at the high chlorophyll concentrations required for efficient energy transfer, by restricting the range of orientations that neighboring chlorophyll pairs can adopt.

Chlorophyll molecules in photosynthetic light-harvesting complexes are surprisingly immune to the rapid fluorescence quenching displayed by concentrated solutions of chlorophyll in molecular solvents (1). The effect has consequently not been given much attention in studies of natural light harvesting (2). However, it highlights a fundamental role of the protein environment in tailoring the behavior of embedded chromophores for efficient energy transfer. Understanding first how this “concentration quenching” occurs and, second, how it is avoided in photosynthetic protein complexes should not be further overlooked in the context of developing new synthetic proteins (3) and biomimetic light-harvesting materials (4, 5), since protection against this unproductive quenching pathway must necessarily be a critical design consideration.Measurements of fluorescence quenching in solutions of chlorophyll in both diethyl ether and acetone, reported in ref. (1), sparked several investigations into possible quenching mechanisms. Careful experiments by Porter et al. excluded the possibility of quenching due to trap states formed by impurities; intersystem crossing to the triplet state; and multiexcitation annihilation or collisions with quenching species (6). Unlike many other porphyrin-based pigments, chlorophyll does not readily aggregate in polar organic solvents (e.g., ethanol, ether) (7), as evidenced by the lack of concentration-dependent spectral changes (6, 8), so this was also ruled out as a potential cause of quenching.The fluorescence decay kinetics of chlorophyll solutions suggest that quenching involves excitation transfer from an excited monomer to a (relatively) nonfluorescent dimer that acts as a trap (9). These nonfluorescent “dimers” are widely accepted to be statistical pairs: pairs of molecules that, in the random distribution in solution, happen to be (temporarily) within some critical distance of each other (8, 10). Beddard and Porter estimated this critical distance as 10Å by fitting experimental quenching data with Monte Carlo simulations of excitations performing random walks around randomly arranged 3D arrays of molecules (10). Their model used experimentally parameterized rates for fluorescence, intersystem crossing, and resonant energy transfer and assumed instantaneous quenching at statistical pairs.The means by which excitations are trapped and quenched at statistical pairs is still unknown. The most promising hypothesis is that the excited chlorophyll pair undergoes a charge transfer (Chl−Chl)∗→Chl+−Chl− to form an ion pair that recombines back to the ground state (2, 7, 11). Electrostatic attraction between the intermediate ions may pull them closer together, helping to stabilize the pair and make it easier to reach the ground state. However, there is not yet any experimental evidence for the existence of an ion-pair intermediate (12).An alternative possibility is that the chlorophyll molecules within a statistical pair form a temporary H-aggregate, where the lowest excited state has an oscillator strength close to zero and acts as a dark trap state (13, 14). Density functional theory calculations on chlorophyll pairs in vacuum support the idea that closely associated cofacial pairs do have a lower energy dark state (13) but the expected shift in absorption wavelength for the higher energy bright state is not observed experimentally (12). For this reason, we consider it the least likely of the two hypothesized mechanisms.Nonradiative decay via an intermediate charge-transfer state is an established quenching mechanism for other small chromophores (15), bound porphyrin-based dyads (16), and chlorophyll in combination with carotenoids (17), where there is a redox potential between the molecules in the statistical pair. It is less clear whether the same mechanism would still operate between homogeneous chlorophyll pairs, although the question of symmetric charge transfer is of great interest in the context of understanding charge separation in the special pair of the photosynthetic reaction center. Theoretical studies of the special pair in vacuum indicate that there is a charge-transfer state energetically close to the first excited state (18, 19), although this could potentially change in the reaction center protein or in solution. Charge transfer has been observed in experimental models of the special pair (covalently linked cofacial porphyrin units) (20, 21) and is known to drive quenching in side-by-side bacteriochlorin or porphyrin dyads (16, 22, 23). However, it is not known to what degree the covalent linking in these models might influence their charge-transfer properties. Charge transfer in these linked systems is more pronounced in solvents with a high dielectric constant, implying that chlorophyll concentration quenching, if it proceeds via a charge-transfer mechanism, should have a solvent dependence. This is loosely in agreement with early experimental results (1), although more data would be needed to confirm the trend.Our aim in this paper is to provide concrete evidence for the feasibility of a quenching mechanism involving charge transfer by calculating the free energy surfaces (FESs) for the photoexcited and intermediate charge-separated state in a solvent environment. We calculate the rate of charge separation at chlorophyll separations around the predicted “critical separation” and compare these to the fluorescence rate to determine the likelihood of observing significant quenching. Finally, we investigate how these surfaces change in a protein environment, revealing how photosynthetic antenna proteins may inhibit quenching to allow efficient energy transport.     

Formation of Shaped Charge Projectile in Air and Water

With the improvement of the antiknock performance of warships, shaped charge warheads have been focused on and widely used to design underwater weapons. In order to cause efficient damage to warships, it is of great significance to study the formation of shaped charge projectiles in air and water. This paper uses Euler governing equations to establish numerical models of shaped charges subjected to air and underwater explosions. The formation and the movement of Explosively Formed Projectiles (EFPs) in different media for three cases: air explosion and underwater explosions with and without air cavities are discussed. First, the velocity distributions of EFPs in the formation process are discussed. Then, the empirical coefficient of the maximum head velocity of EFPs in air is obtained by simulations of air explosions of shaped charges with different types of explosives. The obtained results agree well with the practical solution, which validates the numerical model. Further, this empirical coefficient in water is deduced. After that, the evolutions of the head velocity of EFPs in different media for the above three cases are further compared and analyzed. The fitting formulas of velocity attenuation of EFPs, which form and move in different media, are gained. The obtained results can provide a theoretical basis and numerical support for the design of underwater weapons.