How faceted liquid droplets grow tails

Liquid droplets, widely encountered in everyday life, have no flat facets. Here we show that water-dispersed oil droplets can be reversibly temperature-tuned to icosahedral and other faceted shapes, hitherto unreported for liquid droplets. These shape changes are shown to originate in the interplay between interfacial tension and the elasticity of the droplet’s 2-nm-thick interfacial monolayer, which crystallizes at some T = T_s above the oil’s melting point, with the droplet’s bulk remaining liquid. Strikingly, at still-lower temperatures, this interfacial freezing (IF) effect also causes droplets to deform, split, and grow tails. Our findings provide deep insights into molecular-scale elasticity and allow formation of emulsions of tunable stability for directed self-assembly of complex-shaped particles and other future technologies.Of all same-volume shapes, a sphere has the smallest surface area A. Microscopic liquid droplets are, therefore, spherical, because this shape minimizes their interfacial energy γA for a surface tension γ > 0. Spontaneous transitions to a flat-faceted shape, which increases the surface area, have never been reported for droplets of simple liquids. Here we demonstrate that surfactant-stabilized droplets of oil in water, of sizes ranging from 1 to 100 μm, known as “emulsions” or “macroemulsions” (1), can be tuned to sharp-edged, faceted, polyhedral shapes, dictated by the molecular-level topology of the closed surface. Furthermore, the physical mechanism which drives the faceting transition allows the sign of γ to be switched in a controllable manner, leading to a spontaneous increase in surface area of the droplets, akin to the spontaneous emulsification (SE) (1, 2), yet driven by a completely different, and reversible, process.At room temperature, the spherical shape of our emulsions’ surfactant-stabilized oil droplets indicates shape domination by γ > 0 (oil: 16-carbon alkane, C₁₆; surfactant: trimethyloctadecylammonium bromide, C₁₈TAB, see SI Appendix, Fig. S1). However, the observed shape change to an icosahedron at some T = T_d, below the interfacial freezing temperature T_s (Fig. 1A), demonstrates that γ has become anomalously low and no longer dominates the shape. This γ-decrease upon cooling starkly contrasts with the behavior of most other liquids, where γ increases upon cooling (1). Direct in situ γ-measurements in our emulsions (SI Appendix), as well as pendant drop tensiometry of millimeter-sized droplets, confirm the positive dγ(T)/dT here (Fig. 2). Wilhelmy plate method γ(T) measurements (3, 4) on planar interfaces between bulk alkanes and aqueous C₁₈TAB solutions (blue circles in Fig. 2A) also demonstrate the same dγ(T)/dT > 0 at T < T_s. Thus, the anomalous positive dγ/dT below T_s is confirmed for the C₁₆/C₁₈TAB system by three independent methodologies.Open in a separate windowFig. 1.Buckling in liquid emulsion droplets. (A–D) For small γ, the frozen interface’s elasticity dominates. Small droplets become icosahedra (bright field: A, simulated: B), exhibiting five vertex-emanating edges (lines). Confocal microscopy reveals regular-icosahedron-identifying hexagonal (C) and pentagonal (D) cross-sections. Large droplets, having significant surface-area A and -energy γA, remain spherical (bright field: E, simulated: F), but show protrusions formed by defect buckling (arrows in E).Open in a separate windowFig. 2.Anomalous γ(T) dependence. (A) Alkane–water interfaces, decorated by C₁₈TAB surfactant, undergo IF at T = T_s, yielding a positive temperature slope of the interfacial tension γ at T < T_s. The γ-values of the emulsion droplets (green squares) agree with those measured for a planar interface (blue circles) and for pendant millimeter-sized drops (red triangles). The γ-dominated, the elasticity-dominated, and the SE regimes are schematically marked. γ/γ(T_s) derived from ΔS of surface-frozen C₁₆ alkane melt (5) is shown in purple. A magnified plot at ultralow γ is shown in B.To elucidate the implications of the positive dγ/dT, we note that thermodynamics equates an interface’s γ with its free-energy change upon a unity increase in area: γ = ΔE ? TΔS. ΔE and ΔS are the concomitant internal energy and entropy changes of the molecules (alkanes in our case) transferred from the bulk to the expanded interface. Because an interface is typically less ordered than a bulk, ΔS is positive, yielding dγ/dT < 0. Conversely, our dγ/dT > 0 yields ΔS < 0, suggesting that at T < T_s our droplet’s interface is more ordered than its (liquid) bulk. Indeed, recent X-ray measurements (4) on planar C₁₈TAB-decorated C₁₆–water interfaces showed that the interfacial region freezes at T = T_s, yielding a crystalline monolayer of mixed, fully extended, surface-normal–aligned, C₁₆ and C₁₈TAB molecules. No further structural changes were found upon cooling to C₁₆ bulk freezing. Systematic studies with various combinations of alkane and surfactant chain lengths (4) indicated that interfacial freezing (IF) occurs due to the unique match in molecular geometry between the alkane and the surfactant. The surfactant’s bulky hydrated trimethylammonium headgroups yield a surfactant molecule''s spacing that leaves just sufficient room for alkane molecule interdigitation in between the surfactant tails, thus maximizing the van der Waals contacts. Similar IF has also been observed in alkane monolayers at the planar liquid/air interface of aqueous C₁₈TAB solutions (4, 5), where grazing-incidence X-ray diffraction manifests that the frozen monolayer’s molecules form a 2D quasi–long-range hexagonal crystalline order.Because no changes occur in the liquid bulk phases at T = T_s, the γ(T) slope change at T = T_s is a direct measure of ΔS associated with the monolayer freezing (3–6). Furthermore, the slope of our γ (Fig. 2) matches closely that of the frozen C₁₆ monolayer at the planar surface of its own melt (6) (purple dashed line), where |ΔS| ≈ 0.9 × 10^?3Jm^?2K^?1 was measured. The near-equal dγ(T < T_s)/dT in emulsions and at the planar interfaces discussed above strongly indicates that the IF observed in our droplets indeed forms a hexagonally packed monolayer of fully extended surfactants and alkanes, aligned normal to the interface (Fig. 3 A and B). However, although similar in forming an interfacially frozen monolayer, the planar and spherical interfaces greatly differ. A planar interface can be tiled perfectly by all 2D crystal symmetries, and thus imposes no constraints on the monolayer’s lateral crystalline packing. By contrast, a sphere cannot be perfectly tiled by a planar 2D lattice. Thus, packing defects are necessarily imposed, the number, nature, and symmetry of which depend on the monolayer’s 2D crystalline order. As this study demonstrates, these geometry-imposed defects lead to a rich array of temperature-dependent effects, which have no counterparts in the temperature evolution of a planar interface, where no further changes occur upon cooling once the frozen monolayer is formed.Open in a separate windowFig. 3.Molecular structure cartoons of the droplets’ interface. (A) The interfacial monolayer at T > T_s, comprising mixed C₁₆ alkane and C₁₈TAB surfactant molecules. Yellow, blue, green, and red denote C, H, N⁺, and Br⁻, respectively. C₁₈TAB headgroups are partially water-ionized and hydrated. (B) Only the interface freezes at T = T_s, forming a crystalline, hexagonally packed, monolayer of extended, interface-normal, molecules (4) (Inset). Because a spherical surface cannot be tiled by hexagons, the frozen monolayer includes 12 fivefold defects. (C) At low γ elasticity dominates. The defect-induced strain is partly relieved by buckling (Inset).     

Field-induced superconducting phase of FeSe in the BCS-BEC cross-over

Fermi systems in the cross-over regime between weakly coupled Bardeen–Cooper–Schrieffer (BCS) and strongly coupled Bose–Einstein-condensate (BEC) limits are among the most fascinating objects to study the behavior of an assembly of strongly interacting particles. The physics of this cross-over has been of considerable interest both in the fields of condensed matter and ultracold atoms. One of the most challenging issues in this regime is the effect of large spin imbalance on a Fermi system under magnetic fields. Although several exotic physical properties have been predicted theoretically, the experimental realization of such an unusual superconducting state has not been achieved so far. Here we show that pure single crystals of superconducting FeSe offer the possibility to enter the previously unexplored realm where the three energies, Fermi energy ε_F, superconducting gap Δ, and Zeeman energy, become comparable. Through the superfluid response, transport, thermoelectric response, and spectroscopic-imaging scanning tunneling microscopy, we demonstrate that ε_F of FeSe is extremely small, with the ratio Δ/ε_F ～ 1( ～ 0.3) in the electron (hole) band. Moreover, thermal-conductivity measurements give evidence of a distinct phase line below the upper critical field, where the Zeeman energy becomes comparable to ε_F and Δ. The observation of this field-induced phase provides insights into previously poorly understood aspects of the highly spin-polarized Fermi liquid in the BCS-BEC cross-over regime.Superconductivity in most metals is well explained by the weak-coupling Bardeen–Cooper–Schrieffer (BCS) theory, where the pairing instability arises from weak attractive interactions in a degenerate fermionic system. In the opposite limit of Bose–Einstein condensate (BEC), composite bosons consisting of strongly coupled fermions condense into a coherent quantum state (1, 2). In BCS superconductors, the superconducting transition temperature is usually several orders of magnitude smaller than the Fermi temperature, T_c/T_F = 10^?5–10^?4, whereas in the BEC limit T_c/T_F is of the order of 10^?1. Even in the high-T_c cuprates, T_c/T_F is merely of the order of 10^?2 at optimal doping. Of particular interest is the BCS-BEC cross-over regime with intermediate coupling strength. In this regime the size of interacting pairs ( ～ ξ), which is known as the coherence length, becomes comparable to the average distance between particles ( ～ 1/k_F), i.e., k_Fξ ～ 1 (3–5), where k_F is the Fermi momentum. This regime is expected to have the highest values of T_c/T_F = 0.1 ? 0.2 and Δ/ε_F ～ 0.5 ever observed in any fermionic superfluid.One intriguing issue concerns the role of spin imbalance: whether it will lead to a strong modification of the properties of the Fermi system in the cross-over regime. This problem has been of considerable interest not only in the context of superconductivity but also in ultracold-atom physics (6–8). However, such Fermi systems have been extremely hard to access. In superconductors, the spin imbalance can be introduced through the Zeeman effect by applying a strong magnetic field. Again, in the high-T_c cuprates, the Zeeman energy at the upper critical field at T ? T_c is of the order of only 10^?2ε_F. In ultracold atoms, although several exotic superfluid states have been proposed (9, 10), cooling the systems down to sufficiently low temperature (T ? T_c) is not easily attained.FeSe provides an ideal platform for studying a highly spin-polarized Fermi system in the cross-over regime. FeSe is the simplest iron-based layered superconductor (Fig. 1A, Inset) with T_c of  ～ 9 K (11). The structural transition from tetragonal to orthorhombic crystal symmetry occurs at T_s ≈  90 K and a large electronic in-plane anisotropy appears. In contrast with the other iron-based compounds, no magnetic order occurs below T_s. A prominent feature of the pseudobinary “11” family (FeSe_1−x Te_x) is the presence of very shallow pockets, as reported by angle-resolved photoemission spectroscopy (ARPES). Although a possible BCS-BEC cross-over has been suggested in the bands around the Γ-point (12, 13), it is still an open question whether all bands are in such a cross-over regime. Moreover, it should be noted that high-quality single crystals are highly requisite for the study of the cross-over regime, as exotic superconductivity often is extremely sensitive to impurities. Previous FeSe_x single crystals are strongly disordered, as indicated by large residual resistivity ρ₀ and small residual resistivity ratio RRR, typically 0.1 mΩ cm and  ～ 5, respectively (14).Open in a separate windowFig. 1.Normal and superconducting properties of high-quality single crystals of FeSe. (A) Temperature dependence of the in-plane resistivity ρ of FeSe. The structural transition occurs at T_s ≈  90 K. (Upper Inset) Crystal structure. (Lower Inset) ρ(T) in magnetic field (H‖c). From bottom to top, μ₀H =  0, 2, 4, 6, 8, 10, 12, and 14 T is applied. (B) Temperature dependence of the London penetration depth. (Inset) Superfluid density normalized by the zero temperature value ρs≡λL2(0)/λL2(T). (C) Tunneling conductance spectrum at T = 0.4 K. The peaks at ±2.5 meV (arrows) and shoulder structures at ±3.5 meV (dashed arrows) indicate the multiple superconducting gaps.     

Suppression of phase separation and giant enhancement of superconducting transition temperature in FeSe1?xTex thin films

We demonstrate the successful fabrication on CaF₂ substrates of FeSe_1−xTe_x films with 0 ≤ x ≤ 1, including the region of 0.1 ≤ x ≤ 0.4, which is well known to be the “phase-separation region,” via pulsed laser deposition that is a thermodynamically nonequilibrium method. In the resulting films, we observe a giant enhancement of the superconducting transition temperature, T_c, in the region of 0.1 ≤ x ≤ 0.4: The maximum value reaches 23 K, which is ∼1.5 times as large as the values reported for bulk samples of FeSe_1−xTe_x. We present a complete phase diagram of FeSe_1−xTe_x films. Surprisingly, a sudden suppression of T_c is observed at 0.1 < x < 0.2, whereas T_c increases with decreasing x for 0.2 ≤ x < 1. Namely, there is a clear difference between superconductivity realized in x = 0 ? 0.1 and in x ≥ 0.2. To obtain a film of FeSe_1−xTe_x with high T_c, the controls of the Te content x and the in-plane lattice strain are found to be key factors.Since the discovery of superconductivity in LaFeAs(O,F) (1), many studies concerning iron-based superconductors have been conducted. FeSe is the iron-based superconductor with the simplest crystal structure (2). The T_c of FeSe is ∼8 K, which is not very high in comparison with other iron-based superconductors. However, the value of T_c strongly depends on the applied pressure, and the temperature at which the resistivity becomes zero, Tczero, reaches as high as  ～ 30 K at 6 GPa (3). This suggests that FeSe samples with higher T_c are available by the fabrication of thin films because we can introduce lattice strain. Indeed, we have previously reported that FeSe films fabricated on CaF₂ substrates exhibit T_c values ∼1.5 times higher than those of bulk samples because of in-plane compressive strain (4). On the other hand, superconductivity with T_c of 65 K has recently been reported in a monolayer FeSe film on SrTiO₃ (5, 6). It is unclear whether this superconductivity results from the characteristics of the interface. However, this finding indicates that FeSe demonstrates potential as a very-high-T_c superconductor.The partial substitution of Te for Se in FeSe also raises T_c to a maximum of 14 K at x = 0.5 ? 0.6 (7). In FeSe_1−xTe_x, it is well known that we cannot obtain single-phase samples with 0.1 < x < 0.4 because of phase separation (7). Here, we focus on this region of phase separation. Generally, the process of film deposition involves crystal growth in a thermodynamically nonequilibrium state. Thus, film deposition provides an avenue for the synthesis of a material with a metastable phase. In this paper, we report the fabrication of epitaxial thin films of FeSe_1−xTe_x with 0 ≤ x ≤ 1 on CaF₂ substrates, using the pulsed laser deposition (PLD) method. We demonstrate that single-phase epitaxial films of FeSe_1−xTe_x with 0.1 ≤ x ≤ 0.4 are successfully obtained and that the maximum value of T_c is as large as 23 K, which is higher than the previously reported values for bulk and film samples of FeSe_1−xTe_x (7–12), except for those of the monolayer FeSe films (5, 6). Our results clearly show that the optimal Te content for the highest T_c for FeSe_1−xTe_x films on CaF₂ is different from the widely believed value for this system.Fig. 1A presents the X-ray diffraction patterns of FeSe_1−xTe_x films for x = 0 ? 0.5 on CaF₂. Here and hereafter, the Te content x of our films represents the nominal Te composition of the polycrystalline target. With the exception of an unidentified peak in the FeSe_0.5Te_0.5 film, only the 00l reflections of a tetragonal PbO-type structure are observed, which indicates that these films are well oriented along the c axis. Fig. 1 B and C presents enlarged segments of these plots near the 001 and 003 reflections, respectively. The 2θ values of the peak positions decrease with increasing x in a continuous manner, which is consistent with the fact that the c-axis length increases with increasing x. It should be noted that the values of the full widths at half maximum (FWHM), δ(2θ), of the FeSe_1−xTe_x films with x = 0.1 ? 0.4, which is known as the region of phase separation in the bulk samples (7), are δ(2θ) = 0.2°?0.3° for the 001 reflection and δ(2θ) = 0.4°?0.6° for the 003 reflection, which are nearly the same as the values for the FeSe and FeSe_0.5Te_0.5 films. This result is in sharp contrast to the previously reported result that the FWHM was broad only in films of FeSe_1−xTe_x with x = 0.1 and 0.3 (13), where phase separation has been believed to occur. The results presented in Fig. 1 A–C indicate the formation of a single phase in our FeSe_1−xTe_x films with x = 0.1 ? 0.4.Open in a separate windowFig. 1.(A) Out-of-plane X-ray diffraction patterns of FeSe_1−xTe_x thin films for x = 0 ? 0.5 with film thicknesses of 120–147 nm. The # symbols represent peaks associated with the substrate. The * represents an unidentified peak. Enlarged segments of the plots presented in A near the 001 and 003 peaks are shown in B and C, respectively. (D) The c-axis lengths of FeSe_1−xTe_x films, where the x value indicated on the horizontal axis is the nominal Te content. (E) Relations between the a-axis and c-axis lengths in FeSe_1−xTe_x films. The colors and shapes of the symbols correspond to the Te content x, as shown. The dashed lines are guides for the eye. The data for x = 0 and 0.5 presented in A–E are cited from refs. 4, 9, and 14.In Fig. 1D, the c-axis lengths of 29 films of FeSe_1−xTe_x are plotted as a function of x. The values of the c-axis lengths vary almost linearly with the nominal Te contents of the targets in the whole range of x, including both end-member materials. The evident formation of a single phase and the systematic change in the c-axis length strongly indicate that the nominal Te content of the polycrystalline target is nearly identical to that of the final FeSe_1−xTe_x film. Note that the compositional analysis of grown films using scanning electron microscopy/energy-dispersive X-ray (SEM/EDX) analysis is impossible for FeSe_1−xTe_x films on CaF₂ substrates because the energies of the K edge of Ca and the L edge of Te are very close to each other. The above-mentioned features indicate that phase separation is suppressed in our FeSe_1−xTe_x films with x = 0.1 ? 0.4 on CaF₂ substrates. To our knowledge, this result is the first manifestation of the suppression of phase separation in FeSe_1−xTe_x with x = 0.1 ? 0.4.Fig. 1E presents the relations between the a-axis and c-axis lengths in films of FeSe_1−xTe_x. At first glance, there seem to be no relations between the a-axis length and x, in sharp contrast to the behavior of the c-axis length. The a-axis and c-axis lengths of films with the same x show a weak negative correlation. This behavior cannot be explained by a difference in Te content of a film, which should result in a positive correlation. By contrast, if variations in c are caused by a difference in in-plane lattice strain, this behavior can be explained in terms of the Poisson effect. Indeed, the a-axis lengths of films of FeSe and FeSe_0.5Te_0.5 are smaller than those of bulk samples with the same composition. Thus, we consider that the a-axis length predominantly depends on the in-plane lattice strain rather than the Te content x. One might think that this behavior looks strange, because the lattice constant of CaF₂(aCaF2/2) is longer than the a of FeSe_1−xTe_x, which usually leads to a tensile strain. In a previous paper, the penetration of F⁻ ions from the CaF₂ substrates into the films was proposed as a possible mechanism for nontrivial compressive strain in FeSe_1−xTe_x films on CaF₂ substrates (15). Because of the smaller ionic radius of F⁻ than that of Se²⁻, this peculiar compressive strain can be explained by the partial substitution of F⁻ for Se²⁻ near the interface between a film and a substrate.Fig. 2 A–D presents the temperature dependences of the electrical resistivities, ρ, of 16 films of FeSe_1−xTe_x for x = 0.1 ? 0.4. The value of T_c depends on the film thickness, even in films with the same x. The highest Tconset, which is defined as the temperature where the electrical resistivity deviates from the normal-state behavior, and the Tczero of the FeSe_1−xTe_x films are 13.2 K and 11.5 K, respectively, for x = 0.1; 22.8 K and 20.5 K, respectively, for x = 0.2; 20.9 K and 19.9 K, respectively, for x = 0.3; and 20.9 K and 20.0 K, respectively, for x = 0.4. Compared with the results for bulk samples, a drastic enhancement of T_c is observed in these FeSe_1−xTe_x films. Surprisingly, the values of Tczero in the films with x = 0.2 and 0.4 exceed 20 K. These values are larger than those reported for FeSe_0.5Te_0.5 films (8, 10, 11, 14). In particular, the T_c of the FeSe_0.8Te_0.2 film with a thickness of 73 nm is ∼1.5 times as high as those of bulk crystals of FeSe_1−xTe_x with the optimal composition, x ≈ 0.5 (7). Based on the measurement of the ρ of the FeSe_0.8Te_0.2 film under a magnetic field applied along the c axis, we estimate an upper critical field at 0 K of μ₀H_c2 = 55.4 T, using the Werthamer–Helfand–Hohenberg (WHH) theory (16), which yields a Ginzburg–Landau coherence length at 0 K of ξ_ab(0) ～ 24.4 Å (SI Appendix). This value of μ₀H_c2 is approximately half the value for an FeSe_0.5Te_0.5 film on CaF₂ with a T_c of ∼16 K (9).Open in a separate windowFig. 2.Temperature dependences of the electrical resistivities, ρ, of FeSe_1−xTe_x thin films for (A) x = 0.1, (B) x = 0.2, (C) x = 0.3, and (D) x = 0.4 with different film thicknesses. Insets present enlarged views of the plots near the superconducting transition.Using the data shown above, we present the phase diagram of FeSe_1−xTe_x films on CaF₂ substrates in Fig. 3. For comparison, the data for bulk samples of FeSe_1−xTe_x (7, 17) are also plotted in Fig. 3. In bulk crystals, the optimal Te content to achieve the highest T_c is considered to be x ≈ 0.5, and phase separation occurs in the region of 0.1 ≤ x ≤ 0.4 (7). However, our data clearly demonstrate that this phase separation is absent and that the optimal composition for an FeSe_1−xTe_x film on a CaF₂ substrate is not x ≈ 0.5 but x ≈ 0.2. It should be noted that the dependence of T_c on x suddenly changes at the boundary defined by 0.1 < x < 0.2. Unlike the “dome-shaped” phase diagram that is familiar in iron-based superconductors, the values of T_c in films with 0.2 ≤ x ≤ 1 increase with decreasing x, whereas the strong suppression of T_c is observed at 0.1 < x < 0.2. The behavior in films with x ≥ 0.2 can be explained by the empirical law that shows the relation between T_c and structural parameters. In iron-based superconductors, it is well accepted that the bond angle of (Pn, Ch)-Fe-(Pn, Ch) (Pn, ?Pnictogen; Ch, ?Chalcogen), α (18, 19), and/or the anion height from the iron plane, h (20), are the critical structural parameters that determine the value of T_c. In bulk samples of FeSe_1−xTe_x, α and h approach their optimal values, i.e., α = 109.47° (18, 19) and h = 1.38 Å (20), with decreasing x (down to x = 0), which should be the same in FeSe_1−xTe_x films. Therefore, the increase of T_c in films with 0.2 ≤ x ≤ 1 with decreasing x can be explained by the optimization of α and/or h based on the empirical law. However, the sudden suppression of T_c in films with 0 ≤ x < 0.2 is not consistent with this scenario, and its origin should be sought among other factors. We consider there are two candidates for this origin from the structural analysis of bulk samples of FeSe_1−xTe_x. One is the effect of the orthorhombic distortion. In a bulk sample of FeSe, a structural phase transition from tetragonal to orthorhombic occurs at 90 K (21). However, in bulk samples of FeSe_1−xTe_x with x ～ 0.4 ? 0.6 where T_cs take optimum values, there are papers with different conclusions on the presence/absence of a similar type of structural transition to that of FeSe (22–24). It should be noted that a structural transition temperature is lower and that the orthorhombicity is much smaller than those of FeSe even in the report where the structural transition is present (24). These results on crystal structures suggest that the orthorhombic distortion results in a suppression of T_c. This scenario is applicable to the behavior of our films, if a large orthorhombic distortion is observed only in films with x = 0 ? 0.1. The other candidate is the change in the distance between the layers of Fe-Ch tetrahedra, δ. As shown in SI Appendix, in polycrystalline samples of FeSe_1−xTe_x, the δ value of FeSe is much smaller than those of FeSe_1−xTe_x with x ≥ 0.5 where δ is nearly independent of x (22). We speculate that the decrease of δ in FeSe is related with the suppression of T_c. Indeed, in polycrystalline samples, FeSe exhibits smaller values of δ and T_c than does FeSe_0.5Te_0.5 (7, 22), and the intercalation of alkali metals and alkaline earths into FeSe results in the c-axis length as large as ∼20 Å and T_c as high as 45 K (25, 26). At this moment, the origin of the suppression of T_c at 0.1 < x < 0.2 is unclear. Regardless of its origin, we believe that it is reasonable to distinguish between superconductivity in x = 0 ? 0.1 and in x ≥ 0.2. In other words, our phase diagram in Fig. 3 provides a previously unidentified view for superconductivity in FeSe_1−xTe_x, that is, a discontinuity in superconductivity of FeSe_1−xTe_x. We are able to come to this picture only after the data for x = 0.1 ? 0.4 become available in this study. If we remove a cause for the suppression of T_c in x=0,0.1 in some way, a further increase in T_c can be expected because of the optimization of structural parameters.Open in a separate windowFig. 3.Dependence of T_c on x. The red and blue circles represent the Tconset and Tczero values of the FeSe_1−xTe_x thin films, respectively. The black triangles represent the Tconset values obtained in measurements of the magnetic susceptibility of bulk samples (7, 17). The dashed curve is a guide for the eye.In conclusion, we prepared high-quality epitaxial thin films of FeSe_1−xTe_x on CaF₂ substrates, using the pulsed laser deposition method. We successfully obtained FeSe_1−xTe_x films with 0.1 ≤ x ≤ 0.4, which has long been considered to be the “phase-separation region,” using a thermodynamically nonequilibrium growth of film deposition. From the results of electrical resistivity measurements, a complete phase diagram is presented in this system, in which the maximum value of T_c is as high as 23 K at x = 0.2. Surprisingly, a sudden suppression of T_c is observed at 0.1 < x < 0.2, whereas T_c increases with decreasing x for 0.2 ≤ x < 1. This behavior is different from that of the dome-shaped phase diagram that is familiar in iron-based superconductors.     

Nonergodic metallic and insulating phases of Josephson junction chains

Strictly speaking, the laws of the conventional statistical physics, based on the equipartition postulate [Gibbs J W (1902) Elementary Principles in Statistical Mechanics, developed with especial reference to the rational foundation of thermodynamics] and ergodicity hypothesis [Boltzmann L (1964) Lectures on Gas Theory], apply only in the presence of a heat bath. Until recently this restriction was believed to be not important for real physical systems because a weak coupling to the bath was assumed to be sufficient. However, this belief was not examined seriously until recently when the progress in both quantum gases and solid-state coherent quantum devices allowed one to study the systems with dramatically reduced coupling to the bath. To describe such systems properly one should revisit the very foundations of statistical mechanics. We examine this general problem for the case of the Josephson junction chain that can be implemented in the laboratory and show that it displays a novel high-temperature nonergodic phase with finite resistance. With further increase of the temperature the system undergoes a transition to the fully localized state characterized by infinite resistance and exponentially long relaxation.The remarkable feature of the closed quantum systems is the appearance of many-body localization (MBL) (1): Under certain conditions the states of a many-body system are localized in the Hilbert space resembling the celebrated Anderson localization (2) of single particle states in a random potential. MBL implies that macroscopic states of an isolated system depend on the initial conditions (i.e., the time averaging does not result in equipartition distribution and the entropy never reaches its thermodynamic value). Variation of macroscopic parameters (e.g., temperature) can delocalize the many-body state. However, the delocalization does not imply the recovery of the equipartition. Such a nonergodic behavior in isolated physical systems is the subject of this paper.We argue that regular Josephson junction arrays (JJAs) under the conditions that are feasible to implement and control experimentally demonstrate both MBL and nonergodic behavior. A great advantage of the Josephson circuits is the possibility to disentangle them from the environment, as was demonstrated by the quantum information devices (3). At low temperatures the conductivity σ of JJA is finite (below we call such behavior metallic), whereas as T → 0 JJA becomes either a superconductor (σ → ∞) or an insulator (σ → 0) (4, 5). We predict that at a critical temperature T = T_c JJA undergoes a true phase transition into an MBL insulator (σ = 0 for T > T_c). Remarkably, already in the metallic state JJA becomes nonergodic and cannot be properly described by conventional statistical mechanics.In the bad metal regime the dynamical evolution starting from a particular initial state does not lead to the thermodynamic equilibrium, so that even extensive quantities such as entropy differ from their thermodynamic values. Starting with the seminal paper of Fermi et al. (6) the question of nonergodicity in nonintegrable systems was extensively studied (7). However, the difference between the long time asymptotics of the extensive quantities and their thermodynamic values was not analyzed to the best of our knowledge. We believe that the behavior of JJA that we describe here can be observed in a variety of nonlinear systems.JJA is characterized by the set of phases {?_i} and charges {q_i} of the superconducting islands, ?_i and q_i for each i are canonically conjugated. The Hamiltonian H is the combination of the charging energies of the islands with the Josephson coupling energies. Assuming that the ground capacitance of the islands dominates their mutual capacitances (this assumption is not crucial for the qualitative conclusions) we can write H asH=∑i[12ECqi2+EJ(1−cos(ϕi−ϕi+1))].[1]The ground state of the model Eq. 1 is determined by the ratio of the Josephson and charging energies, E_J/E_C, that controls the strength of quantum fluctuations: JJA is an insulator at E_J/E_C < η and a superconductor at E_J/E_C>η (4, 5) with η ≈ 0.63 (Supporting Information, section 1 and Fig. S1). The quantum transition at E_J/E_C = η belongs to the Berezinsky–Kosterlitz–Thouless universality class (8). Away from the ground state in addition to E_J/E_C there appears dimensionless parameter U/E_J, where U is the energy per superconducting island (U = T in the thermodynamic equilibrium at T ? E_J).Open in a separate windowFig. S1.(Inset) The fidelity of the ground state as a function of E_J/E_C for different sizes L. The minimum of the fidelity as a function of [ln(L)]^?2 is in the main panel. The red line fits the data to a line. The location of the critical point can be extracted from the value of the fitting at origin E_J/E_C = 0.63 ± 0.04.The main qualitative finding of this paper is the appearance of a nonergodic and highly resistive “bad metal” phase at high temperatures, T/E_J > 1, which at T≥Tc≈EJ2/EC undergoes the transition to the MBL insulator. In contrast to the T = 0 behavior, these results are robust (e.g., they are insensitive to the presence of static random charges). The full phase diagram in the variables E_J/E_C, T/E_J is shown in Fig. 1. We confirmed numerically that the bad metal persists in the classic (E_J ? E_C) limit although T_c → ∞; it is characterized by the exponential growth of the resistance with T and violation of thermodynamic identities. We support these findings by semiquantitative theoretical arguments. Finally, we present the results of numerical diagonalization and tDMRG [time density matrix renormalization group (9)] of quantum systems that demonstrate both the nonergodic bad metal and the MBL insulator.Open in a separate windowFig. 1.Phase diagram of 1D Josephson junction array. The MBL phase transition separates the nonergodic bad metal with exponentially large but finite resistance from the insulator with infinite resistance. Cooling the nonergodic bad metal transforms it into a good ergodic metal. The points show approximate positions of the effective T/E_J for the quantum problem with a finite number of charging states. In bad metal as well as in insulator the temperature is ill-defined. So, T_eff should be understood as the measure of the total energy of the system. The red stars indicate insulator, blue circles bad metal, and squares good metal. The dashed line shows the estimate of MBL for large E_J/E_C discussed in the text. The superconductor is formed at T = 0 at E_J/E_C > 0.62, as indicated by the thick blue line. This phase diagram is not sensitive to the presence of disorder for T ? E_C.It is natural to compare the nonergodic state of JJA with a conventional glass characterized by infinitely many metastable states. The glass entropy does not vanish at T = 0, that is, when heated from T = 0 to the melting temperature the glass releases less entropy than the crystal [Kauzmann paradox (10)]. Similarly to glasses JJA demonstrates nonergodic behavior in both quantum and classical regimes. However, the ergodicity violation emerges as high rather than low temperatures transforming the Kauzmann paradox that arises at low temperatures into an apparent temperature divergence at high temperatures (discussed below).     

Plethora of transitions during breakup of liquid filaments

Thinning and breakup of liquid filaments are central to dripping of leaky faucets, inkjet drop formation, and raindrop fragmentation. As the filament radius decreases, curvature and capillary pressure, both inversely proportional to radius, increase and fluid is expelled with increasing velocity from the neck. As the neck radius vanishes, the governing equations become singular and the filament breaks. In slightly viscous liquids, thinning initially occurs in an inertial regime where inertial and capillary forces balance. By contrast, in highly viscous liquids, initial thinning occurs in a viscous regime where viscous and capillary forces balance. As the filament thins, viscous forces in the former case and inertial forces in the latter become important, and theory shows that the filament approaches breakup in the final inertial–viscous regime where all three forces balance. However, previous simulations and experiments reveal that transition from an initial to the final regime either occurs at a value of filament radius well below that predicted by theory or is not observed. Here, we perform new simulations and experiments, and show that a thinning filament unexpectedly passes through a number of intermediate transient regimes, thereby delaying onset of the inertial–viscous regime. The new findings have practical implications regarding formation of undesirable satellite droplets and also raise the question as to whether similar dynamical transitions arise in other free-surface flows such as coalescence that also exhibit singularities.Drop formation is ubiquitous in daily life, industry, and nature (1–3). The phenomenon is central to inkjet printing (4, 5), dripping from leaky faucets (6, 7), measurement of equilibrium and dynamic surface tension (8, 9), DNA arraying and printing of cells (10, 11), chemical separations and analysis (12, 13), production of particles and capsules (14, 15), printing of wires and transistors (16, 17), and mist formation in waterfalls and fragmentation of raindrops (18, 19). Fig. 1A shows an experimental setup for studying the dynamics of a drop of an incompressible Newtonian fluid of density ρ, viscosity μ, and surface tension σ forming from a tube of radius R (Fig. 1B and Drop Formation from a Tube and Filament Thinning). A salient feature of, and key to understanding, drop formation is the occurrence of a thin filament that connects an about-to-form primary drop to the rest of the fluid that is attached to the tube (Fig. 1 B and C). Thus, it often proves convenient to study filament thinning in the idealized setup depicted in Fig. 1D (Drop Formation from a Tube and Filament Thinning). As time t advances and the filament radius decreases, curvature and capillary pressure, both of which are at leading order inversely proportional to radius, increase and fluid is expelled with increasing velocity from the neck. At the instant t = t_b when the neck radius vanishes, a finite time singularity occurs and the filament breaks. When the filament breaks, one or more satellite droplets may also form. These satellites are typically much smaller than the primary drop (20) and almost always undesirable in applications (2).Open in a separate windowFig. 1.Methods used for studying and new phase diagram for capillary thinning and pinch-off. (A) Experimental setup used to capture images of the thinning neck of a drop forming from a nozzle. The images are then postprocessed to obtain the minimum neck radius as a function of the time remaining until breakup. (B) Snapshot of a drop forming from a tube that highlights the pinching zone in the vicinity of the pinch point. (C) Two series of images that focus on the pinching zones and depict the evolution in time of thinning filaments for drops of two fluids with different viscosities (i.e., different Oh). (D) Setup for simulating filament thinning and pinch-off: periodically perturbed jet (Left) and domain of axial length λ/2 used in simulations (Right). (E) Phase space showing trajectories taken by filaments of a slightly viscous (Oh < 1) and a highly viscous (Oh > 1) fluid. The large squares indicate the starting states at t = 0. The arrows along each trajectory show the direction of evolution.For Newtonian filaments, three theories have been developed to describe the dynamics in the vicinity of the pinch-off singularity (Scaling Theories of Pinch-Off). When viscous effects are weak, thinning and pinching occur in an inertial (I) regime (21–23) where inertial and capillary forces balance, and the minimum filament radius h_min (Fig. 1B) and the instantaneous Reynolds number Re(t) vary with dimensionless time τ to breakup ashminR∼τ2/3, Re(t)∼1Ohτ1/3,[1]where τ ≡ (t_b ? t)/t_I and tI≡ρR3/σ (Scaling Theories of Pinch-Off). For real liquids, the Ohnesorge number Oh=μ/ρRσ is not identically zero no matter how small the viscosity. Thus, for low-viscosity liquids, Oh ? 1 and Eq. 1 shows that regardless of how large the Reynolds number is initially, as τ → 0 and breakup is approached, Re(t) → 0. Therefore, the inertial regime cannot persist all of the way to breakup and can only describe the initial dynamics for low-viscosity fluids. Similarly, when viscous effects are dominant, thinning and pinching occur in a viscous (V) regime (24) where viscous and capillary forces balance, and h_min and Re(t) vary with τ ashminR∼τ, Re(t)∼1Oh2τ2β−1,[2]where τ ≡ (t_b ? t)/t_V, t_V ≡ μR/σ, and β = 0.175 (Scaling Theories of Pinch-Off). For real liquids, Oh cannot be infinite no matter how large the viscosity. Thus, for high-viscosity liquids, Oh ? 1 and Eq. 2 shows that regardless of how small the Reynolds number is initially, as τ → 0 and breakup is approached, Re(t) → ∞. Therefore, the V regime cannot persist all of the way to breakup and hence can only describe the initial dynamics even for high-viscosity fluids. Therefore, as the filament radius tends to zero, a transition occurs from either the I or the V regime to a final inertial–viscous (IV) regime in which all three forces, i.e., inertial, viscous, and capillary, balance and the instantaneous Reynolds number Re(t) ～ 1 (25). From Eqs. 1 and 2, the transition from the I to the IV regime and that from the V to IV regime can be calculated by setting Re(t) to be order one. Thus, transition from the I to the IV regime should occur when (2, 18)h_min/R ～ Oh², [3]and that from the V to the IV regime should occur when (2, 18)h_min/R ～ Oh^2/(2β?1).[4]However, whereas careful simulations and experiments have shown that the transition from the I to the IV regime does indeed occur, it has been found to take place for values of h_min that are about an order of magnitude smaller than that predicted from theory (Eq. 3) (26). Furthermore, the transition from the V to the IV regime has not yet been demonstrated to occur from simulation and an attempt for an experimental demonstration of the transition (27) was perhaps at too small a value of Oh (Oh = 0.49) to be conclusively in the V regime. In this paper, we demonstrate that in contrast with the conventional wisdom that the dynamics of capillary pinching should exhibit a transition from either the I to the IV regime or the V to the IV regime, the transition from either of the two initial regimes to the final IV regime is in fact more complex and, unexpectedly, can be delayed by the occurrence of a number of intermediate transient regimes as shown in Fig. 1E. The possibility of such complexity has been anticipated in part by Eggers (18) but no study has yet been carried out to explore the existence of these intermediate regimes or contemplate its implications in other free-surface flows exhibiting finite time singularities.In this work, the dynamics of filament thinning is studied both numerically and experimentally. Simulations are performed to track how sinusoidal perturbations on a liquid cylinder cause it to break, which have been successfully used in the past to study pinch-off and scaling for Newtonian (26, 28) as well as non-Newtonian fluids (29, 30) (Fig. 1D and Simulations). In the experiments, high-speed imaging and image analysis are used to measure the evolution in time of the minimum filament radius for liquids dripping from a tubular nozzle. Glycerol–water mixtures are used as working fluids to explore systems with different values of Oh.Fig. 2A shows the computed variation of h_min with τ for a slightly viscous liquid of Oh = 0.23. The simulations make plain that after sufficient time has passed so that the initial transients have decayed, the filament first thins in the I regime, where h_min ～ τ^2/3, as expected. According to conventional wisdom, the thinning dynamics is expected to transition from the I regime to the IV regime when h_min ～ Oh² = 0.053; thenceforward, the thinning is to follow IV scaling where h_min = 0.0304τ/Oh. However, the simulation results show that this transition is delayed and does not take place until h_min has fallen below a value that is about an order of magnitude smaller than that predicted by the theoretical estimate. Moreover, the simulations show unexpectedly that the dynamics switches over from the I to the IV regime only after passing through an intermediate V regime, where h_min = 0.0709τ/Oh. The existence of these regimes can be verified by plotting the local Reynolds number Re_local (Simulations) in the thinning filament in the vicinity of the pinch point as a function of h_min (Fig. 2B). Fig. 2B clearly shows that at early times when h_min ≈ 0.2, Re_local ? 1, confirming the existence of the I regime. However, when h_min has fallen to  ≈ 0.03, Re_local ? 1, which clearly demonstrates that the dynamics has entered the newly discovered intermediate V regime. Finally, as the filament asymptotically approaches breakup, i.e., for values of h_min ≈ 10^?3 or smaller, Re_local ～ 1, demonstrating that near the singularity, all three forces (viz., inertial, viscous, and capillary) balance each other and the dynamics lies in the IV regime. To confirm the correctness of these computationally made predictions, dripping experiments have been carried out with two liquids of Oh = 0.23 and Oh = 0.55. For the former, Fig. 2C shows the transition from the initial I regime to the intermediate V regime, with the latter regime lasting nearly over two decades in h_min. When Oh = 0.23, it is not possible to observe the transition from the V regime to the final IV regime because that transition occurs for neck radii smaller than a micrometer, which is the lower limit of length scales that can be imaged using visible light. The experimental results for Oh = 0.55 depicted in Fig. 2D, on the other hand, do show the transition to the final IV regime, albeit with an intermediate V regime of much shorter duration.Open in a separate windowFig. 2.Simulations and experiments demonstrating the existence of an intermediate viscous regime between the initial inertial regime and the final IV regime for slightly viscous fluids (Oh < 1). (A) Variation of minimum neck radius with time until breakup when Oh = 0.23 obtained from simulations. (B) Computed variation with minimum neck radius of the local Reynolds number in the neighborhood of the pinch point verifies the existence of all three regimes: Re_local ? 1 in the I regime, Re_local ? 1 in the V regime, and Re_local ～ 1 in the IV regime. (C) Experimental confirmation of the existence of an intermediate V regime when Oh = 0.23. The IV regime is not attained here because of optical limitations. (D) At a slightly higher value of Oh than that in C (Oh = 0.55), the V to IV transition is observed experimentally. (C and D, Insets) Same data as in the main figures are presented but use linear rather than logarithmic axes.Having demonstrated the existence of the intermediate V regime, we now turn our attention to understanding the reason for its occurrence, which is facilitated by examining flow fields within thinning filaments. To do so, we turn our attention to a filament of Oh = 0.07 which, as shown in Fig. 3A, clearly depicts the existence of all three scaling regimes. The instantaneous streamlines and pressure contours at three different times when the dynamics lies in each of these three regimes are shown in Fig. 3 B–D over 0 ≤ z ≤ λ/2 ≡ 4. At early times, the minimum in the filament radius is located at z = λ/2 ≡ 4, i.e., halfway between two swells, one located at z = 0 and the other at z = λ ≡ 8 (Fig. 3B). As the filament continues to thin, the fluid accelerates as it flows from the neck, where pressure is highest, toward the two swells, where pressure is lowest. On account of this effect, which is attributable to finite fluid inertia (20, 22), the filament begins to thin fastest at two locations that are located on either side of z = λ/2. Within the computational domain, this leads to a shift in the minimum radius from the end of the domain (z = 4) to its interior, i.e., z ≈ 1.95. As shown in Fig. 3C, the occurrence of this new minimum gives rise to a new stagnation zone in the interior of the domain in the vicinity of which the flow has slowed down considerably and even reversed. This shift in the location of h_min and the accompanying slowing down of the flow then takes the dynamics into the V regime. Although the new stagnation zone persists for some time, the filament does not break while in the V regime. The capillary pressure which continues to rise as the filament continues to thin accelerates fluid out of the thinning neck and causes inertia to become significant once again, thereby taking the filament into the IV regime. Hence, with the simulation and experimental results shown in Figs. 2 and ​and3,3, the thinning and breakup dynamics of slightly viscous filaments for which Oh < 1 are seen to exhibit I to V to IV scaling as τ → 0. Furthermore, these results at long last shed light on the reason for the delay in the transition to the final IV regime that had remained perplexing and unexplained for over a decade.Open in a separate windowFig. 3.Simulation results when Oh = 0.07 highlight the formation of a stagnation zone within the filament and help explain why the intermediate viscous regime exists. (A) Variation of minimum neck radius with time until breakup that shows occurrence of all three regimes and transition from I to IV regime through an intermediate V regime. (See below for the explanation of the arrow.) (B) Instantaneous streamlines and pressure contours within the thinning filament when the dynamics lies in the I regime. As shown in the figure, this I regime has a slender geometry (21) rather than a fully developed double-cone structure (23). The legend on the top right identifies contour values of the pressure. At this instant in time, the minimum neck radius is located at z = λ/2 = 4 and the fluid accelerates as it flows from the neck to the swell. (C) Instantaneous streamlines and pressure contours in the filament at a later time than in B where the acceleration of the fluid has resulted in shifting of the neck from the top end of the domain (z = 4) to a location between the two ends (0 < z < 4). The time and minimum filament radius when this shift commences is identified by the arrow in A. (Inset) A new stagnation zone has formed away from the two ends, resulting in a region of reversed flow and the slowing down of the flow in the vicinity of the new minimum in filament radius. (D) The stagnation zone persists but because of the large capillary pressures that develop as the neck continues to thin, fluid is once more accelerated as it flows away from the neck. Thus, inertial forces come into play again and compete with viscous and capillary forces in setting the final fate of the filament.Having clarified the heretofore inadequately understood thinning dynamics of slightly viscous fluids of Oh < 1, we next show that highly viscous fluids of Oh > 1 exhibit even more subtle behavior during capillary thinning. Fig. 4 shows results of simulations and experiments for a fluid of Oh = 1.81. As expected, both simulations (Fig. 4A) and experiments (Fig. 4B) reveal that the initial and final scaling regimes are the V and IV regimes. Conventional wisdom dictates that the transition from the V to the IV regime should occur when h_min ～ Oh^2/(2β?1) = 0.162, which is contradicted by both simulations and experiments. The simulations show (Fig. 4A), and experiments confirm (Fig. 4B), that there exists an intermediate I regime that follows the initial V regime. Local Reynolds number calculations near the pinch point from the simulations are yet even more revealing (Fig. 4C): they show the existence of an intermediate V regime that lies between the intermediate I and the final IV regime. Therefore, according to Fig. 4, the capillary thinning of highly viscous filaments for which Oh > 1 is seen to transition from V to I to V to IV regimes as τ → 0. Furthermore, it is worth noting that Fig. 4A depicts, to our knowledge, the first demonstration by simulation of the transition from an initial V regime to the final IV regime.Open in a separate windowFig. 4.Simulations and experiments demonstrating the existence of several intermediate regimes between the initial viscous regime and the final IV regime for a highly viscous fluid of Oh = 1.81. (A) Computations show that as the filament thins, the dynamics transitions from an initial V regime to the final IV regime through an intermediate I regime (but see C). (B) Experiments accord with the predictions from simulations and exhibit the same transition dynamics. (C) However, local Reynolds number calculations from the simulations reveal more information about the transitions. Whereas the initial V, intermediate I, and final IV regimes are confirmed from the computed variation of Re_local with h_min, this analysis also indicates the existence for a very short time of an intermediate V regime after the I regime.In conclusion, our analysis provides, to our knowledge, the first correct trajectories in the phase space of (h_min, Re_local) that are taken by filaments as they undergo capillary pinching. A particularly interesting finding is that the dynamics cannot reach the asymptotic universal IV regime directly from the I regime without passing through an intermediate transient V regime even though this latter regime may be very short-lived. The presence of the intermediate V regime indicates that even for a low-viscosity fluid, at some stage viscous force (along with capillary force) will dominate the dynamics during filament thinning and breakup. The existence of intermediate regimes has several practical implications as occurrence of slender threads that pinch symmetrically at their midpoints is associated with breakup of highly viscous filaments undergoing creeping flow, whereas occurrence of satellites is associated with inviscid fluids (20, 23, 24, 30). Therefore, the presence of the intermediate I regime makes plain that a visible satellite drop may form even during breakup of highly viscous filaments that reach the IV regime for values of h_min below the limit set by visible light. Additionally, the existence of multiple regime transitions before a filament enters the final IV regime helps explain why it has heretofore proven difficult to observe this regime during pinch-off of highly viscous filaments.The unexpected findings of this work raise a number of questions. Two issues that have not been addressed here are that the amount of time spent by filaments in each regime remains unclear and that similar transitions that may take place during capillary pinching of complex fluids (29, 30) remain unexplored. Moreover, it is well known that there are a number of other free-surface flows that exhibit finite time singularities. Chief among these is the coalescence singularity that arises when two drops are just allowed to touch and then merge into one (31). Whether transitions of the sort uncovered in this work exist in problems like coalescence are worthy topics for future study and may help explain why it took over a decade to uncover the true asymptotic regime of coalescence (31, 32).     

Swallowing Outcomes Following Unilateral STN vs. GPi Surgery: A Retrospective Analysis

The adverse effects of deep brain stimulation (DBS) surgery on swallowing could potentially exacerbate the natural deterioration of airway protection associated with Parkinson’s disease (PD) degeneration and increase the incidence of aspiration pneumonia and associated death. There are no studies that compare swallowing outcomes associated with subthalamic nucleus (STN) versus globus pallidus interna (GPi) DBS surgery; therefore, we completed a retrospective study comparing swallowing outcomes in a cohort of patients with PD who underwent unilateral DBS surgery in either the STN or GPi. A chart review was completed to identify all patients with a diagnosis of PD who received videofluoroscopic swallowing evaluations before DBS and after unilateral DBS in the STN or GPi. The retrospective search yielded 33 patients (STN = 14, GPi = 19) with idiopathic PD who met the inclusion criteria. Mean penetration–aspiration (PA) scores did not change significantly for participants who underwent GPi surgery (z = –.181, p = .857), but mean PA scores significantly worsened for participants who underwent STN DBS (z = –2.682, p = .007). There was a significant improvement in Unified PD Rating Scale (UPDRS) scores off medication before surgery, to off medication and on stimulation after surgery for both groups (F = 23.667, p < .001). Despite the limitations of a retrospective analysis, this preliminary study suggests that unilateral STN DBS may have an adverse effect on swallowing function, while unilateral GPi DBS does not appear to have a similar deleterious effect. This study and other future studies should help to elucidate the mechanisms underpinning the effects of DBS on swallowing function.     

Weak lasing in one-dimensional polariton superlattices

Bosons with finite lifetime exhibit condensation and lasing when their influx exceeds the lasing threshold determined by the dissipative losses. In general, different one-particle states decay differently, and the bosons are usually assumed to condense in the state with the longest lifetime. Interaction between the bosons partially neglected by such an assumption can smear the lasing threshold into a threshold domain—a stable lasing many-body state exists within certain intervals of the bosonic influxes. This recently described weak lasing regime is formed by the spontaneously symmetry breaking and phase-locking self-organization of bosonic modes, which results in an essentially many-body state with a stable balance between gains and losses. Here we report, to our knowledge, the first observation of the weak lasing phase in a one-dimensional condensate of exciton–polaritons subject to a periodic potential. Real and reciprocal space photoluminescence images demonstrate that the spatial period of the condensate is twice as large as the period of the underlying periodic potential. These experiments are realized at room temperature in a ZnO microwire deposited on a silicon grating. The period doubling takes place at a critical pumping power, whereas at a lower power polariton emission images have the same periodicity as the grating.The application of artificial periodic potentials to electrons and photons causes a rich variety of phenomena, from electronic minibands in semiconductor superlattices to characteristic stop bands in photonic crystals (1–4). These phenomena form the basis for further developments of optoelectronics. Cavity polaritons (5, 6) (quasi-particles formed by the strong coupling of confined photons with excitons) attracted much attention in recent years due to the remarkable coherent effects linked to their half-matter, half-light nature (7–11). As a result, a new area of physics at the boundary between solid-state physics and photonics has emerged.Experiments on spatially inhomogeneous polariton condensation are usually interpreted assuming that all one-particle states have the same lifetime (12, 13). Lifting off this assumption leads to the prediction (14) of the “weak lasing” state of interacting polaritons: a type of condensate stabilized by the spontaneous reduction of the symmetry rather than by the dissipation nonlinearities due to, e.g., reservoir depletion. In this work we report, to our knowledge, the first experimental observation of room-temperature polariton condensation in 1D superlattices, which brings clear evidence for the weak lasing state.The polariton superlattice was assembled using a ZnO microrod with a hexagonal cross-section: a natural whispering gallery resonator to efficiently confine exciton–polaritons (15, 16). Setting the microrod on a silicon slice with periodically arranged channels (Fig. 1) allowed us to avoid the intrinsic structural diffraction typical for the structures with periodic patterns deposited on top of microcavities (17).Open in a separate windowFig. 1.Illustration of the assembled polaritonic superlattice based on a ZnO–Si microstructure. (A) Schematic representation of the 1D polaritonic crystal. (B) Scanning electron microscope image (top view) of a ZnO microrod with hexagonal cross-section placed on a periodic Si grating. The 1-μm-wide silicon channels equally spaced with the internal distance a = 2 μm apply a static periodic potential to the polaritons with amplitude ReU ∼ 2 meV. The “s” and “f” mark the silicon-contacting parts and the freestanding parts of the microcavity, respectively. (C and D) ARPL spectral images taken under continuous He–Cd laser (325-nm) excitation. TE (electric field component of light along the z axis) polariton modes are shown. (C) Emission from a freestanding ZnO microrod. The white dashed curves are theoretical fits of the lower polariton branches. (D) The same ZnO microwire lying on a flat silicon surface. The peak position and the lineshape at k_|| = 0 are identified. The horizontal dashed lines indicate the lower polariton energy shift (ReU). The incidence angle ϕ is linked with the in-plane wave vector of light by k_|| = (E/ħc)sin ϕ, where E is the photon energy.The polaritons in this structure were created by nonresonant continuous wave (or long pulse) optical pumping at room temperature, and they were characterized by angle-resolved and spatially resolved photoluminescence (ARPL and SRPL) from the top of the microcavity. The periodic potential caused by the silicon grating manifested itself by a characteristic folded dispersion of the lower polariton branch, revealed in the ARPL images (Fig. 2). One can see the avoided crossing of the polariton dispersion branches resulting in a distinct band gap near the Bragg plane. At strong enough pumping the polariton condensation demonstrates a striking feature: the condensate is formed at the excited polariton states near the energy gaps (states A, Fig. 3A) rather than at the ground state: the π-state condensate (17).Open in a separate windowFig. 2.Dispersion of exciton–polaritons in momentum space demonstrating formation of a polariton superlattice. (A) Photoluminescence mapping (second derivative) in k space under continuous excitation at room temperature. White dashed curves display the calculated dispersion with a band gap (ΔE = 0.7 meV). (B and C) Enlarged regions identified by the dashed rectangles in A, respectively, exhibiting the anticrossing dispersion and well-resolved energy gap.Open in a separate windowFig. 3.Polariton condensation at the π-states. (A) Polariton lasing at the edges of the mini-Brillouin zone in k space at room temperature. White dashed curves display the calculated dispersion with a band gap ΔE = 0.7 meV. (B) Wave vector and energy dependence of the exciton–polariton radiative lifetime (calculated). The color scale is in units of picoseconds (red color represents longer lifetime). (C) (Top and Middle) Real and imaginary parts of the complex potential induced by the Si grating for the microrod polaritons. The “air” and “silicon” represent, respectively, the freestanding parts and silicon-contacting parts of the microcavity. (Bottom) Sketch of the probability amplitude distribution (|Ψ|²) for the states labeled as A and A′ in Fig. 3B. (D) Spatially resolved PL image obtained by exciting the microrod step-by-step along the z axis. The full width at half maximum of the pulsed excitation laser is about 1 μm. The emission bright regions are pinned to the period potential. (E–G) Spatial coherence analysis from a Michelson interferometer. (E and F) Real-space PL mappings measured by each arm of the interferometer. (F) The second arm with a retroreflector flips the image E. The excitation source in this experiment was a pulsed laser (wavelength: 355 nm, diameter of the laser spot: 10 μm). (G) The coherent overlapping between the two images of the condensates forms the interference pattern. The interference fringes appear because of a small inclination angle between the images from the two arms of the interferometer.We checked the spatial coherence of condensates at state A by interferometry experiments. Fig. 3 E and F shows two SRPL images of the condensates coming from the two arms of the Michelson interferometer; Fig. 3F shows the inversion of the pattern Fig. 3E by a retroreflector. Fig. 3G shows the interference pattern created by the superposition of the two images with the relative time delay smaller than the coherence time of the polariton condensate (∼3 ps). The arrows indicate unambiguous interference fringes between two condensates separated by 6 μm. The interference patterns can be observed even for a separation as large as 10 μm, i.e., the π-state condensate (17) in our superlattices indeed demonstrates a long-range coherence.Besides creating the potential wells for polaritons, the contacts of the ZnO microrod with the patterned Si substrate affect the polariton dissipation: In the contact regions (inside the wells) the losses are stronger. This effect naturally explains the π-state condensation: as it is shown in Fig. 3C the minima of the probability amplitude of the state A are at the contact regions, i.e., the polaritons in this state live longer than in the other state A′ at the edge of the Brillouin zone, which has maxima at the contact regions. The ground state D with zero wave vector k is distributed between the wells and the barriers more equally and thus possesses an intermediate lifetime. The presence of long-living states at the Bragg gap edges has been previously observed in experiments on X-ray diffraction in crystals (18) and referred to as the Borrmann effect. A similar effect is known to suppress light localization in disordered photonic crystals (19). We calculated the lifetime of polariton states for the different folded dispersion branches in the simplified Kronig–Penney model (see SI Text for details) and found that in the A state the polaritons indeed live longer than in the states A′ and D, as shown in Fig. 3B (red color corresponds to the longer lifetime): τ_A > τ_D > τ_A^′.It is safe to assume that all polariton states within the Brillouin zone have approximately the same influx rate W, which is proportional to the external pumping rate P. Accordingly, as W increases, the lasing condition Wτ ≥ 1 is satisfied for the A state first. Given the period of the structure a, the emission from this state contains plane waves with wave vectors k_∥ = ±π(2n + 1)/a, where n = 0, ±1, ±2, … .However, the condensate in the A state becomes unstable for interacting polaritons at the second threshold W2>W1=τA−1 (see SI Text for details). The condensed state for W > W₂ is stabilized by the gradually increasing admixture of the D state to the A state. The experimental values of pumping that correspond to the first and the second threshold are P₁ ≈ 10?nW and P₂ ≈ 20?nW. The admixture of the D state is the manifestation of the weak lasing regime (14) characterized by a spontaneous symmetry breaking. Indeed, the condensate wave function can be written asΨ(z) = C_Aψ_A(z) + C_Dψ_D(z), [1]where ψ_A,D are the single-polariton wave functions of the A and D states, and the polariton density |Ψ(z)|² in this state is not periodic with the period a of the underlying lattice. Instead, it is periodic with the period 2a: because the signs of ψ_A(z) are opposite in the neighboring barriers, while the signs of ψ_D(z) are the same, the amplitudes of |Ψ(z)| are different in odd and even barriers. As we show in SI Text, the weak lasing condensate can be formed in two equivalent states, with coefficients C_A,D having the same signs in one state and opposite signs in the other. In both cases, the condensate acquires the double period 2a, and in addition to the pure A-state emission pattern there appears an emission line at k_∥ = 0 (with possible weak satellites at k_∥ = ±2πn/a).We observed the period doubling of the polariton condensate in both ARPL and SRPL images (Fig. 4). At low pumping, the emission has the same periodicity in real space as the superlattice, whereas for pumping above the second threshold of about 20 nW the emission pattern doubles its period. We have checked that for P > 20?nW the three peaks at k_∥ = 0, ?±π/a in Fig. 4C indeed correspond to the same frequency and are mutually coherent. At large pumping, the ratio of the intensity of the k_∥ = 0 peak to the intensity of the k_∥ = ±π/a peaks saturates at about 0.6, which is substantially smaller than the theoretically expected saturation value 1.5 for an ideal 1D lattice. This discrepancy is presumably due to the strong disorder present in the ZnO microrod, which is clearly seen from fluctuations in the amplitudes of the peaks in Fig. 4D.Open in a separate windowFig. 4.Phase transition from π-state condensation to weak lasing. (A) The evolution of polariton condensates in momentum space with increasing pump power. The experiments were carried out on another sample. (B) The power dependence of the PL intensity at the lasing frequency in k space. At the threshold of about 20 nW, a small peak appears at k_|| = 0. Its frequency corresponds to the edges of the Brillouin zone. (C) The evolution of the polariton condensate’s distribution in real space with increasing excitation power. (D) The power dependence of the PL intensity at the lasing frequency in real space. At the threshold of 20 nW, the period of the polariton condensate is doubled compared with the period of the superlattice.Previous low-temperature experiments on GaAs-based polariton superlattices (17) evidenced polariton lasing from the edge of the Brillouin zone but no period doubling. We believe that the weak lasing phase in ZnO polariton superlattices is robust because both real and imaginary parts of the periodic potential are modulated much stronger than in the planar GaAs microcavity with a metallic pattern on the top studied in ref. 17.In conclusion, by the nonresonant optical pumping of a ZnO microrod–Si grid superlattice we created a condensate of exciton–polaritons at room temperature and proved its long-range phase coherence. At sufficiently strong pumping the spatial period of the condensate turned out to be twice as long as the period of the superlattice. This spontaneous symmetry breaking strongly suggests that the weak lasing regime of polariton condensation has been achieved in our experiments.     

Compositional landscape for glass formation in metal alloys

A high-resolution compositional map of glass-forming ability (GFA) in the Ni–Cr–Nb–P–B system is experimentally determined along various compositional planes. GFA is shown to be a piecewise continuous function formed by intersecting compositional subsurfaces, each associated with a nucleation pathway for a specific crystalline phase. Within each subsurface, GFA varies exponentially with composition, wheres exponential cusps in GFA are observed when crossing from one crystallization pathway to another. The overall GFA is shown to peak at multiple exponential hypercusps that are interconnected by ridges. At these compositions, quenching from the high-temperature melt yields glassy rods with diameters exceeding 1 cm, whereas for compositions far from these cusps the critical rod diameter drops precipitously and levels off to 1 to 2 mm. The compositional landscape of GFA is shown to arise primarily from an interplay between the thermodynamics and kinetics of crystal nucleation, or more precisely, from a competition between driving force for crystallization and liquid fragility.The glass-forming ability, or GFA, of a liquid metal alloy is not an intrinsic material attribute, but rather defined by the absence of a viable crystallization pathway as the liquid is undercooled below its thermodynamic melting temperature (1, 2). Crystallization is typically triggered by nucleation of a particular crystalline phase, followed by other competing phases, often catalyzed by the presence of the first phase. Crystal nucleation rates depend not only on temperature, pressure, and alloy composition, but also on extrinsic factors such as the presence of chemical impurities, trace crystalline debris (e.g., oxide inclusions), container wall effects, or shear flow conditions in the liquid, to name a few (3–7). Variations in these extrinsic factors often lead to inconsistent and nonreproducible GFA.The classical nucleation theory of crystals in undercooled liquids was originally developed by Turnbull (1) to account for the substantial undercooling observed in elemental liquid metals. He later extended his theory to explain metallic glass formation in rapidly cooled low melting eutectic Au–Si and Au–Ge–Si alloys (8, 9). Below the liquidus temperature T_L, the liquid viscosity, η(T), rises steeply with falling temperature. A liquid ultimately freezes at a glass transition temperature T_g, where the viscosity reaches a solid-like value of ∼10¹² Pa⋅s. Turnbull considered the “reduced glass transition temperature” t_rg = T_g/T_L as a characteristic material parameter. He argued that crystal nucleation rates should fall precipitously as t_rg increases, becoming immeasurably small for t_rg ≈ 2/3. This is widely referred to as Turnbull’s criteria for bulk glass formation; it has been proven to be a valuable, albeit rough, guide in the development of bulk metallic glasses (10–12).In the present work, a systematic experimental approach is developed to quantify the intrinsic dependence of GFA on composition for near-eutectic multicomponent metal alloys. The optimization of GFA for quinary Ni–Cr–Nb–P–B alloys is presented as a case study wherein bulk glasses of centimeter thickness are achieved. This quinary system is based on the low melting binary Ni₈₁P₁₉ eutectic alloy with small additions of Cr and Nb as substitutes for Ni, and B as a substitute for P.Binary Ni–P and ternary Ni–Cr–P alloys have long been known to form glassy ribbons of 20 to 40-μm thickness on quenching from the melt at cooling rates of 10⁵ to 10⁶ K/s using rapid melt quenching approaches such as planar flow casting (13, 14). Following the discovery of bulk metallic glasses, Hashimoto and coworkers (15) as well as Inoue and coworkers (16) investigated bulk glass formation in quinary Ni–Cr–Nb–P–B alloys and identified specific alloy compositions capable of forming metallic glass rods with diameters of 1 to 2 mm. In the current investigation, by using an efficient and reproducible GFA assessment and optimization strategy, we report that maximum attainable metallic glass rod diameters in the same Ni–Cr–Nb–P–B system are an order of magnitude larger (1–2 cm) than reported in prior work.To accurately quantify the intrinsic composition dependence of GFA requires (i) precisely controlling alloy composition and impurity content, (ii) quantitatively and reproducibly determining GFA at a specific composition by controlling the sample cooling history, and (iii) minimizing the influence of extrinsic factors such as heterogeneous nucleation sites (foreign oxide inclusions, the container wall, etc.) and melt flow conditions during cooling. In this work, alloys with precisely controlled composition were produced from high-purity starting elements. GFA was determined by melting the alloys in silica tubes and subsequently water quenching to form metallic glass rods. The silica tubes exhibit no detectable reaction with the present alloys. Moreover, being a glass, the silica tube is not expected to induce heterogeneous nucleation at the inner wall. Finally, the melt being confined inside the tube during quenching does not undergo significant shear flow. As such, the cooling history of the sample is governed almost solely by conduction without any significant convection, and is therefore expected to be reproducible. GFA is characterized by a critical rod diameter, d_cr, defined as the largest diameter rod that can be quenched into a fully glassy structure without detectable crystallinity, as verified by X-ray diffraction. Additional details on the alloy preparation, cooling history, and determination of GFA are presented in Materials and Methods.The quinary Ni–Cr–Nb–P–B system has a 4D composition space with independent variables, w, x, y, and z, where composition is expressed as Ni_{100−w−x−y−z}Cr_wNb_xP_yB_z. The variables are in atomic percentages. In Fig. 1A, we present a detailed 2D GFA contour map associated with composition variation along 2 degrees of freedom, w and x, while keeping y and z constant at 16.5 and 3, respectively. This high-resolution contour map is based on the measured d_cr for 42 alloys. Two distinct local maxima with d_cr ≥ 10 mm are clearly evident in the contour map. More specifically, along the compositional line x = 4.0625−0.125w (where y and z are held constant at 16.5 and 3), a ridge interconnecting the two peaks is observed in the GFA landscape. The compositional dependence of GFA along this compositional line is presented in Fig. 1B. Along this ridge and within 4.5 < w < 10.5 (which corresponds to 2.75 < x < 3.5), d_cr is found to vary between 8 and 10 mm, whereas a precipitous dip in the GFA is observed for w < 4.5 and w > 10.5. At w = 5.6 and 8.5 (corresponding to x = 3.4 and 3), the compositions Ni_71.5Cr_5.6Nb_3.4P_16.5B₃ and Ni₆₉Cr_8.5Nb₃P_16.5B₃ are seen to exhibit local maxima with d_cr ≥ 10 mm.Open in a separate windowFig. 1.(A) Two-dimensional GFA contour map for Ni_80.5−w−xCr_wNb_xP_16.5B₃ alloys plotting the critical rod diameter d_cr against the Cr and Nb atomic concentrations w and x, while keeping the P and B atomic concentrations y and z constant at 16.5% and 3%, respectively. (B) One-dimensional GFA plot for Ni_{77.4375−0.875w}Cr_wNb_{4.0625−0.125w}P_16.5B₃ alloys plotting the critical rod diameter d_cr against the Cr atomic concentration w along the compositional line x = 4.0625 − 0.125w associated with the GFA ridge in the w–x domain shown in A. The dotted line is a trend line through the experimental data (open circles).In Fig. 2A we present a second 2D GFA contour map associated with composition variation in the x–z plane. This map was generated from GFA data on 58 separate alloy compositions. Another ridge in the GFA landscape is identified in this compositional plane along the compositional line x = z where the sum of transition metals (w + x) and metalloids (y + z) are held constant at 11.5 and 19.5, respectively. The GFA compositional dependence along this line is presented in Fig. 2B. Along this ridge with 3< (x, z) < 4, d_cr varies between 9 and 10 mm, whereas it gradually degrades outside this range. A ridge in GFA along x = z suggests that the GFA dependence on Nb and B contents is strongly correlated. This suggests Nb and B atoms tend to occupy associated sites in the short-range configurational order of the glass structure. It is also worth noting that another shallower peak is identified in this 2D plane, isolated from the x = z ridge near x = 2 and z = 5, where d_cr ≈ 7 mm.Open in a separate windowFig. 2.(A) Two-dimensional GFA contour map for Ni₆₉Cr_11.5−xNb_xP_19.5−zB_z alloys plotting the critical rod diameter d_cr against the Nb and B atomic concentrations x and z, while keeping the sum of Cr and Nb and the sum of P and B atomic concentrations (w + x) and (y + z) constant at 11.5% and 19.5%, respectively. (B) One-dimensional GFA plot for Ni₆₉Cr_11.5−zNb_zP_19.5−zB_z alloys plotting the critical rod diameter d_cr against the B atomic concentration z along the compositional line x = z associated with the GFA ridge in the x–z domain shown in A. The dotted line is a trend line through the experimental data (open circles).None of the local maxima identified in the compositional planes of Figs. 1 and ​and22 necessarily represent a global maximum for the overall GFA. Other higher maxima, including an absolute global maximum, may exist in the 4D composition space along different planes; however such maxima are not expected to be far from the common peaks appearing in the planes of Figs. 1A and ​and2A.2A. If a global GFA maximum exists in the current compositional neighborhood it would be hard to predict it given the steepness of the GFA composition maps. In the present work, alloys with an even higher GFA have in fact been discovered for compositions in the neighborhood surrounding Ni₆₉Cr_8.5Nb₃P_16.5B₃. For example, a six-component alloy Ni_68.6Cr_8.7Nb₃P₁₆B_3.2Si_0.5, which includes a minority addition of Si in its metalloid moiety, demonstrates d_cr ≈ 20 mm when processed by a high-temperature fluxing process (Materials and Methods). A fully amorphous 17-mm-diameter rod of Ni_68.6Cr_8.7Nb₃P₁₆B_3.2Si_0.5, along with an X-ray diffraction pattern and a calorimetric scan verifying its amorphous structure, are shown in Fig. 3.Open in a separate windowFig. 3.(A) A sectioned 17-mm fully amorphous Ni_68.6Cr_8.7Nb₃P₁₆B_3.2Si_0.5 rod (section shown next to a dime for comparison). (B) X-ray diffractogram taken along the rod cross-section verifying the amorphous structure of the rod. (C) Differential calorimetry scan taken at 20 K/min scan rate. Arrows from left to right designate the glass transition, crystallization onset, solidus, and liquidus temperatures of 678 K, 722 K, 1119 K, and 1157 K, respectively.To map the detailed compositional dependence of GFA in the neighborhood of a peak, we evaluated the GFA along a family of straight lines in the composition space that intersect the peak, where each line is referred to as an “alloy series.” The detailed compositional dependence of GFA for four alloy series (labeled I–IV) intersecting the common composition Ni₆₉Cr_8.5Nb₃P_16.5B₃ is presented in Fig. 4. Series I–IV respectively correspond to varying w, x, z, and (y + z) around the composition Ni₆₉Cr_8.5Nb₃P_16.5B₃, which corresponds to a local maximum in GFA in the 4D compositional space. For series I–IV one observes a steeply rising GFA followed by a rapidly decaying GFA as the peak at d_cr is traversed. These GFA functions along the composition lines generally consist of piecewise continuous curves, or branches. The curves meet at cusps. It was observed that the individual branches of the GFA map have a roughly exponential dependence on composition (see SI Materials and Methods for a list of the exponential fitting parameters.). We argue below that the atomic rearrangement and formation of a critical nucleus within a liquid are both thermally activated processes with associated barrier heights that should depend linearly, to leading order, on composition. As such, an exponential dependence of GFA on composition ensues.Open in a separate windowFig. 4.Compositional dependence of GFA along four series intersecting a GFA peak at composition Ni₆₉Cr_8.5Nb₃P_16.5B₃. Solid lines are exponential fits to the experimental data (open circles) on each side of the peak. (A) Critical rod diameter d_cr plotted against the Cr atomic concentration as a substitute for Ni according to Ni_77.5−wCr_wNb₃P_16.5B₃. (B) Critical rod diameter d_cr plotted against the Nb atomic concentration as a substitute for Cr according to Ni₆₉Cr_11.5−xNb_xP_16.5B₃. (C) Critical rod diameter d_cr plotted against the B atomic concentration as a substitute for P according to Ni₆₉Cr_8.5Nb₃P_19.5−zB_z. (D) Critical rod diameter d_cr plotted against the atomic concentration of metalloids substituting for metals according to (Ni_0.8541Cr_0.1085Nb_0.0374)_100−(y+z)(P_0.8376B_0.1624)_(y+z).From transition state theory, the nucleation of a phase α (α refers to one of the competing crystalline phases) is expected to be an activated process involving the crossing of a temperature and composition dependent nucleation barrier, ΔG_α(T, c) (5, 6, 8, 9). The rate at which atomic configurations are sampled in an undercooled liquid is taken to be proportional to the liquid fluidity, or inverse viscosity η^?1. Viscous flow is taken to be an activated process that may be characterized by a temperature and composition dependent barrier, W(T, c) (17, 18). The characteristic time for the nucleation of an α-crystal takes the general formτ_α = ν^?1?exp[{W(T, c) + ΔG_α(T, c)}/kT], [1]where ν is an attempt frequency taken to be a characteristic atomic vibrational frequency in the liquid. For a fixed c, a plot of ln?τ_α vs. T produces the well-known C-shaped time-temperature-transformation (TTT) diagram for the α-crystal nucleation. Below the α-liquidus temperature where the crystal becomes thermodynamically stable, competition between a rising W vs. a falling ΔG_α determines the TTT diagram. The TTT diagram exhibits a minimum crystallization time scale, τα∗, at an associated “nose” temperature, Tα∗, for which the nucleation time is minimized. One obtains τα∗ and Tα∗ by requiring dτ_α/dT = 0 at fixed c. Assuming τα∗ to be a well-behaved function of c and expanding τα∗ vs. c in a Taylor series around an initial composition c_o, one obtains the compositional dependence of τα∗ for a small composition change, c − c_o, to leading order asτα∗(c)−τα∗(co)≈exp[λα⋅(c−co)],[2]where λα=∇(ln⁡τα∗(c))=∇[{W(T∗,c)+ΔGα(T∗,c)}/kT∗] is formally the gradient vector with respect to composition of ln⁡τα∗(c) evaluated at a fixed composition, c_o, and at the nose temperature T^?. Eq. 2 predicts that for the α-crystallization pathway, ln⁡τα∗ should vary exponentially with c for small compositional displacements, c − c_o. The exponential composition dependence is a consequence of crystal nucleation being a thermally activated process. The minimum crystallization time, τα∗, is a fundamental measure of GFA. According to the Fourier heat flow equation, one expects the time scale for cooling a molten sample to scale with the square of the lateral dimension of the sample, i.e., τα∗≈dcr2. Accordingly, a GFA composition map that reflects the composition variation of τα∗ can be constructed by mapping dcr2 as a function of composition. This map would reflect the composition variation of τα∗ for the α-crystallization event where such event has the shortest (i.e., the limiting) crystallization time. Collectively considering the set of all competing crystallization pathways for the various crystalline phases (e.g., α, β, γ, etc.), a global GFA composition map is constructed determined at each composition by the competing phase which has the shortest crystallization time. The overall GFA map will consist of piecewise continuous exponential subsurfaces, indexed by α, β, γ, etc., which intersect to form exponential cusps at compositions associated with the cross-over in the nucleation pathway (e.g., from α to β, etc.).According to Eq. 1, GFA is determined by the competition of two thermally activated processes: that of forming the critical nucleus of the crystalline phase and that of configurationally rearranging the liquid, having respective activation barriers W(T^?, c) and ΔG_α(T^?, c). In our analysis, we only assume that these barriers are smooth functions of temperature and composition. The rising η with increasing undercooling can be well described by the liquid fragility parameter m, defined as m = [d(log?η)/d(T_g/T)]_Tg together with the value of T_g (19). To lowest order, the driving force for crystallization with increasing liquid undercooling scales with 1 ? T/T_L, so that ΔG_α scales according to (1?T/T_L)^?2 for modest undercooling (2). So at the Kauzmann temperature T_K where the entropy of the liquid is assumed to match the entropy of the crystal (also referred to as the ideal or thermodynamic glass transition) (20), the leading term in ΔG_α would be of order (1?T_K/T_L)^?2. Here we assume that the calorimetric glass transition temperature T_g, which is readily accessible experimentally, adequately approximates T_K. As such, the leading term in ΔG_α is taken here to be of order (1?t_rg)^?2. Both t_rg and m are experimentally accessible material properties that provide a quantitative measure of the respective activation barriers along with their variation with composition. One can describe mathematically the dependence of the crystallization time scale τα∗ on the independent parameters t_rg and m by examining the dependence of the activation barriers W(T, c) and ΔG_α(T, c) on these parameters (see SI Materials and Methods for a more detailed discussion). Below we argue that the observed compositional dependence of GFA in the present work is attributable almost entirely to the combined effects of varying t_rg and m with composition.To clarify the origin of the composition dependence of GFA, we performed detailed calorimetric and rheometric measurements to evaluate both t_rg and m as functions of composition along the representative alloy series III (Fig. 4C), Ni₆₉Cr_8.5Nb₃P_19.5−zB_z (see Materials and Methods for details on measurements of m and t_rg, and SI Materials and Methods for calorimetry and viscosity plots). A similar analysis can be applied to the other composition series. The GFA composition data along this series are plotted in terms of dcr2 vs. z in Fig. 5A; the data were fitted by two exponential functions of composition for the two branches of the GFA curve. For a local GFA maximum at composition c_o, like the one at 3 atomic percent B (i.e., at z_o = 3), the τα∗ values for two different competing crystallization pathways cross over and are mutually equal at z_o. We arbitrarily refer to the two crystal branches (low z and high z) of the GFA curve using crystal labels “α” and “β ”. Firstly, within the error of our calorimetry data, we observe that c_o is located quite precisely at a eutectic composition for which the alloy liquidus is minimum (see SI Materials and Methods for melting data). This is actually true for all of the alloy series I–IV (Fig. 4). GFA is therefore optimized very close to a quinary eutectic composition. The d_cr data for series I–IV, therefore, describe GFA along lines in composition space that all pass through this eutectic composition. From the calorimetric liquidus measurements along series III, the α region z < z_o is a hypoeutectic (with a falling liquidus curve as z increases) region whereas the β region z_o > z_o is hypereutectic (with a rising liquidus curve as z increases). Because t_rg depends inversely on the liquidus temperature, its composition dependence demonstrates a cusp-like maximum with discontinuous slope at z_o. The plot of t_rg along series III shown in Fig. 5B reveals this sharp discontinuity. Specifically, t_rg is highest at the eutectic composition and drops precipitously for z > z_o. In contrast, t_rg increases very slightly as z approaches z_o in the hypoeutectic region. On the other hand, the experimental liquid fragility, m, being a property of the liquid phase alone and independent of the crystal/liquid phase equilibria, is shown in Fig. 5C to be a continuous and monotonically decreasing function of c. Specifically, m drops steeply as z increases with an approximately exponentially decaying trend. A decreasing m corresponds to a higher viscosity throughout the undercooled liquid region, i.e., from T_g to T_L, thereby implying a greater viscosity at the nucleation temperature, Tα∗. One therefore expects GFA to increase with falling m.Open in a separate windowFig. 5.(A) Compositional dependence of GFA plotted in terms of dcr2 against the B atomic concentration as a substitute for P according to Ni₆₉Cr_8.5Nb₃P_19.5-zB_z. Solid lines are exponential fits to the experimental data (open circles) on each side of the peak (at 3 at. % B). (B) Reduced glass transition temperature t_rg plotted against the B atomic concentration as a substitute for P according to Ni₆₉Cr_8.5Nb₃P_19.5-zB_z. Solid lines are polynomial fits to the experimental data (open circles) on each side of the cusp (at 3 at.% B). (C) Liquid fragility m plotted against the B atomic concentration as a substitute for P according to Ni₆₉Cr_8.5Nb₃P_19.5-zB_z. The solid line is an exponential fit to the experimental data (open circles).By qualitatively analyzing the plots of Fig. 5 B and C, one can gain critical insight into the origin of the piecewise continuous form of the GFA function (Fig. 5A). The drop in t_rg for z > z_o is steep and should lead to rapid exponential decay of GFA. This drop in t_rg in the hypereutectic β-region arises from a steep rise in T_L (SI Materials and Methods). In turn, this increases the crystallization driving force and steeply reduces ΔG_β such that GFA decays rapidly. For z < z_o, one has nearly constant or slightly rising t_rg as z increases. This slight rise in t_rg cannot explain the sharply rising GFA. Rather, the sharp GFA rise in the hypoeutectic α-region can be plausibly explained by the rapidly falling fragility, m. As the liquid structure becomes stronger (decreasing m with increasing B content), the rate at which atomic configurations are sampled at Tα∗ slows, increasing the time to achieve a critical crystal nucleus, τα∗. The hypo- and hypereutectic branches of the GFA map are apparently the result of an interplay of the t_rg and m variations with composition.To quantify the analysis above, consider Eq. 2 and the exponential decay parameters λ_α and λ_β obtained from the exponential fits to the GFA data for the respective hypo- and hypereutectic branches of the GFA curve. According to Eq. 2, the discontinuity in the slope of ln⁡dcr2 vs. z at the GFA cusp is simply given by the difference λ_β ? λ_α. Without loss of generality, one can show that λ_α and λ_β can be separated into contributions arising from variations in t_rg and m, and potentially any other relevant material parameters that enter the expression for ln⁡τα∗(c) (the mathematical details of this separation are presented in SI Materials and Methods). Using the chain rule to evaluate λα=∇(ln⁡τα∗(c)), one obtainsλ_α = λ_m ? λ_trg,α[3a]λ_β = λ_m ? λ_trg,β.[3b]Here λ_m being a property of the liquid phase only, is identical for both branches; its composition dependence given by λm={[d(ln⁡dcr2)/dm](dm/dc)}trg. The λ_trg parameters depend on the nucleating crystal and are different for α and β; they are given by λtrg={[d(ln⁡dcr2)/dtrg](dtrg/dc)}m. The λ_trg jumps discontinuously with varying c on going from the α- to the β-branch of the curve due to the change in the slope dt_rg/dc (as seen in Fig. 5B). It was already pointed out that the composition c_o of the GFA cusp coincides precisely with the eutectic composition (at which the α- and β-liquidus curves cross). From this, one can conclude that any other parameter that influences GFA (e.g., the melt–crystal interfacial energy for the α- and β-crystalline phases) must be nearly equal for both branches of the GFA curve. Were this not the case, the GFA cusp would be shifted off the eutectic composition, c_o. The above facts suggest that λ_α and λ_β in Eqs. 3a and 3b are determined mainly by the measured composition dependences of t_rg and m. In this case, Eqs. 3a and 3b become a pair of linear equations with two unknowns, λ_m and d(ln⁡dcr2)/dtrg. At the cusp composition, experimental fitting yields λ_α = 1.43, λ_β = ?0.693, dt_rg,α/dc = 0.00211, and dt_rg,β/dc = ?0.0213 (where c is in units of z, i.e., in atomic percentages of B, and d_cr is in millimeters; Fig. 5 A and B and SI Materials and Methods). Solving for the two unknowns one obtains λ_m = 1.24 and d(ln⁡dcr2)/dtrg=90.6, which give λ_trg,α = 0.191 and λ_trg,β = ?1.93. One finds that λ_α consists of a fragility contribution λ_m/λ_α = 87%, and a contribution from Turnbull’s parameter λ_trg,α/λ_α = 13%. In the hypoeutectic α-region, therefore, the decreasing liquid fragility parameter (i.e., the increasing liquid viscosity at the nucleation nose) dominates the exponential rise in GFA. By contrast, the negative λ_β consists of an overwhelming negative contribution from Turnbull’s parameter λ_trg,β/λ_β = 278%, and a positive contribution from fragility λ_m/λ_β = ?178%. Hence, the exponential GFA decay in the hypereutectic β-region would have been significantly greater (by a factor of 2.78) had the fragility of the liquid not been decreasing with increasing c.As follows from the analysis above, the Turnbull parameter and fragility alone give a plausible and self-consistent account of the composition variations of GFA. Specifically, the large variation in fragility over a fairly narrow compositional change has a dramatic effect on the compositional dependence of GFA; it steepens the GFA rise in the hypoeutectic region and offsets the GFA drop in the hypereutectic region. Finally, we note that from our fit for m(z) in Fig. 5C, we have dm/dc = ?5.4 as evaluated at z_o. Using our value of λ_m, one obtains the intrinsic dependence of GFA on m, i.e., d(ln⁡dcr2)/dm=−0.23. For a fixed t_rg, this value implies that a decrease in the fragility parameter m of about 4.5 is associated with a remarkable 65% increase in d_cr. Clearly, the liquid fragility m plays a very important role in determining glass formation. Mukherjee et al. (21) directly measured the TTT diagrams for a series of compositionally distinct Zr-based glasses all having nearly the same t_rg but varying liquid fragility. The measured τα∗, which was found to vary by more than 1 order of magnitude among these alloys, was shown to be directly proportional to the liquid viscosity at the nose temperature Tα∗. It was argued that the GFA variation among these Zr-based glasses arises mainly from the variation in fragility m. In a separate work, Na et al. (22) studied the fragility of compositionally distinct Fe-based metallic glasses, and showed that their GFA and fragility obey a fairly tight correlation that extended over nearly 2 orders of magnitude in τα∗. These studies are consistent with the key findings of the present work. One is naturally led to consider the numerous parameters and criteria proposed in the literature to guide the discovery of bulk metallic glasses (3, 6, 10–12, 21). Based on the present work, it is clear that the successful prediction of GFA requires, at a minimum, properly describing the roles of Turnbull’s parameter and liquid fragility.     

Quantum dimer model for the pseudogap metal

We propose a quantum dimer model for the metallic state of the hole-doped cuprates at low hole density, p. The Hilbert space is spanned by spinless, neutral, bosonic dimers and spin S = 1/2, charge + e fermionic dimers. The model realizes a “fractionalized Fermi liquid” with no symmetry breaking and small hole pocket Fermi surfaces enclosing a total area determined by p. Exact diagonalization, on lattices of sizes up to 8 × 8, shows anisotropic quasiparticle residue around the pocket Fermi surfaces. We discuss the relationship to experiments.The recent experimental progress in determining the phase diagram of the hole-doped Cu-based high-temperature superconductors has highlighted the unusual and remarkable properties of the pseudogap (PG) metal (Fig. 1). A characterizing feature of this phase is a depletion of the electronic density of states at the Fermi energy (1, 2), anisotropically distributed in momentum space, that persists up to room temperature.Open in a separate windowFig. 1.Schematic phase diagram of hole-doped cuprates (apart from those with La doping) as a function of temperature (T) and hole density (p). The antiferromagnetic (AF) insulator is present near p = 0, and the d-wave superconductor (dSC) is present below a critical temperature T_c. The pseudogap (PG) is present for T < T^* and acquires density wave (DW) order at low T. The metallic states are the PG metal, the conventional Fermi liquid (FL), and the strange metal (SM). The dimer model of the present paper describes only the PG metal as a fractionalized Fermi liquid (FL*).Attempts have been made to explain the pseudogap metal using thermally fluctuating order parameters; we argue below that such approaches are difficult to reconcile with recent transport experiments. Instead, we introduce a new microscopic model that realizes an alternative perspective (3), in which the pseudogap metal is a finite temperature (T) realization of an interesting quantum state: the fractionalized Fermi liquid (FL*). We show that our model is consistent with key features of the pseudogap metal observed by both transport and spectroscopic probes.The crucial observation that motivates our work is the tension between photoemission and transport experiments. In the cuprates, the hole density p is conventionally measured relative to that of the insulating antiferromagnet (AF), which has one electron per site in the Cu d band. Therefore, the hole density relative to a filled Cu band, with two electrons per site, is actually 1 + p. In fact, photoemission experiments at large hole doping observe a Fermi surface enclosing an area determined by the hole density 1 + p (4), in agreement with the Luttinger relation. In contrast, in the pseudogap metal, a mysterious “Fermi arc” spectrum is observed (5–7), with no clear evidence of closed Fermi surfaces. However, despite this unusual spectroscopic feature, transport measurements report vanilla Fermi liquid properties, but associated with carrier density p, rather than 1 + p. The carrier density of p was indicated directly in Hall measurements (8), whereas other early experiments indicated suppression of the Drude weight (9–11). Although the latter could be compatible with a carrier density of 1 + p but with a suppressed kinetic term, the Hall measurements indicate the suppression of the Drude weight is more likely due to a small carrier density. Two recent experiments displaying Fermi liquid behavior at low p are especially notable: (i) the quasiparticle lifetime τ(ω, T) determined from optical conductivity experiments (12) has the Fermi liquid-like dependence 1/τ ∝ (?ω)² + (cπk_BT)², with c an order unity constant; and (ii) the in-plane magnetoresistance of the pseudogap (13) is proportional to τ^?1(1 + bH²τ² + …) in an applied field H, where τ ～ T^?2 and b is a T-independent constant; this is Kohler’s rule for a Fermi liquid.It is difficult to account for the nearly perfect Fermi liquid-like T dependence in transport properties of the pseudogap in a theory in which a large Fermi surface of size 1 + p (14) is disrupted by a thermally fluctuating order. In such a theory, we expect that transport should instead reflect the T dependence of the correlation length of the order.Moreover, a reasonable candidate for the fluctuating order has not yet been identified. The density wave (DW) order present at lower temperature in the pseudogap regime has been identified to have a d-form factor (15–18), and its temperature dependence (19–25) indicates that it is unlikely to be the origin of the pseudogap present at higher temperature. Similar considerations apply to other fluctuating order models (26) based on AF or d-wave superconductor.We are therefore led to an alternative perspective (3), in which the pseudogap metal represents a new quantum state that could be stable down to very low T, at least for model Hamiltonians not too different from realistic cuprate models. The observed low-T DW order is then presumed to be an instability of the pseudogap metal (27–31). An early discussion (32) of the pseudogap metal proposed a state that was a doped spin liquid with “spinon” and “holon” excitations fractionalizing the spin and charge of an electron: the spinon carries spin S = 1/2 and is charge neutral, whereas the holon is spinless and carries charge + e. However, this state is incompatible with the sharp “Fermi arc” photoemission spectrum (7) around the diagonals of the Brillouin zone: the spin liquid has no sharp excitations with the quantum number of an electron and so will only produce broad multiparticle continua in photoemission.Instead, we need a quantum state that has long-lived electron-like quasiparticles around a Fermi surface of size p, even though such a Fermi surface would violate the Luttinger relation of a Fermi liquid. The fractionalized Fermi liquid (FL*) (33) fulfills these requirements.     

Systematic review and meta-analysis of published, randomized, controlled trials comparing suture anastomosis to stapled anastomosis for ileostomy closure

The objective of this article is to systematically analyze the randomized, controlled trials comparing the effectiveness of suture anastomosis (SUA) versus stapled anastomosis (STA) in patients undergoing ileostomy closure. Randomized, controlled trials comparing the effectiveness of SUA versus STA in patients undergoing ileostomy closure were analyzed using RevMan^®, and combined outcomes were expressed as odds risk ratio (OR) and standardized mean difference (SMD). Four randomized, controlled trials that recruited 645 patients were retrieved from electronic databases. There were 327 patients in the STA group and 318 patients in the SUA group. There was significant heterogeneity among included trials. Operative time (SMD ?1.02; 95 % CI ?1.89, ?0.15; z = 2.29; p < 0.02) was shorter following STA compared to SUA. In addition, risk of small bowel obstruction (OR 0.54; 95 % confidence interval (CI), 0.30, 0.95; z = 2.13; p < 0.03) was lower in the STA group. Risk of anastomotic leak (OR 0.87; 95 % CI 0.12, 6.33; z = 0.14; p = 0.89), surgical site infection, reoperation and readmission were similar following STA and SUA in patients undergoing ileostomy closure. Length of hospital stay was also similar between STA and SUA groups. In ileostomy closure, STA was associated with shorter operative time and lower risk of postoperative small bowel obstruction. However, STA and SUA were similar in terms of anastomotic leak, surgical site infection, readmission, reoperations and length of hospital stay.     

Latent structure in random sequences drives neural learning toward a rational bias

People generally fail to produce random sequences by overusing alternating patterns and avoiding repeating ones—the gambler’s fallacy bias. We can explain the neural basis of this bias in terms of a biologically motivated neural model that learns from errors in predicting what will happen next. Through mere exposure to random sequences over time, the model naturally develops a representation that is biased toward alternation, because of its sensitivity to some surprisingly rich statistical structure that emerges in these random sequences. Furthermore, the model directly produces the best-fitting bias-gain parameter for an existing Bayesian model, by which we obtain an accurate fit to the human data in random sequence production. These results show that our seemingly irrational, biased view of randomness can be understood instead as the perfectly reasonable response of an effective learning mechanism to subtle statistical structure embedded in random sequences.People are prone to search for patterns in sequences of events, even when the sequences are completely random. In a famous game of roulette at the Monte Carlo casino in 1913, black repeated a record 26 times—people began extreme betting on red after about 15 repetitions (1). The gambler’s fallacy—a belief that chance is a self-correcting process where a deviation in one direction would induce a deviation in the opposite direction—has been deemed a misperception of random sequences (2). For decades, this fallacy is thought to have originated from the “representativeness bias,” in which a sequence of events generated by a random process is expected to represent the essential characteristics of that process even when the sequence is short (3).However, there is a surprising amount of systematic structure lurking within random sequences. For example, in the classic case of tossing a fair coin, where the probability of each outcome (heads or tails) is exactly 0.5 on every single trial, one would naturally assume that there is no possibility for some kind of interesting structure to emerge, given such a simple and desolate form of randomness. And yet, if one records the average amount of time for a pattern to first occur in a sequence (i.e., the waiting time statistic), it is significantly longer for a repetition (head–head HH or tail–tail TT, six tosses) than for an alternation (HT or TH, four tosses). This is despite the fact that on average, repetitions and alternations are equally probable (occurring once in every four tosses, i.e., the same mean time statistic). For both of these facts to be true, it must be that repetitions are more bunched together over time—they come in bursts, with greater spacing between, compared with alternations. Intuitively, this difference comes from the fact that repetitions can build upon each other (e.g., sequence HHH contains two instances of HH), whereas alternations cannot. Statistically, the mean time and waiting time delineate the mean and variance in the distribution of the interarrival times of patterns, respectively (4). Despite the same frequency of occurrence (i.e., the same mean), alternations are more evenly distributed over time than repetitions (i.e., different variances). Another source of insight comes from the transition graph (Fig. 1A), which reveals a structural asymmetry in the process of fair coin tossing. For example, when the process has the same chance to visit any of the states, the minimum number of transitions it takes to leave and then revisit a repetition state is longer than that for an alternation state. Let p_A denote the probability of alternation between any two consecutive trials; despite the same mean time at p_A = 1/2, repetitions will have longer waiting times than alternations as long as p_A > 1/3 (Fig. 1B). (See SI Text for the calculation of mean time and waiting time statistics.)Open in a separate windowFig. 1.Time of patterns described by the probability of alternation between consecutive trials (p_A). (A) Transitions between patterns of length 2. At p_A = 1/2, the process has the same chance to visit either a repetition state (HH or TT) or an alternation state (HT or TH). However, it takes a minimum of three transitions for the process to leave and then revisit a repetition state (e.g., HH  →  HT  →  TH  →  HH), but only two for an alternation state (e.g., HT  →  TH  →  HT). (B) Equilibriums by p_A values. A repetition (R) and an alternation (A) have the same mean time E[T_R] = E[T_A] at p_A = 1/2, the same waiting time E[TR∗]=E[TA∗] at p_A = 1/3, and the same sum E[TR]+E[TR∗]=E[TA]+E[TA∗] at p_A = 3/7.Is this latent structure of waiting time just a strange mathematical curiosity or could it possibly have deep implications for our cognitive-level perceptions of randomness? It has been speculated that the systematic bias in human randomness perception such as the gambler’s fallacy might be due to the greater variance in the interarrival times or the “delayed” waiting time for repetition patterns (4, 5). Here, we show that a neural model based on a detailed biological understanding of the way the neocortex integrates information over time when processing sequences of events (6, 7) is naturally sensitive to both the mean time and waiting time statistics. Indeed, its behavior is explained by a simple averaging of the influences of both of these statistics, and this behavior emerges in the model over a wide range of parameters. Furthermore, this averaging dynamic directly produces the best-fitting bias-gain parameter for an existing Bayesian model of randomness judgments (8), which was previously an unexplained free parameter and obtained only through parameter fitting. We also show that we can extend this Bayesian model to better fit the full range of human data by including a higher-order pattern statistic, and the neurally derived bias-gain parameter still provides the best fit to the human data in the augmented model. Overall, our model provides a neural grounding for the pervasive gambler’s fallacy bias in human judgments of random processes, where people systematically discount repetitions and emphasize alternations (9, 10).     

From the Cover: Evidence for superconductivity in Li-decorated monolayer graphene

Monolayer graphene exhibits many spectacular electronic properties, with superconductivity being arguably the most notable exception. It was theoretically proposed that superconductivity might be induced by enhancing the electron–phonon coupling through the decoration of graphene with an alkali adatom superlattice [Profeta G, Calandra M, Mauri F (2012) Nat Phys 8(2):131–134]. Although experiments have shown an adatom-induced enhancement of the electron–phonon coupling, superconductivity has never been observed. Using angle-resolved photoemission spectroscopy (ARPES), we show that lithium deposited on graphene at low temperature strongly modifies the phonon density of states, leading to an enhancement of the electron–phonon coupling of up to λ ? 0.58. On part of the graphene-derived π^?-band Fermi surface, we then observe the opening of a Δ ? 0.9-meV temperature-dependent pairing gap. This result suggests for the first time, to our knowledge, that Li-decorated monolayer graphene is indeed superconducting, with T_c ? 5.9 K.Although not observed in pure bulk graphite, superconductivity occurs in certain graphite intercalated compounds (GICs), with T_c values of up to 11.5 K in the case of CaC₆ (1, 2). The origin of superconductivity in these materials has been identified in the enhancement of electron–phonon coupling induced by the intercalant layers (3, 4). The observation of a superconducting gap on the graphitic π^?-bands in bulk CaC₆ (5) suggests that realizing superconductivity in monolayer graphene might be a real possibility. This prospect has, indeed, attracted intense theoretical and experimental efforts (6–12). In particular, recent density functional theory calculations have suggested that, analogous to the case of intercalated bulk graphite, superconductivity can be induced in monolayer graphene through the adsorption of certain alkali metals (8).Although the Li-based GIC—bulk LiC₆—is not known to be superconducting, Li-decorated graphene emerges as a particularly interesting case with a predicted superconducting T_c of up to 8.1 K (8). The proposed mechanism for this enhancement of T_c is the removal of the confining potential of the graphite C₆ layers, which changes both the occupancy of the Li 2s band (or the ionization of the Li) and its position with respect to the graphene layer. These modifications lead to an increase of the electron–phonon coupling constant from λ = 0.33 to λ = 0.61, in going from bulk to monolayer LiC₆. It has been argued that the LiC₆ monolayer should exhibit the largest values of both λ and T_c among all alkali–metal–C₆ superlattices (8). Nevertheless, although there is thorough experimental evidence for adatom-enhanced electron–phonon coupling in graphene (7, 11, 13), superconductivity has not yet been observed in decorated monolayer graphene.Angle-resolved photoemission spectroscopy (ARPES) measurements of the electronic dispersion of pristine and Li-decorated graphene at 8 K, characterized by the distinctive Dirac cones at the corners of the hexagonal Brillouin zone (Fig. 1E), are shown in Fig. 1 A and B. Li adatoms electron-dope the graphene sheet by charge transfer doping, leading to a shift of the Dirac point to higher binding energies. As evidenced by the evolution of the graphene sheet carrier density in Fig. 1F, this trend begins to saturate after several minutes of Li deposition. Concomitantly, we observe the emergence of a new spectral weight (Fig. 1E) at the Brillouin zone center (the comparison of the Γ-point ARPES dispersion for pristine and 10-min Li-decorated graphene is shown in Fig. 1 C and D). The origin of this spectral weight is probably the Li-2s band expected for this system (8) superimposed with the folded graphene bands caused by a Li superstructure, which were observed in Li and Ca bulk GIC systems (5, 14). This spectral weight, which disappears above  ～ 50 K and is not recovered on subsequent cooling, is associated with the strong enhancement of electron–phonon coupling (discussed later, see Fig. 3 and SI Appendix).Open in a separate windowFig. 1.Charge transfer doping of graphene by lithium adatoms. Dirac-cone dispersion measured by ARPES at 8 K (A) on pristine graphene and (B) after 3 min of Li evaporation along the K-point momentum cut indicated by the white line in the Fermi surface plot in E. The Dirac cone–Fermi surface was measured at this specific K point and then replicated at the other K points by symmetry (note that high-symmetry points are here defined for the Brillouin zone of pristine graphene and not of 3×3R30° reconstructed Li-graphene, which is, instead, the notation in ref. 8). The point at which the spectroscopic gap is studied is indicated by the shaded white circle. The Dirac point, (A) already located below E_F on pristine graphene because of the charge transfer from the SiC substrate, further shifts to higher energies with (B) Li evaporation. The presence of a single well-defined Dirac cone indicates a macroscopically uniform Li-induced doping. (C) Although no bands are present at the Γ-point on pristine graphene, spectral weight is detected on 10-min Li-decorated graphene in D and E. As illustrated in the 8 K sheet carrier density plot vs. Li deposition time in F, which accounts for the filling of the π??? Fermi surface, the spectral weight at Γ is observed for charge densities n_2D ? 9 × 10¹³ cm^-2 (but completely disappears if the sample temperature is raised above  ～ 50 K and is not recovered on subsequent cooling) (SI Appendix).Open in a separate windowFig. 3.Analysis of electron–phonon coupling in Li-decorated graphene. (A) Dirac dispersion from 3-min Li-decorated graphene along the k-space cut indicated in the Fermi surface plot in E that exhibits kink anomalies caused by electron–phonon coupling (white line indicates MDC dispersion). (B–D) MDC dispersion and bare bands obtained from the self-consistent Kramers–Kronig bare-band fitting (KKBF) routine (20, 21) for several Li coverages (Methods and SI Appendix); the real part of the self-energy Σ′ is shown in Right (orange indicates Σ′ from the KKBF routine analysis, and black indicates Σ′ corresponding to the Eliashberg function presented below). (F–H) Eliashberg function α²F(ω) from the integral inversion of Σ′(ω) (22) and electron–phonon coupling constant λ = 2∫dω?α²F(ω)/ω (Methods and SI Appendix); in H, the theoretical results from ref. 8 for a LiC₆ monolayer are also shown (gray shading). (I) Experimentally determined contribution to the total electron–phonon coupling (black circles) from phonon modes in the energy ranges 100–250 meV (blue shading and white circles) and 0–100 meV (orange shading); the coupling of low-energy modes strongly increases with Li coverage.Next, we use high-resolution, low-temperature ARPES to search for the opening of a temperature-dependent pairing gap along the π^?-band Fermi surface as a direct spectroscopic signature of the realization of a superconducting state in monolayer LiC₆. To increase our experimental sensitivity, as illustrated in Fig. 2A, using the approach introduced for FeAs (17) and cuprate (18) superconductors, we perform an analysis of ARPES energy distribution curves (EDCs) integrated in dk along a 1D momentum–space cut perpendicular to the Fermi surface. This method also provides the added benefit that the integrated EDCs can be modeled in terms of a simple Dynes gap function (19) multiplied by a linear background and the Fermi–Dirac distribution, all convolved with a Gaussian resolution function (Methods and Eq. 4). As shown in Fig. 2A and especially, Fig. 2B, for data from the k-space location indicated by the white circles in Figs. 1E and ​and3E,3E, a temperature dependence characteristic of the opening of a pairing gap can be observed near E_F. The leading-edge midpoints of the Li-graphene spectra move away from E_F (Fig. 2B) in cooling from 15 to 3.5 K, which is at variance with the case of Au spectra crossing precisely at E_F according to the Fermi–Dirac distribution (Fig. 2D). Fitting these data with Eq. 4 returns a gap value of Δ = 0.9 ± 0.2 meV at 3.5 K (with Γ ? 0.09 meV). [Note that the parameter Γ in the Dynes fitting function is not treated as a free-fitting parameter, because the broadening of the coherence peaks and filling in of the gap are dominated by the experimental energy resolution. However, setting this parameter to small realistic values (Γ ～ 0.1Δ) improves the fit at the center of the gap (i.e., at E = 0 in the symmetrized data) without affecting the value of the gap itself.] Given its small value compared with the experimental resolution, the gap opening is best visualized in the symmetrized data in Fig. 2C, which minimize the effects of the Fermi function. Finally, we note that the gap appears to be anisotropic and is either absent or below our detection limit along the K ? M direction (SI Appendix, Fig. S4).Open in a separate windowFig. 2.Spectroscopic observation of a pairing gap in Li-decorated graphene. (A) Dirac dispersion from 10-min Li-decorated graphene measured at 15 and 3.5 K at the k-space location indicated by the white circles in Figs. 1E and ​and3E;3E; the temperature dependence is here evaluated for EDCs integrated in the 0.1-Å^-1 momentum region about k_F shown by the white box in Lower, with (Upper) the only changes occurring near E_F. (D) Although Au spectra cross at E_f as described by the Fermi–Dirac distribution, (B) the crossing points of the Li-graphene spectra are shifted away from E_F (cyan dashed line) because of the pull back of the leading edge at 3.5 K. A fit to the Dynes gap equation (Methods) yields a gap of Δ ? 0.9 meV at 3.5 K (and 0 meV at 15 K). The superconducting gap opening is best visualized in the symmetrized data in C [i.e., by taking I(ω) + I(?ω), which minimizes the effects of the Fermi function, even in the case of finite energy and momentum resolutions (15, 16); blue and red symbols in C represent the smoothed data, whereas the light shading gives the rmsds of the raw data]. The qualitatively similar behavior observed on polycrystalline niobium—and returning a superconducting gap Δ ? 1.4 meV—is shown in E and F.The detection of a temperature-dependent anisotropic gap at the Fermi level with a leading-edge profile described by the Dynes function—with its asymmetry about E_F and associated transfer of spectral weight to just below the gap edge—is suggestive of a superconducting pairing gap. The phenomenology would, in fact, be very different in the case of a Coulomb gap, which is typically observed in disordered semiconductors (23–25) because of the combination of disorder with long-range Coulomb interactions. A Coulomb gap would lead to a rigid shift of the spectra leading edge (isotropic in momentum) and result in a vanishing of the momentum-integrated density of states at E_F. Similarly, the observed gap is unlikely to have a charge density wave origin, because the observed gap is tied to the Fermi energy as opposed to a particular high-symmetry wavevector (the latter might occur at the M points, when graphene is doped all of the way to the Van Hove singularity, resulting in a highly nested hexagonal Fermi surface; or the K points, in the case of a 3 ×3 R 30° reconstruction, leading to a Dirac point gap). Finally, we note that these measurements do not allow us to speculate on the precise symmetry of the gap along a single Dirac cone–Fermi surface or the relative phases of the gap on the six disconnected Fermi pockets. As such, our results do not rule out any of the recent proposals for a possible unconventional superconducting order parameter in graphene (9, 26, 27).To further explore the nature of the gap observed on Li-decorated graphene (and also show our ability to resolve a gap of the order of 1 meV), in Fig. 2 E and F, we show as a benchmark comparison the analogous results from a bulk polycrystalline niobium sample—a known conventional superconductor with T_c ? 9.2?K. The Dynes fit of the integrated EDCs Fermi edge in Fig. 2E determines the gap to be Δ = 1.4 ± 0.2 meV (with Γ ? 0.14 meV), in excellent agreement with reported values (28). Although the leading-edge shift (Fig. 2E) and the dip in the symmetrized spectra (Fig. 2F) are more pronounced than for Li-graphene owing to the larger gap, the behavior is qualitatively very similar. This similarity provides additional support to the superconducting origin of the temperature-dependent gap observed in Li-decorated graphene.If the spectroscopic gap observed in Li-graphene is, indeed, a superconducting gap, the responsible mechanism may likely be electron–phonon coupling, which was predicted by the theory for monolayer Li-graphene (8) and also, seen experimentally for the bulk GIC CaC₆ (5). In direct support of this scenario, we present a detailed analysis of the graphene π^?-bands in Fig. 3, showing that the Li-induced enhancement of the electron–phonon coupling is, indeed, sufficient to stabilize a low-temperature superconducting state. Graphene doped with alkali adatoms always shows a strong kink in the π^?-band dispersion at a binding energy of about 160 meV (11). For the Li-graphene studied here, the same effect is seen in the momentum distribution curve (MDC) dispersions and the corresponding real part of the self-energy Σ′ in Fig. 3 B–D. This structure stems from the coupling to carbon in-plane (C_xy) phonons (4, 8). Despite the apparent strength of this kink, the interaction with these phonon modes contributes little to the overall coupling parameter because of their high energy (note that ω is a weighting factor in the integral calculation of λ) (Methods). As illustrated by the white circles in Fig. 3I, the contribution to λ from these high-energy (100–200 meV) modes is determined to be 0.14 ± 0.05, and it remains approximately constant for all Li coverages studied here. This value is, however, too small to stabilize a superconducting state in this system (8, 11).With increasing Li coverage and the appearance of the spectral weight at Γ, significant modifications to the low-energy part of the dispersion ( ? 100 meV) become apparent (Fig. 3 B–D). With 10 min of Li deposition (Fig. 3D), an additional kink is visible at a binding energy of ∼30 meV along with the associated peak in the real part of the self-energy Σ′. The extracted (Methods) Eliashberg functions and energy-resolved λ(ω) in Fig. 3 F–H show that, at high Li coverage, phonon modes at energies below 60 meV are coupling strongly to the graphene electronic excitations. The phonon modes in this energy range are of Li in-plane (Li_xy) and C out-of-plane (C_z) character (4, 8). This assignment is in agreement with predictions (8) as shown by the direct comparison between theory and experiment in Fig. 3H. [As for the theoretical and experimental Eliashberg functions α²F(k, ω) in Fig. 3H, the agreement may, at first glance, appear not as good as the one for λ(ω). We note, however, that, in this regard, the relevant information is in the macroscopic energy distribution of the α²F(k, ω) weight rather than in its detailed structure.] As for the total electron–phonon coupling λ for each coverage (black circles in Fig. 3I), our values measured on the π^?-band Fermi surface at an intermediate location between Γ ? K and K ? M directions (Fig. 3E) provide an effective estimate for the momentum-averaged coupling strength. [The electron–phonon coupling parameter increases monotonically along the π^?-band Fermi surface in going from the Γ ? K to the K ? M direction as observed in both decorated graphene (11) and intercalated graphite (29). Empirically, the value measured at the intermediate Fermi crossing corresponds to the momentum-averaged coupling strength along the π^?-band Fermi surface.] Remarkably, the value λ = 0.58 ± 0.05 observed at the highest Li coverage (Fig. 3I) is comparable with λ = 0.61 predicted for monolayer LiC₆ (8) as well as λ ? 0.58 observed for bulk CaC₆ (29)—it is, thus, large enough for inducing superconductivity in Li-decorated graphene. It is also significantly larger than the momentum-averaged results previously reported for both Li and Ca depositions on monolayer graphene [λ ? 0.22 and λ ? 0.28, respectively (11)]. We note that achieving such a large λ-value is critically dependent on the presence of the spectral weight observed at Γ when Li is deposited on graphene at low temperatures, presumably forming an ordered structure on the surface and not intercalating. As shown in SI Appendix, we find λ = 0.13 ± 0.05 after the same sample is annealed at 60 K for several minutes, destroying the Li order and associated Γ-spectral weight.Taken together, our ARPES study of Li-decorated monolayer graphene provides evidence for the presence of a temperature-dependent pairing gap on part of the graphene-derived π^? Fermi surface. The detailed evolution of the density of states at the gap edge as well as the phenomenology analogous to the one of known superconductors, such as Nb—as well as CaC₆ and NbSe₂, which also show a similarly anisotropic gap around the K point (30–34)—indicate that the pairing gap observed at 3.5 K in graphene is most likely associated with superconductivity. Based on the Bardeen–Cooper–Schrieffer gap equation, Δ = 3.5?k_b?T_c, we estimate a superconducting transition temperature T_c ? 5.9 K, remarkably close to the value of 8.1 K found in density functional theory calculations (8). This work constitutes the first, to our knowledge, experimental realization of superconductivity in graphene—the most prominent electronic phenomenon still missing among the remarkable properties of this single layer of carbon atoms.