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Research ArticleCONDENSED MATTER PHYSICS
Unusual behavior of cuprates explained by heterogeneous charge localization
D. Pelc1,2,
View ORCID ProfileP. Popčević3,4,
View ORCID ProfileM. Požek1,*,
M. Greven2,* and
View ORCID ProfileN. Barišić1,2,3,*
1Department of Physics, Faculty of Science, University of Zagreb, Bijenička cesta 32, HR-10000 Zagreb, Croatia.
2School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA.
3Institute of Solid State Physics, TU Wien, 1040 Vienna, Austria.
4Institute of Physics, HR-10000 Zagreb, Croatia.
↵*Corresponding author. Email: mpozek{at}phy.hr (M.P.); greven{at}umn.edu (M.G.); neven.barisic{at}tuwien.ac.at (N.B.)
Science Advances 25 Jan 2019:
Vol. 5, no. 1, eaau4538
D. Pelc
P. Popčević
ORCID record for P. Popčević
M. Požek
ORCID record for M. Požek
For correspondence: mpozek@phy.hr greven@umn.edu neven.barisic@tuwien.ac.at
M. Greven
N. Barišić
ORCID record for N. Barišić
The discovery of high-temperature superconductivity in cuprates ranks among the major scientific milestones of the past half century, yet pivotal questions regarding the complex phase diagram of these materials remain unanswered. Generally thought of as doped charge-transfer insulators, these complex oxides exhibit pseudogap, strange-metal, superconducting, and Fermi liquid behavior with increasing hole-dopant concentration. Motivated by recent experimental observations, here we introduce a phenomenological model wherein exactly one hole per planar copper-oxygen unit is delocalized with increasing doping and temperature. The model is percolative in nature, with parameters that are highly consistent with experiments. It comprehensively captures key unconventional experimental results, including the temperature and the doping dependence of the pseudogap phenomenon, the strange-metal linear temperature dependence of the planar resistivity, and the doping dependence of the superfluid density. The success and simplicity of the model greatly demystify the cuprate phase diagram and point to a local superconducting pairing mechanism.
The parent cuprate compounds are antiferromagnetic charge-transfer insulators that evolve into conductors and superconductors upon doping. The superconducting transition temperature is highest at an optimal level of p ≈ 0.16 doped holes per planar CuO2 unit, and therefore, the phase diagram is divided into underdoped and overdoped regions. The highly overdoped materials behave as conventional Fermi-liquid metals, with a large Fermi surface that corresponds to 1 + p holes per CuO2 unit and a planar resistivity that exhibits quadratic temperature dependence (1). The underdoped region features the pseudogap regime below a characteristic temperature T* that decreases linearly with doping and extrapolates to zero around pc ≈ 0.20 (1). The pseudogap is associated with a depletion of electronic states at the Fermi level and with myriad charge and magnetic ordering tendencies. Above T*, in the unusual "strange-metal" regime, the resistivity is approximately linear in an extended temperature range.
Recent transport and optical conductivity experiments revealed that the itinerant carriers below a characteristic temperature T** < T* behave like a Fermi liquid (2–7), with a carrier density that is equal to the nominal concentration p. The itinerant carrier concentration therefore evolves from p in the pseudogap regime to 1 + p in the overdoped regime. As would be expected from these observations, the inverse Hall mobility [i.e., the cotangent of the Hall angle, cot(ΘH)], a measure of the transport scattering rate, also exhibits Fermi-liquid behavior below T**. The quadratic temperature dependence of cot(ΘH) continues uninterrupted beyond T*, into the strange-metal regime, with virtually no doping or compound dependence (5, 6). These observations constitute a crucial constraint that has not been captured theoretically. Interpreted in the simplest possible manner (5), the transport data imply that the carrier density acquires temperature dependence in the strange-metal regime and that the same underlying Fermi-liquid physics describes the itinerant carriers throughout the entire phase diagram.
A second ubiquitous feature of the cuprates is structural and electronic inhomogeneity on multiple length scales (8–11). Surface-sensitive probes such as scanning tunneling microscopy (STM) (12) reveal broad distributions of local electronic gaps, and bulk local probes such as nuclear magnetic resonance (NMR) (13) provide evidence of intrinsic electrostatic inhomogeneity. Recent conductivity and magnetization experiments show that the superconducting precursor regime is dominated by inhomogeneity, leading to percolation (14–16). Yet, inhomogeneity is often disregarded in modeling the salient features of the cuprates.
Here, we present a phenomenological model of the cuprate phase diagram that respects the experimental observation of universal underlying Fermi-liquid behavior and combines it with spatially inhomogeneous (de)localization-induced changes in the itinerant hole density. The model does not involve specific assumptions about microscopics but rather provides a broad framework to understand the cuprates, and it parametrizes their common behavior with a small number of experimentally constrained constants. The model is thus comparable to well-known phenomenological approaches in science, such as the Standard Model of particle physics, the Landau theory of phase transitions, and models of population growth. Along with a comprehensive description of the normal state, our model gives fresh insight into key aspects of the superconducting state that are at odds with existing theories, and it therefore paves the way toward a microscopic understanding of cuprate superconductivity. In particular, we show that the unexpected decrease of the superfluid density in the overdoped part of the phase diagram (17) follows naturally if we simply assume that the pairing glue is associated with the localized holes. The superfluid density is thus obtained from the evolution of the normal-state properties.
The above-listed experimental facts provide the foundation of our model, leading to three general premises. First, two electronic subsystems coexist within the unit cell: itinerant and localized holes, with the p holes introduced via doping always being itinerant. Along with the aforementioned transport and optical experiments (see the Supplementary Materials) (18), local-probe evidence for the two components comes from NMR (19, 20), and specific heat results are consistent with this scenario (21). Second, we use the experimental fact that the itinerant component exhibits a universal Fermi-liquid transport scattering rate throughout the phase diagram (5, 6). The third premise is that the localized subsystem consists of exactly one hole per CuO2 unit, separated from the Fermi level by a spatially inhomogeneous, doping-dependent localization gap Δ (11, 12). In photoemission experiments, this is manifested as a partially gapped Fermi surface that is repopulated from arcs to the full carrier density of 1 + p with increasing doping as the individual gaps close with doping, there is no essential change in the shape of the underlying Fermi surface (22). The states on the arcs therefore are exactly the same as those found at high doping levels (5), where a Fermi liquid exists. We associate the localization gap with the strong electronic correlations that cause the charge-transfer gap of the undoped parent insulators. As itinerant carriers are introduced into the material, they are expected to influence the electronic interactions. The gap should therefore decrease with doping and, eventually, close beyond optimal doping, where the carrier density approaches 1 + p. Moreover, the gap is taken to be inhomogeneous, i.e., to vary from one CuO2 unit to another, again in line with experiments. To examine the predictions of the model, we quantify the above statements in the simplest possible manner. The basic quantity is the effective density peff(p,T) of itinerant carriers, which depends on temperature and doping. Doping provides p itinerant carriers per per CuO2 unit. Each CuO2 unit also contains one localized hole that can be thermally activated, giving rise to an e−Δ/2kT term (k = 1.38 × 10−23 J/K is the Boltzmann constant). Once the local gap Δ closes, the corresponding hole is added to the Fermi sea. Since the gap is inhomogeneous, the effective hole density is
(1)where peff (p,0) is the effective density at zero temperature and g(Δ) is the doping-dependent normalized gap distribution function. We start by considering a simple Gaussian function for the distribution of local environments, with a doping-independent width δ and a mean Δp that decreases linearly with doping, Δp = Δ0 (1 – p/pc), where Δ0 and pc are constants. Our results are not very sensitive to the precise choice of the distribution function or to the form of its doping dependence (see extended discussion in the Supplementary Materials). The portion of the Gaussian with nominally negative gaps simply represents those CuO2 units whose gaps have closed and thus have their holes added to the Fermi sea (Fig. 1A). The effective density at zero temperature then is simply
Fig. 1 Gap inhomogeneity and phase diagram of the cuprates.
(A) Gaussian gap distribution function g at several doping levels, shown as a function of energy. The parameters are pc = 0.2, Δ0 = 4000 K, and δ = 800 K. The fraction of the distribution that has reached the Fermi level (the portion above zero energy indicated with a dashed line) is added to the p delocalized doped charge carriers at temperature T = 0. (B) Effective density of delocalized carriers per CuO2 unit at T = 0, obtained as the sum of the doped hole concentration p and delocalized holes from the distributions shown in (A) (full line). The dashed line corresponds to the (skewed Gaussian) gap distribution parameters used for comparison with experiments on La2−xSrxCuO4 [LSCO; see Figs. 2 (C and D) and 4, Materials and Methods, and Supplementary Materials], with pc = 0.22, Δ0 = 3900 K, δ = 800 K, and skew parameter α = 2. (C) Second derivative of the normal-state resistivity, multiplied by peff at T = 0. This result is obtained by combining the effective carrier density from Eq. 1, obtained with the gap distributions in (A), with the experimentally determined Fermi liquid scattering rate of itinerant carriers (5). The characteristic features of the phase diagram are apparent: a quadratic resistive regime at both low and high doping, the temperatures T** and T*, and the linear-T-like regime around optimal doping.
The thermal activation term is approximate, since it assumes a constant prefactor (of unity), whereas the true density of states may be temperature dependent. We note that the same form was used in previous analyses of the Hall density (23, 24), and we use it for simplicity. Equation 1 has physically compelling limits: peff is 1 + p both at high temperatures and high doping levels. Furthermore, the density of localized holes is ploc = 1 + p − peff.
Neglecting compound-specific Fermi-surface complications that can cause a failure of the effective-mass approximation but not of the applicability of Fermi-liquid concepts (see the Supplementary Materials) (5), we take ρ = C2T2/peff and RH = 1/(epeff) for the resistivity and Hall constant (per CuO2 unit), respectively. Notably, the universal temperature dependence cot(ΘH) = C2T2 is embedded in these calculations, i.e., the experimentally established value C2 = 0.0175 K−2 (5, 6) is used to obtain the absolute value of ρ. We note that this experimental value is consistent with estimates for conventional Umklapp electron-electron scattering (25, 26).
We model the superfluid density ρs0 by assuming that pairing only occurs in the vicinity of localized holes
This expression consists of two parts: (i) the conventional dirty Bardeen-Cooper-Schrieffer (BCS) expression for itinerant holes (Homes' law) (27),
, where γ = 35.2 cm/K·microhm is a universal numerical constant, Tc is taken as a measure of the superconducting gap as per BCS theory, and
is the residual normal-state conductivity, a measure of (pair-breaking) disorder; (ii) ploc, obtained directly from modeling the normal state. The simplest possible, linear dependence is assumed between ρs0 and ploc. Homes' law fails to give the correct dependence of ρs0 on doping (or Tc) for overdoped compounds (17). Note also that Homes' law breaks down in the limit of low levels of pair-breaking (point) disorder (27), which some cuprates, such as HgBa2CuO4+δ (Hg1201), might approach.
Modification of the model at low superfluid densities
In a local pairing scenario with short coherence lengths and underlying spatial inhomogeneity, superconducting gap inhomogeneity should also play an important role when bulk superconductivity is not fully established. It was recently shown (for p < pc) that superconductivity appears in a percolative fashion upon cooling toward Tc (14–16). The underlying inhomogeneity induces a superconducting gap distribution of nearly universal width Ξ0 ~ 30 K. Close to the critical doping levels pc1 ≈ 0.06 and pc2 ≈ 0.26 that define the zero-temperature extent of the bulk superconducting state, we thus expect the simple relation Eq. 3 to be modified by a percolative term. We will show that this provides an excellent description of the superfluid density near pc2.
Figure 1 shows a generic calculation with a Gaussian distribution with pc = 0.2, Δ0 = 4000 K, and δ = 800 K (Fig. 1A). The density of itinerant holes at T = 0 obtained from Eq. 1 is shown in Fig. 1B: peff(T = 0) begins to deviate from p around optimal doping and smoothly crosses over to 1 + p holes at high doping/temperature. Motivated by experimental work (7), the temperature and doping dependence of the resistivity curvature, d2ρ/dT2·peff(T = 0), is plotted to obtain the phase diagram in Fig. 1C. All defining normal-state features are captured as follows: the T2 regime in the underdoped region that ends at T**; the characteristic temperature T*; an extended linear-T-like regime around optimal doping; and a smooth crossover to Fermi-liquid behavior on the overdoped side. This is achieved with a mere three parameters with values consistent with experiments: δ is consistent with the widths of features seen in optical conductivity and STM (figs. S3 and S4) (18, 28), pc is roughly the doping level where T* extrapolates to zero, and Δ0 is broadly consistent with the charge-transfer gap scale (24). As noted, an additional parameter, the experimentally determined universal scattering-rate coefficient C2, is needed to obtain numerical values of the resistivity (5, 8). The results are not sensitive to details of the distribution shape (see the Supplementary Materials).
Figure 2 demonstrates excellent quantitative agreement of the model with transport data for Hg1201 and La2−xSrxCuO4 (LSCO). Motivated by STM and nuclear quadrupole resonance work (see the Supplementary Materials and fig. S3) (12, 13), we use a slightly different, skewed Gaussian gap distribution, although the generic Gaussian (Fig. 1) leads to similar results (see the Supplementary Materials and figs. S1 and S2). Figure 2 (A and B) shows the resistivity and Hall constant of underdoped Hg1201 (5) along with the model results. The doping level (p ≈ 0.1) was chosen because all characteristic features are observed there, including T**, T*, and the linear-T resistivity regime. Figure 2 (C and D) demonstrates that the model captures the doping and temperature dependence of the resistivity curvature of LSCO. It also captures the "anomalous criticality" observed in the resistivity above optimal doping (fig. S7) (29), the temperature dependence of the Hall constant at all doping levels (fig. S6) (23), and the universal dependence of the linear and quadratic sheet resistance coefficients on doping (fig. S8) (2).
Fig. 2 Comparison of the model to experiments.
(A) Resistivity and (B) Hall constant for underdoped Hg1201 [p ≈ 0.1; blue circles; data from (5)] compared with the gap disorder model (red lines). A skewed Gaussian gap distribution and linear mean gap doping dependence are used, with pc = 0.2, Δ0 = 4000 K, δ = 600 K, and α = 2 (see Materials and Methods). The asymptotic value of the measured RH at T = 0 corresponds to ~90% of the nominal concentration of itinerant carriers, and the calculated curve is thus multiplied by the same factor. Note that this value is within the experimental uncertainty due to sample size and shape uncertainty (2, 5). (C) Resistivity curvature map of LSCO normalized at each doping level to ρ(300 K), from (7). In addition to the characteristic temperatures T** and T*, LSCO features a structural transition, which is detected in the resistivity measurement (red downward sloping line) (5). (D) Calculated resistivity curvature map, with added percolative superconducting precursor regime (14) close to Tc and parameters pc = 0.22, Δ0 = 3900 K, δ = 800 K, and α = 2 (same as for the dashed line in Fig. 1B). Resistivity is normalized by peff(T = 0), which is nearly equivalent to the experimental normalization and multiplied by a factor of 4·10−6 to have the same absolute color scale as (C).
Charge transport is a good probe of the itinerant subsystem, since it is sensitive to an energy window ~kT around the Fermi level, yet it is only an indirect probe of the localized-carrier subsystem below the Fermi level. More detailed insight into energy scales related to the charge-transfer gap can be obtained from spectroscopic techniques, and we compare our model to these results. Experimental signatures of the gap scale (see the Supplementary Materials) (11, 18) include broad features in tunneling and photoemission spectroscopy, as well as a characteristic mid-infrared peak in optical spectroscopy (fig. S4). As shown in Fig. 3, the agreement between the model parameters obtained from transport and the spectroscopic techniques is remarkable in the superconducting doping range (p > 0.05). Not surprisingly, at low doping, a simple linear doping dependence of the mean gap scale is inadequate, as it extrapolates to a value that is substantially smaller than the charge-transfer gap (24). To correct for this deviation, in agreement with spectroscopic results, alternative parametrizations of the doping dependence are possible (see Materials and Methods) that extrapolate to the correct charge-transfer gap value and yield even better agreement with transport results. Microscopically, the nonlinear dependence is expected, since the gap decreases because of the influence of the itinerant subsystem, and the density of itinerant holes, in turn, depends on the value of the mean gap.
Fig. 3 Mean localization gap.
Comparison of the characteristic high-energy scale for different compounds: high-energy pseudogap scale in photoemission [angle-resolved photoemission spectroscopy (ARPES)], superconductor-insulator-superconductor (SIS) tunneling spectra, and mid-infrared peak in optical conductivity data. The solid green line is our generic parameterization of the localization gap, and the shaded green band indicates the gap distribution width (Fig. 1). The dashed line is an alternative phenomenological form of the doping dependence (see Materials and Methods), which features an upward curvature and extrapolates to the transport charge-transfer gap of ~1 eV at zero doping (24). ARPES and SIS data for LSCO and Bi2Sr2CaCu2O8+δ (Bi2212) are adapted from (11); the data are from multiple experiments [see (11) for original references]. For LSCO optical conductivity peak extraction, see the Supplementary Materials. YBCO data are from (18).
Once the gap distribution parameters are known for the normal state, the superfluid density naturally follows—crucially, no additional free parameters are introduced (except for the description of the narrow region close to pc2; see below). To calculate ρs0 for LSCO, we approximate
in Eq. 2 by a small constant in the overdoped regime [
= 15 microhm·cm, consistent with experiments (7, 29)], use doping-dependent values of
from experiment below optimal doping (7), and calculate ploc from Eq. 1. As shown in Fig. 4, the result of this calculation is in excellent agreement with the experimentally determined doping dependence of ρs0. On the overdoped side, ρs0 is limited by ploc, whereas on the underdoped side, ploc = 1, and Homes' law is recovered.
Fig. 4 Superfluid density, inhomogeneous localization, and superconducting percolation.
(A) Zero-temperature superfluid density ρs0 across the cuprate phase diagram calculated from Eq. 4 (orange line) and compared with experiments. The same parameters as in Figs. 1B and 2D are used (pc = 0.22, Δ0 = 3900 K, δ = 800 K, and α = 2). At low doping, ρs0 is limited by the density of mobile holes (red circles) and by pair-breaking impurities, whereas at high doping, the limiting factor is the density of localized holes (green squares). Diamonds are data from (17), whereas the other symbols are data for La2−xBaxCuO4 (at p = x = 0.095) and LSCO from optical conductivity [empty circles (97)], penetration depth [full circles (98)], and muon spin rotation [squares (51)]. The insets are schematic, and one green square represents approximately four planar CuO2 units. A percolation regime is expected on both ends of the superconducting (SC) dome due to spatial inhomogeneity of superconducting gaps (14–16). (B) ρs0 for overdoped LSCO (17) in units of holes CuO2 units cell (blue diamonds). Between optimal doping and Tc ~ 12 K (0.17 < p < 0.24), ρs0 is determined by the density of localized holes ploc (dash-dotted line), whereas at lower Tc superconducting gap inhomogeneity causes percolation [dash-dotted green line; see Materials and Methods for the form of f(Tc)]. The product of the two (full line) gives a reasonable description of the entire curve. Inset: Log-log plot of ρs0 at low Tc demonstrates good agreement with percolative scaling ρs0 ~ Tc1.6 (see the Supplementary Materials) (dashed dotted line) compared with the previously used quadratic scaling (full line) (17). A power-law fit ρs0 ~ Tcγ below 9 K gives γ = 1.64 ± 0.07.
As noted, in the narrow regions at the edges of the superconducting dome, Eq. 3 ought to be modified by a percolative correction due to intrinsic superconducting gap disorder. We take this corrective term from previous work on granular superconductors (see Materials and Methods) and use the detailed measurements of ρs0(p) for overdoped LSCO (17) to test this idea. As seen from Fig. 4B, we again find excellent quantitative agreement with experiments: The model captures the kink at Tc ~ 12 K and, in particular, the low-Tc regime is consistent with superconducting percolation scaling. We emphasize that this scaling, ρs0 ~ Tc1.6, directly follows from the data of (17) and, hence, constitutes independent support for percolation (note that it is independent of the particular model for the percolation correction). The width of the superconducting gap distribution, Ξ0, is introduced as a free parameter, and the data are best fit with Ξ0 = 23 ± 1 K, remarkably close to the value 27 ± 2 K obtained in previous studies of the superconducting precursor as a function of temperature (14, 16). Alternatively, no additional free parameter is necessary if we take this previous result as input, and we achieve nearly the same good agreement in the percolation regime. Signatures of granular superconductivity have also been observed (30) in experiments on underdoped thin films of YBa2Cu3O6+δ (YBCO), which mirror the percolative regime discussed here.
Our minimalistic phenomenological model captures both the normal- and superconducting-state behavior at a quantitative level, yet it provides neither the microscopic origin of the inhomogeneous gap nor the exact nature of the pairing glue. Nevertheless, the model enables crucial insight into several salient aspects of cuprate physics—the origin of the pseudogap and related unconventional magnetism, the universal intrinsic inhomogeneity, the nature of the strange-metal state, and superconductivity—which we briefly discuss in what follows. Notably, the myriad pseudogap features and the superconducting glue must be related to the strong correlation physics associated with the localized hole. Our simple model does not explicitly include short-range interunit cell correlations (which ought to be important for the hole localization) and should thus be viewed as coarse grained.
Within the proposed picture, the density of itinerant carriers at the Fermi level starts to decrease upon cooling at temperatures comparable to the localization gap scale, which corresponds to the opening of local (pseudo)gaps. These temperatures are very high in underdoped compounds but approach zero at high doping. Once a significant fraction of the carriers is localized, the pseudogap is manifested in spectroscopic experiments (Fig. 3), with a characteristic energy/temperature somewhat dependent on the experimental probe (11). Since the localization is a residue of Mott physics, it will be accompanied by the emergence of antiferromagnetic correlations. Neutron scattering experiments show a distinct decrease of antiferromagnetic fluctuations above T* (31) and a vanishing of low-energy antiferromagnetic fluctuations typically associated with local moments above p ~ 0.28 (32). Moreover, the universal unconventional intraunit cell magnetism (33) and related nematicity (34) below T* might already emerge at higher temperatures along with the localization of holes, through a coupling to phonons (35) or the possible formation of loop currents (33, 36). Kerr effect measurements indicate broken time-reversal symmetry for T < T* yet show a memory effect that stems from temperatures far above T* (37). Likewise, magnetization anisotropy measurements detect nematicity with a predetermined direction below T* (34). This can be attributed to the broad distribution of localization energies, with incipient order at high temperatures, and T* a lower, emergent scale. There should exist a higher emergent temperature scale Tp (Tp > T* > T**) where the localized holes percolate. This percolation scale may already have been observed in susceptibility and resistivity measurements on LSCO (see the Supplementary Materials).
An important feature of our model is the universal gap disorder, which might originate from either electronic correlations (38) or intrinsic structural effects (8, 9). Inhomogeneity is an intrinsic feature of many perovskites (8–10). For the cuprates, this is well documented through structural data (8) and various direct local probes, particularly STM (12, 28), NMR (13), and x-ray absorption fine structure (see also the Supplementary Materials) (39). We emphasize that the inhomogeneity is at best weakly correlated with the level of (chemical) point disorder introduced by doping, since the model parameters are similar for LSCO and Hg1201, two cuprates with distinctly different doping chemistry and point defect severity (40). Yet, it is a distinct possibility that mechanical strain and strain accommodation associated with the perovskite-based structure are the primary causes of the gap distribution (8, 9, 39). This would result in prominent electronic features, since charge and spin degrees of freedom naturally couple to strain (and vice versa). For example, in colossal magnetoresistance manganites, strong coupling between elastic and electronic degrees of freedom is understood to be the cause of the observed multiscale inhomogeneity (41). Furthermore, recent nonlinear response experiments have found similar superconducting inhomogeneity in different perovskite-based materials that are otherwise very different (42), which indicates a common structural origin of the inhomogeneity. We find that the doping-dependent structural phase transition temperature in LSCO corresponds to nearly full charge localization (ploc ~ 98%), which suggests that this transition is an emergent phenomenon as well (see the Supplementary Materials and fig. S5) and that the coupling of charge carriers to the lattice is considerable. In agreement with our picture, state-of-the-art ultrafast photoemission results (43) were interpreted in terms of both correlation-induced localization physics and intrinsic inhomogeneity associated with the antinodal regions of the Fermi surface. We emphasize that ours is an effective low-energy model that packages in a simple fashion the complicated infrared (and higher energy) physics of the doped cuprates. For example, the optical spectra are complex (18, 44), and electronic correlations up to energies of several electron volts manifest themselves upon cooling below Tc (44). Lattice anharmonicity may play a role and affect the relative importance of covalent and ionic characteristics (45).
In the proposed picture, two highly unconventional features—linear-T resistivity and nonmonotonic superfluid density—originate from the same underlying gap distribution. The linear-T-like resistivity appears whenever the temperature is high enough compared with a significant fraction of the gaps. This behavior is not limited to optimal doping and is observed between p ≈ 0.05 and pc ≈ 0.20, with ρ ∝ T/p (2, 46). Close to optimal doping, the gap distribution extends up to the Fermi energy, and thus, the linear-T-like resistivity also extends to low temperatures, again consistent with experiments (29). Because of the integration in Eq. 1, any featureless distribution will give a broad region of approximately T-linear resistivity up to temperatures determined by the distribution width (δ ~ 800 K). This should be contrasted with power-law resistivity dependences due to quantum criticality, e.g., in heavy-fermion materials (47). Although often conjured to explain cuprate properties, quantum criticality results in a number of scaling laws—e.g., for the Grüneisen ratio and the dynamical spin susceptibility (47)—that have not been convincingly demonstrated, especially at the lowest temperatures/energies (29). In contrast, the existence of a universal Fermi-liquid transport scattering rate and related scaling laws are well documented throughout the phase diagram (2–5). Another possible source of linear-T resistivity was suggested to be a bad-metal incoherent transport (48), which would imply a short electronic mean-free path that violates the Mott-Ioffe-Regel limit. However, it was shown that this conventional, semiclassical limit is a serious underestimate (2, 49) and that the cuprates mostly lie in the coherent regime. As noted in (6), the experimental value of C2 corresponds to a room-temperature mobility of μ = [Hcot(ΘH)]−1 ≈ 10 cm2 V−1 s−1, which is not unusual and comparable to that of ordinary metals such as aluminum (5). Nevertheless, it is reasonable to expect deviations from simple Fermi-liquid behavior at very low doping and/or high temperatures, since other/additional scattering processes might become significant (e.g., optical phonons) or kT becomes comparable to the effective Fermi energy associated with arcs (2, 50), causing saturation (49). Our calculations demonstrate that an extended linear-T resistivity can be obtained in a simple manner, without invoking bad-metal transport or quantum criticality. However, we note that a quantum critical point associated with the emergent unconventional magnetism below T* may well be present but, without a significant influence on transport and superconducting properties, especially at high temperatures.
The doping dependence of the superfluid density is captured using the extremely simple Eq. 2. The well-known Uemura relation ρs0 ∝ Tc for underdoped cuprates (51) directly follows from Eq. 2 if
is doping independent, which for some compounds is rather well satisfied (7, 51). In contrast,
of underdoped LSCO exhibits considerable doping dependence (7). The essential ingredients in the calculation of ρs0 are a local superconducting mechanism [as also suggested in (17)] and an underlying spatially inhomogeneous pairing strength, which naturally explains the deviations from both Leggett's theorem and Homes' law for overdoped compounds. Notably, the physical picture is in remarkable agreement with previous STM work that shows strong association between the local pairing gap and normal-state electronic correlations (28).
While our model gives an overarching picture of the normal state and captures the doping dependence of the superfluid density, it is less obvious how to treat the pairing mechanism (and thus Tc itself). Nevertheless, it is clear that the cuprate phenomenology is consistently explained by assuming two-component physics—Fermi liquid and localized—with the pairing caused by the localized component. This suggests that the mechanism belongs to the broad class of electron-electron–mediated pairing that is fast compared with the electron-phonon time scales (52). More specifically, the pairing glue is plausibly related to the mechanism first proposed by Little (53) in the context of organic conductors and, only recently, tested explicitly in a carbon nanotube system (54). This electron-electron mechanism involves (virtual) oscillations of localized charge that provide an interaction between itinerant carriers (see also the Supplementary Materials). Several experiments give evidence of the importance of oxygen-oxygen charge transfer (see the Supplementary Materials), which points to the relevance of O–Cu–O charge fluctuations (45). Qualitatively, the superconducting transition temperatures are high because of the large energy scales involved. For this pairing, the relevant energy scale should be related to the localization gap, which increases monotonically with decreasing doping, yet Tc(p) is dome shaped. This can be understood by considering the experimental fact that electron-electron interactions are not instantaneous but retarded (55): For any local superconducting glue with retarded interactions, two electrons/holes must be at the same location within a given time scale. This will not occur frequently in underdoped compounds, where the carrier density is low, which leads to a decrease in Tc (see the Supplementary Materials). In overdoped compounds, the magnitude of the localization gaps decreases, which causes a concomitant decrease of Tc.
If inhomogeneous nanoscale charge localization is a generic property of perovskites, then our model could be relevant to a wide class of doped charge-transfer or Mott insulators. A Fermi-liquid scattering rate, pseudogap effects, and nontrivial resistivity have been detected in titanates (56), whereas iridates show local gap disorder, Fermi arcs, and unconventional magnetism similar to the underdoped cuprates (57). Our model provides a unifying description of low-energy cuprate physics and captures the most relevant features of the phase diagram. All this simply follows from a spatially inhomogeneous hole (de)localization process, Fermi-liquid behavior of the itinerant (delocalized) carriers, and a local superconducting mechanism associated with the localized holes.
Superfluid density in the percolation regime
At doping levels close to the critical values pc1 and pc2, where bulk superconductivity disappears, we expect the underlying superconducting gap disorder [seen in temperature-dependent experiments at lower doping (14–16)] to influence the superfluid density: When the gap distribution is close to zero energy, patches of superconducting material will form in the material at T = 0. Quantitatively, this effect can be included into the superfluid density by modifying Eq. 2
(4)where the function f(Tc) is the gap distribution correction. The shape of f(Tc) is taken from a previous study of a diluted granular superconductor (58), where it was found that the (normalized) superfluid density is equal to the normal-state conductivity of the equivalent percolating resistor network (59). To a good approximation, the dependence of the superfluid density on the superconducting fraction (at zero temperature) is then f = [(P − PC)/(1 − PC)]γ, where P is the fraction of superconducting patches, PC the percolation threshold, and γ an exponent that depends on the dimensionality of the percolation. Note that the function f is normalized to the value at P = 1. To obtain the link between P and the mean Tc needed for Fig. 4, we must specify the local gap distribution function [similar to the calculations of different responses in dependence on temperature (14, 16)]. P then is the integral of the distribution function. Similar to previous work, we chose a Gaussian distribution with width Ξ0, leading to
(5)where E and E−1 denote the direct and inverse error function, respectively. The exponents are γ ≈ 1.6 and γ ≈ 1.0 for three-dimensional (3D) and 2D percolation, respectively (59). The data are compatible with the 3D case (see inset of Fig. 4B), in agreement with temperature-dependent experiments that probe superconducting percolation above Tc in multiple cuprates (14–16). The corresponding critical concentration is then PC = 0.3 (59).
Phase diagram and skewness of distribution
The gap distribution is usually skewed to the high-energy side (see also Supplementary Text) (12, 60). To quantify and assess the impact of this tendency, we parametrized the distribution as a Gaussian multiplied by its integral
(6)where α is a dimensionless parameter that controls the skewness of the distribution, and ϕ is a normalized Gaussian distribution with mean Δp and full width at half maximum δ. The skewness can be continuously varied by changing α. For α = 0, the distribution reduces to a simple Gaussian.
In fig. S1, we tested different values of the skew parameter and different functional forms for the gap distribution to demonstrate that this yields essentially the same phase diagrams. The main features are rather insensitive to the choice of distribution shape, mainly due to the fact that the resistivity calculation involves an integral over energy. Along with skewed Gaussians, we tested a shifted gamma distribution that features a heavier tail at high energies and gives a broader linear-T resistivity region on the overdoped side of the phase diagram.
Choice of doping dependences of distribution parameters
The simplest assumption of a linear decrease of the mean energy and doping-independent distribution width may be relaxed by introducing additional parameters and assumptions. While these assumptions are purely phenomenological (and detract somewhat from the main message of the simplest possible calculation), an eventual microscopic theory should be capable of providing the true doping dependences, at least in principle. To test the robustness of our calculation, we introduced a curvature into the dependence of the mean gap on doping, using the function of the form
This function still crosses zero at p = pc (and contains no additional parameters) but has an upward curvature at higher doping. A similar function can be used for the dependence of the distribution width on doping, but with a more general form
(8)where β is a numerical constant. The more constrained form with β = 1/tanh(1) cannot be used for δ, since it would lead to a zero distribution width at pc and nonphysical divergences in the calculations. We chose β ~ 0.4, but again, it turned out that the exact value is not very important. The main features of the phase diagram were unchanged, but introducing these nontrivial doping dependences of Δp and δ somewhat broadened the region of the phase diagram where the resistivity is linear in temperature on the overdoped side (fig. S2). To introduce curvature in Δp(p) at low doping, in line with some experiments (Fig. 3 in the main text), we further modified Eq. 7 and cast it in the form
(9)without introducing additional free parameters. This form gives better agreement between calculated and measured Hall coefficients for strongly underdoped LSCO (see the Supplementary Materials), as well as a better match between the modeled mean energy (dashed line in Fig. 3) and the characteristic high-energy scale from experiments. Moreover, it has the physically appealing feature that Δ0 is approximately 1 eV, the charge-transfer gap at zero doping as determined from Hall-effect measurements (24). Yet again, this introduces no considerable changes to the overall picture.
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/1/eaau4538/DC1
Fig. S1. Normal-state phase diagram for different gap distributions.
Fig. S2. Normal-state phase diagrams for two doping dependences of the gap distribution parameters.
Fig. S3. Local probes of disorder in cuprates.
Fig. S4. High-energy gap scale in cuprates.
Fig. S5. Characteristic temperature and localization.
Fig. S6. Temperature and doping dependence of Hall constant for LSCO.
Fig. S7. Doping dependence of linear and quadratic resistivity coefficients of LSCO.
Fig. S8. Doping dependence of sheet resistance coefficients.
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Acknowledgments: N.B. is grateful to the late S. Barišić for extensive discussions. We wish to acknowledge J. M. Tranquada and G. Yu for helpful comments on the manuscript. Funding: D.P. and M.P. acknowledge funding by the Croatian Science Foundation under grant no. IP-2018-01-2970. P.P. acknowledges funding by the Croatian Science Foundation under grant no. IP-2016-06-7258. The work at the University of Minnesota was funded by the Department of Energy through the University of Minnesota Center for Quantum Materials under DE-SC-0016371. The work at the TU Wien was supported by FWF project P27980-N36 and the European Research Council (ERC Consolidator Grant no. 725521). Author contributions: D.P., M.G., and N.B. conceived the model. D.P. and P.P. performed the calculations and analyzed the data. M.P., M.G., and N.B. supervised the research. D.P., M.G., and N.B. wrote the manuscript with input from all the other authors. Competing interests: All authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
You are going to email the following Unusual behavior of cuprates explained by heterogeneous charge localization
By D. Pelc, P. Popčević, M. Požek, M. Greven, N. Barišić
Science Advances 25 Jan 2019 : eaau4538
A phenomenological model comprehensively captures the defining features of the cuprate high-temperature superconductors.


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Location:59 Moo6, Cochasit, Nongkae, Saraburi. 18250 Thailand.


text:
Back When
More to the beginning of western civilization than naked Greeks and clothed Christians
Thirty Years Ago
I would not complain that Jeannette De Wyze's story on nudists ("What You See," February 2) was completely one-sided. But still, De Wyze never once treated any of the serious objections to nudism.
For example, De Wyze must realize that there is more to the beginning of western civilization than naked Greeks and clothed Christians. Does she really think that non-Christian Hebrews and Romans, not to mention Egyptians, Etruscans, Chaldeans, were nudists?
— LETTERS: "ONE TOGA OVER THE LINE," Abe Tashkow, February 5, 1978
Twenty-Five Years Ago
But the Hedgecock staff is more upset with O'Connor's absence from a January 30 debate before the gay San Diego Democratic Club. Having refused to appear at two previous Democratic Club functions, O'Connor pledged to attend the recent session. She never showed, leaving Hedgecock alone to defend gay rights. "It's okay to be with the pack [of other candidates] in backing gays," commented one Hedgecock advisor, "but it's dangerous to go it alone."
— THE INSIDE STORY, Paul Krueger, February 10, 1983
It was seven o'clock in the morning, cold as a dog's nose, and I was down on my knees, crawling through the brush on Volcan Mountain, looking for something that I knew damn well didn't exist.
I had ditched my mountain bike in the bushes down by the locked gate at the base of Volcan Mountain and started up the hillside on foot. There were patches of sharp-smelling chinquapin that always give me sneezing fits, and thick stands of cedars and oaks. A covey of worried quail led me up a steep creek; when I got too close, they would fly on ahead.
Just about every tree on Volcan Mountain taller than eight feet has a sign tacked to it: "No trespassing — Violaters Liable To Arrest! Signed, E.C. Rutherford." I didn't expect Mr. Rutherford to understand that I was trying to find the headwaters of the San Dieguito River.
— "FIFTY MILES OF RIVER," Steve Sorensen, February 11, 1988
Fifteen Years Ago
The sight of a bloated Fleetwood Mac dueting with martian-like Michael J., the unctuous strains of Loggins and Messina's horrid old chesnut "Your Mama Don't Dance," the incongruous presence of Lou Reed and Los Lobos: a mockery of the ideals that real rock 'n' roll cherishes.
— MUSIC SCENE: "THE TEENAGE WASTELAND LIVES ON IN A CHULA VISTA GARAGE," Gina Arnold, February 11, 1993
My experience at Horizon Christian Fellowship was not positive. I disliked the music. I disliked the fact that there were neither hymnals nor overhead projections of hymn lyrics. This might not have mattered so much if the musical portion of the service hadn't lasted for 45 minutes. While I understand why a "non-denominational" church might not want to tie itself to traditional hymns, the result — no hymnals, no overhead projections — was a clanish, insider/outsider feel. Although I can't imagine why anyone — even the most ardent born-again Christian — would want to commit any of these contemporary hymns to memory, much less print them on paper.
— SHEEP AND GOATS, Abe Opincar, February 12, 1998
Five Years Ago
"The image is of dignity, but the real picture of Native Americans is not such a beautiful picture," Fallbrook resident Rita Coolidge tells me over the phone. "There are a lot of people who don't benefit from casino money. Alcoholism is high; education is poor.
"Initially I came with Kris [Kristofferson] and thought, 'What a gem; it's so family oriented.' Fallbrook has a little main street. I can wear my pajama bottoms out in public with my jacket. Who cares?
"I walk around my neighborhood with my Rottweiler. Sometimes I do an aerobic walk and go over to Rocky Peak and spend half my mornings there."
— "GRACE AMONG AVOCADOS,"Jill Underwood, February 6, 2003
Onewheel E-Hoverboard rentals
12/3 SOOW 100' spool wire
Dog e treats
Fender FM65R guitar amplifier
A tale of two Thai restaurants
Bahn Thai knows how to sling spicy take-out
San Diego County's waterfalls
San Diego Unified revokes then reinstates benefits for Gompers Prep founders
San Diego cracks down on anyone living in his car
B.B. King Loved My Hat
A Good Old San Diego Political Garden Party
Pedal It Somewhere Else
Back When Clinton Brame talked about Tarawa
The Doctor Is Dead
"Nothing is wasted from the pig"
Liquid Blue Live from the Whiskey, UCSD Artist Lecture: Cortez, Saturday Morning at the Zoo
Events January 21-January 23, 2020

redpajama_set_name: RedPajamaCommonCrawl


Ladies and gentlemen, this recipe takes the cake. Errrr cookie. It takes the cookie.
As a gal, I know that we all have our moments when we need chocolate -- ASAP. There will be times where I can't wait to have a bite of chocolate in my mouth. So, most of the time I try to eat healthy and I hate feeling guilty about indulging. This recipe will make you feel a little better about calorie-splurging.
This recipe is probably one of the healthiest cookie recipes you can find. It's vegan/dairy-free, gluten-free, paleo/primal, oil-free, and nut-free. You can't find very many recipes that fit into all of those categories! If a recipe is paleo and gluten-free, it most likely has nuts in it. But not this one. Plus, these cookies don't even taste healthy -- I pinky swear! They're decadent and chocolaty; and they won't make you feel guilty, like you ate a whole pan of brownies.
1. Use a ripe avocado. If you're looking for an avocado at the store, you want one that is slightly soft to the touch. If they're all too firm, left it ripen on the counter for a few days. If it's too ripe, you will taste it. If it has too many brown spots, you will taste it. This recipe will turn out perfectly when your avocado is bright green and ripe.
2. Use good-quality chocolate chips and cocoa powder. It really does make a difference. I like either Guittard's Cocoa Rouge or Hershey's Special Dark. There are many good quality ones out there; take a chance on them. Consider making your own chocolate chips too -- the recipe link is below!
3. Choose your liquid element. You don't need to use bourbon unless you want to. You can substitute water or coffee to mix with the flax seed.
These cookies are delicious and easy to make!
Preheat the oven to 350°. Line a cookie sheet with parchment paper or a Silpat.
In a small cup, combine the flax and bourbon (water). Allow the mixture to thicken for a few minutes.
In a large mixing bowl, combine the avocado, sugar, chocolate chips, cocoa powder, instant coffee, and cinnamon. Cream the ingredients until they are mushed together. Add in the flax/bourbon mixture and baking soda. You will have a beautiful, fudgy mixture.
Using a cookie scoop, scoop the batter onto your prepared baking tray. I got about eight cookies from my batter. Space the cookies out. Leave the batter in round heaps; they will melt and spread a little. Bake the cookies for 12-15 minutes.
Enjoy these fudgy delights. They taste decadent and there's no way you can tell that they're gluten-free or (somewhat) healthy.


Blood Cancer Prevention Tips to Avoid it for life!
We all know that our body comprises three types of Blood Cells – Red Blood Cells, White Blood Cells, and Platelets. All these three types of cells have different functions to perform like Red Blood Cells are responsible for carrying oxygen to the different parts of the body, white blood cells are the fighters of the body, whereas platelets help in clotting the blood.
The white cells are responsible for fighting against the foreign cells which are responsible for the abnormal activities in the body. But there arises some kind of situation in which the body experiences abnormal division, growth, and spreading of blood cells which leads to the generation of blood cancer.
There are three different types of blood cancer from which a normal person suffers i.e Leukemia, Lymphoma, and Myeloma. Also, it is being observed that Leukemia is mostly diagnosed in small children whereas, in the case of Lymphoma, the kids between the age of 16 to 24 years are at a higher risk. Note that these stats are category specific, so in general we say that people of age 60 or above are more prone to blood cancer. Therefore, adults must be very conscious of their health to prevent blood cancer.
By now, it must be clear in your mind that blood cancer can occur at any stage and to any person. Hence, all should take special care of their health and for that, we are now going to provide you with the tips which will avoid blood cancer for life. Note that these tips are compiled by interviewing the best doctor for cancer treatment in Jaipur and are the best tips for blood cancer prevention.
Eating a healthy diet is quite a common suggestion but most of us are found lacking in here. Our eating habit is directly related to our health. Therefore, one should look to lower the intake of high-calorie foods in order to prevent blood cancer. Other than this, people should also look to avoid or minimize the intake of processed meat.
Drinking is a great home remedy for cancer prevention. Increasing the intake of water up to a great extent is found to be effective in diluting and removing the cancer-causing chemicals from the body in the form of urine. Also, there are some cases in which it is found that the water which the people are seen drinking is of poor quality. So, drink a good quality of water. Boiled water is most recommended.
In a survey, it was being found that Tobacco is the leading cause of most of the cancer types including blood cancer. So, quit smoking before its too late. Also, the passive smoking or second-hand smoking is injurious to the health of a person. Don't smoke and keep yourself away from the smokers.
To carry out dry cleaning, the workers use a chemical known as the carcinogenic agent called Perc and according to the research and study of the National Academies of Sciences, it was found that Perc is one of the significant factors behind the cause of blood cancer. Therefore, it is suggested to use soaps and detergents for cleaning instead of dry cleaning.
We all know about the harmful radiations which are coming for cell phones, but still, the time spent on the phone is increasing day by day. So, reduce or minimize the use of cell phones or you can opt to use hands-free mobiles.
Make a habit of performing moderate exercises for a minimum of 2 hours per week. This practice will help in lowering the risk of blood cancer by a huge margin. Exercising in the morning is the most beneficial.
Weight is found to be the cause behind a number of diseases and blood cancer is one of them. So, maintain a proper body weight according to your height and other factors. Regular exercising can help you in achieving so.
Along with skin cancer, the UV rays from the sun can take you to blood cancer. The use of sunscreens and protective clothing is recommended in order to protect yourself from the sun.
Screening Tests are quite beneficial in diagnosing the presence of cancer at the early stage. Although they are not beneficial in all the cases, still they are useful. So, go for screening tests after the prescription from the doctor.
The person who drinks alcohol is 30% more prone to cancer compared to the one who doesn't drink. So, quit smoking now. It is even riskier if you have a habit of smoking and drinking both.
The blood cancer diagnosis at the early stage supports a higher survival rate. So, by going for the regular health checkups, one can diagnose cancer at the initial stage.
These are our 11 tips which will avoid blood cancer for life if you follow them strictly. Hope you enjoyed reading here. Do share your valuable feedback by commenting in the below section.
What Are Cell Stem Transplant Side Effects?
Know About Diagnosis Of Men And Women Fertility?


Whether you have years of experience in sewing and are looking for a lightweight machine, or just learning your way around a sewing machine, the Brother XM2701 Lightweight Sewing Machine is the perfect choice for you.
Not only does the machine come with 27 unique, built-in stitches great for your everyday sewing needs, the easy-to-use features make it a great machine to expand your skills while saving on your bottom line. Take a look at this durable, affordable sewing machine and imagine the projects you can start. With how easy the XM2701 is to use, you won't just be starting projects, you'll be finishing them before you know it.
The XM2701 is overall a great, versatile, easy-to-use sewing machine that is backed by a 25-year warranty. The combination of available stitches at your disposal will keep you busy for a long time, and the convenience of a lightweight machine offers in ease of transport is top notch.
Don't be discouraged by the low price; this is not a cheaply made sewing machine that will break down on you in a few years. Brother has been an industry leader in lightweight sewing machines for years, and you have the peace of mind associated with purchasing a product that is made with quality components by a company that has been in business since 1954.
Basically, this product has a lot of upside with little in the way of negative qualities that would deter you from buying it.Whether you are looking to repair and maintain garments and fabrics around the household, or if you're just in the market for a new lightweight sewing machine,the Brother XM2701 Lightweight Sewing Machine is one of the best options available.
Super easy to use, inexpensive, and ultra-lightweight are some of the best features about the Brother XM2701 Lightweight Sewing Machine. Couple that with the 25 year warranty and the stellar customer service from Brother and you have the whole package when it comes to home and lightweight sewing jobs.Great for those everyday repairs, or if you wish to jump into hobby projects, this machine can do the trick.


Habitat preservation is the foundation of Plant.Grow.Fly. and host plants are the nurseries of your garden, as they are vital in helping pollinator numbers. Butterflies are uniquely connected to their host plants, laying their eggs on or next to specific species of plants. Female adult butterflies lay their eggs on many surfaces of the host plant and generally, in 1 to 2 weeks, a caterpillar emerges from each egg and begins feeding on the plant. Holes in the leaves of your garden are a very good sign: this means caterpillars are present! If you're concerned about the aesthetics of holey leaves, perhaps strategically place the host plants intermixed with nectar plants.
$$. Perennial. Bloom April-June. Grow up to 1 ft tall with a ¾-1½ ft spread. Low maintenance ground cover. Host to Painted Lady.
$$. Perennial. Bloom in fall. Grow 1-5 ft tall with a 1-4 ft spread. Used in containers and beds. Host to Pearl Crescent and Silvery Checkerspot.
$$. Perennial. Blooms late spring. Grows 2-4 ft tall with a 2-4 ft spread. Used in containers and beds. Host to Silver-Spotted Skipper and Wild Indigo Duskywing.
$$$. Deciduous shrub. Blooms May-July. Grows 2-4 ft tall with a 3-5 ft spread. Host to Spring Azure, Summer Azure, and Mottled Duskwing.
containers and beds. Host to the Black Swallowtail.
$$$. Deciduous shrub. Blooms August-September. Grows 4- 5 ft tall with a 4-10 ft spread. Host to Eastern Tailed Blue.
$$. Perennial. Blooms early summer-midsummer. Grows 2-3 ft tall with a 1-2 ft spread. Used in containers, beds, and on slopes. Deer resistant. Host to Eastern Tailed blue and Gray Hairstreak.
$$. Perennial. Blooms throughout summer. Grows 2-4 ft tall with a 2-3 ft spread. Used in containers and beds. Deer resistant. Host to Monarch.
$$. Perennial. Host to Wood Nymph, Northern Wood Satyr, Northern Pearly Eye, Crossline Skipper, Delaware Skipper, Least Skipper, Little Glasswing, Sachem, Northern Broken Dash, and Peck's Skipper.
$$. Annual and perennial varieties. Bloom late spring-late summer. Grow 8 in-6 ft tall with an 8-20 inch spread depending on variety. Used in containers, beds, slopes, and as ground cover. Host to the Buckeye.
$. Annual and perennial varieties. Bloom throughout summer. Grow 3-10 ft tall with a 3-4 ft spread. Used in beds and borders. Host to Painted Lady and Silvery Checkerspot.
$. Perennial. Bloom spring, fall, and winter. Grows 1-12 inches tall with a 6 inch spread. Used in containers, beds, and groundcover. Deer resistant. Host to Great Spangled Fritillary and Variegated Fritillary.
Species to select include Bur, Pin, Swamp White, Red, Chinquapin, or Black. Mature heights can reach up to 100' depending on species. As a whole, 532 different pollinator species depend on oaks making them an excellent host plant.


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Q: Can not delete unSelected dates from array in material calendar view I am using material calendar view in my android project. I am adding the selected date in array using the date method 'setOnDateChangedListener' and here is the code of my function
 materialCalendarView.setSelectionMode(MaterialCalendarView.SELECTION_MODE_MULTIPLE);
    materialCalendarView.setCurrentDate(today);
    materialCalendarView.state().edit()
            .setMinimumDate(CalendarDay.from(today))
            .setMaximumDate(CalendarDay.from(calendar))
            .setCalendarDisplayMode(CalendarMode.MONTHS)
            .commit();

 materialCalendarView.setOnDateChangedListener(new OnDateSelectedListener()
    {
        @Override
        public void onDateSelected(@NonNull final MaterialCalendarView widget, @NonNull final CalendarDay date, final boolean selected) {

            for(int i=0;i<widget.getSelectedDates().size();i++)
            {

                my_Dates1.add(date.getDay());

            }

            Log.e("Datess", String.valueOf(my_Dates1));

        }
    });

i want to remove a date from array if it is 'unSelected' from the calendar. There is a boolean selected, Can anybody tell me how i remove a date from array if it is 'unSelected' form the calendar using select variable. Thank you in advance.

redpajama_set_name: RedPajamaStackExchange


The life-changing moment for Bill and Dinah Vogel came 14 years ago, when their youngest son Charlie—only two at the time—was diagnosed with Spinal Muscular Atrophy, a neuromuscular disease that would confine him to a wheelchair. Both Bill and Dinah had busy careers, but after learning the seriousness of their son's condition, Dinah decided to leave hers and devote herself full-time to her family.
Both the Vogels had entrepreneurial spirit. They were always coming up with inventions to make life a little easier, such as the shaving mirror Bill had designed that wouldn't fog up in the shower and the potty chair liner Dinah had devised that made potty training a toddler easier. So they decided to take a leap. Bill left his corporate career, and the Vogels went into business for themselves as a husband and wife team, manufacturing and selling their inventions. They stayed in the place they had come to love, Knoxville, Tennessee, and chose Amazon to help get their business off the ground.
"I picked Amazon because I just think it's great place to start," says Dinah. Her instincts were good. Within a year Amazon customers made the Vogels' "The Shave Well Company" fog-free shower mirror the top-selling shower mirror on the site.
Best of all, the Vogels use Fulfillment by Amazon to store their inventory and fulfill customer orders. This gives them the freedom to run their business when and where they need.
When asked how this flexibility has affected his family, Bill begins to talk again about Charlie. Charlie was recently invited to participate in an experimental drug study that required nine extended visits to New York. Bill, Dinah, and Charlie have all been able to go to each visit, working together to overcome the extra challenges that a wheelchair adds to air travel and getting around New York City.
Charlie has also inspired an important aspect of the Vogel's business: the way all of those shaving mirrors and potty chair liners are assembled. Bill and Dinah pay to have the work done here in the U.S.A. by Sunshine Industries, a Knoxville non-profit that provides jobs for people with disabilities.
Bill marvels at what he and Dinah have accomplished.


For those looking for a small SUV that packs a punch, there can be no other option better than the new Chevrolet Trax, available in Macedon. Drivers, this vehicle is more handsome than its predecessor - this one enmeshes modernity with utility with immaculate ease. The new Chevrolet Trax, offered in Macedon, is capable of turning heads on the road, while you easily fit eight passengers and all that you require for an exciting day ahead. Moreover, this vehicle lets you stay connected on the go, with the 4G LTE Wi-Fi hotspot. Ten airbags and a high-strength steel safety cage add to the safety of the new Chevrolet Trax, featured at our Macedon dealership. A competitive handling and braking performance ensures a hassle-free and smooth drive. We have a huge selection of new vehicles and we cordially invite you to test-drive the new Chevrolet Trax in Macedon today!


Reading inspirational quotes is a soothing experience and if the quotes are about love and relationships, they're going to touch your heart and enhance your comprehension of this warm subject.
Love relationships are one of the most prominent desires we humans have and the need for them cannot be undermined.
Relationships are the heart of life, and love is the soul. A love relationship is what gives us the feeling of completeness in our lives.
Reading these inspiring quotes and sayings on the topic of love and relationships can help you learn about the subject and become a better partner if you're already in a love relationship or motivate you to get into one if you're single.
And if you are single, you would definitely like our guide on how to start a conversation with your crush.
Below we've compiled a beautiful list of quotes and sayings to inspire you about love and relationships.
A true relationship is two imperfect people refusing to give up on each other.
If you want a relationship that looks and feels like the most amazing thing on earth, you need to treat it like it is the most amazing thing on earth.
True lovers are never apart. Maybe in distance but never in heart.
Love isn't finding the perfect person; it's seeing an imperfect person perfectly.
Relationships are built upon effective communication. A relationship cannot survive without a proper amount of communication. Read these Communication and Relationships quotes to learn the importance of communication in a relationship.
Choose your relationships wisely. It's better to be alone than to be in a bad company.
Love shouldn't be a secret. If you love someone, tell them. You will always regret if you don't.
The purpose of a relationship is not to have another who might complete you, but to have another with whom you share your completeness.
We have to realize that there cannot be relationships unless there is commitment, unless there is loyalty, unless there is love, patience, and persistence.
Even though love cannot be described in words but hopefully by reading these love and relationship quotes, you've learned a thing or two about the topic.
Enjoyed reading these quotes and sayings? Have we missed any that you like? Write in the comments below to share your thoughts.
What a lovely post. Thank you for these quotes. They remind me why I love my hubby so much!
Although I am not a lovey-dovey person nor do I believe in love and stuff but these quotes are good to unread. Nice post.
Although I am not a relationship person, or lovey-dovey, there were a few nice quotes in here.
Wow you have a great compilation of quotes here. I simply love quotes like these, they give me some sort of a positive mood.
"The most desired gift of love is not diamonds or roses or chocolate. It is focused attention" I love this post the most. I do believe, once you realize that you are not valued and attention is not paid, we lose our passion. Thanks for sharing.
You're most welcome. And you're absolutely right, attention is one of the default expectations a partner has and rightfully so. Glad you liked it, Cheers.
Quotes are my favorite thing! I absolutely love this one, " A true relationship is two imperfect people refusing to give up on each other." That quote describes my relationship best!
All of these are really great quotes. Didn't know there were so many. I really do like the one which talks about love taking off the masks we fear we cannot live without but know we cannot live within. So very true.


You wish you had followed your instincts. If only you had not ventured down into the subway tunnel, in pursuit of the masked stranger you had unfortunately encountered less than forty-eight hours ago. You quickly become adjusted to the dim white light, as you look down the seemingly endless tunnel. All too quickly, you feel a quick push and then become aware of the cold rails beneath you. A blinding light gives your eyes cause to squint, as the stars that cloud your eyesight clear. A loud horn rings through the subway tunnel, reverberating off of the solid concrete walls, when you fully reach the realization of the multi-ton train rushing toward you. It feels almost like a dream, to the point where your senses are diluted and your brain is in a fog. You find it ironically funny, as your life is essentially flashing before your eyes, right back to that moment when you bumped into that stranger.
You had just exited the bank at 52nd street. You had cashed your paycheck for the week, and stepped out into the bright Friday mid-afternoon sun. You had to squint, and just as you had, you bumped into some person wearing jeans and a large orange hooded sweatshirt with the hood up. You found this to be odd because, for a normal autumn day, it was quite warm.
You quickly murmured, "Sorry," under your breath. As you walked away, you had the strange feeling that this person's eyes were following you down the street. You stole a swift look behind you, and found that this person had disappeared. You carried on with your daily tasks, but couldn't shake the feeling of someone's eyes constantly following you. You visited the grocery store, the gas station, and a small fast food restaurant for a quick dinner, and then ventured home. As you looked out your window, you thought for a second that you saw the same orange sweatshirt that you had seen earlier. You moved closer to the window, and decide that whatever or whomever you had thought was there no longer was. You have dealt with paranoia issues in the past, but after taking medications for it, you believed it was gone.
You live all alone in a large old house, in a small township a couple minutes' drive from the city where you work. After being kicked out of your parents' house, you hastily bought the first house the realtor had shown you. With no utilities yet to power your house, you had quickly grown accustomed to snuggling in bed by candlelight, and continued to do so, even after the power was connected. You rapidly fell asleep after a long day's tasks, yet just as fast you woke up to the screeching of your house alarm. When you had moved in, the realtor offered remove the system, but you had insisted it be left in to satisfy your slight paranoia. You had jumped out of bed and rushed downstairs with a flashlight and a baseball bat you always left in your room, to find the front door wide open and a cold breeze blowing inside. You had shut the door, reset the alarm, and quietly crept through the house looking for, what you hoped you would not find, an intruder. After looking for what felt like an eternity, your nerves seemed to calm down and the adrenaline wore off. You crawled back into your cozy bed, yet kept your weapon close, just to reassure yourself.
You woke the next day without having encountered any more disturbances during the night. Being the weekend, you did not have much to accomplish. You looked out and, deciding it was a nice-looking Saturday afternoon, left the house to go to the local park for a walk. You arrived at your favorite park where you used to play as a child. You sat on a swing, remembering the fun times you used to have there. You began to swing and, just as you reached the apex, something caught your attention out of the corner of your eye. Someone with a bright orange hoodie was standing some ways off; motionless and watching you swing back and forth. You felt a terrible sensation in your stomach, as if you knew something horrible was about to happen. You quickly jumped off the swing set of your childhood and hurriedly left the playground. You continued to take quick glances over your shoulder, but the fact that this person was still in the same place did not ease your terror. Millions of thoughts raced through your head at that second. Your pulse quickened and your breath became short and rushed. Chills spread through your body; you felt as if you could not move fast enough.
"Leave me alone!" you had finally called out in desperation. The only response this creepy being gave was a slow movement toward you. You exited the park, half-jogging down the street. The being in the orange hoodie seemed to quicken his step as well. You had looked above you and saw that the sky had darkened with storm clouds. You felt a couple raindrops and a light mist at first, which soon became a heavy downpour of cold rain. Your shirt was soon soaked and you were gasping for breath, but the man in the orange hoodie still was in pursuit. Every single nerve in your body was now on fire. You could hear your heartbeat in your own ears. Your blood pumped through your veins. Your senses were heightened to new levels. You were no longer in control of your body; animal-like instincts had completely taken over your every move. You soon found that you were randomly wandering the streets of the city. Every glance in a window seemed to present you with another terrifying image of the orange hoodie. Finally, you turned left at the end of the block to find that the street was completely deserted. Every shop was closed and there was no sign of life to be found. Still believing your pursuer was behind you, you continued down the street. You took another worried glance behind you, and saw no one there. You had breathed a small sigh of relief, only to see your follower venture down into a dark subway staircase. You were oddly filled with a strong burst of courage to start pursuing him, to see who he is and why he is following you.
You started down the staircase, calling "Hello?" and "Where are you?" No response came back, as you entered the subway station. The fact that the place was completely empty did not help your already tense nerves. The desire to run away slowly crept back into your body, but you suppressed that under your curiosity mixed with anger. You looked around in the dim station with the flickering lights, and felt a push from behind.
Finishing your flashback, you open your eyes, only for a second. The screech of the subway brakes mixing with the loud horn is almost deafening. Your eyes look up and find that the man in the orange hoodie is standing over you. His face is lit up, and time seems to stand still. His face is covered with scars. His eyes seem as black as night. Flaring nostrils and a wicked smile only add to your fear of this unknown man. You look back into the blinding light of the approaching subway train, and know that there is nothing you can do to save yourself. Slowly, your eyes close, your hands curl into fists, and you tense your entire body, knowing your doom is imminent. The last thing you hear before feeling the slam of the subway train is a bone-chilling laugh that echoes in your ears.
This story is really good and about my favorite character. As you can tell. Second person takes away from the story tho. I would of used Third or First.
You would have said what 5 people allready said but five people allready said what you were thinking .. You think.
Second person really takes away from this story. I would have chosen either third, or first person for it, like most pastas. Second person is very hard to pull off, and this is a great example. Sorry bud..
I think this story would've benefited from being written other than in Past Perfect tense. Also, it's a pity about dropping most of the connective verbs, it give the impression of a rush job. I liked the subject matter a lot and enjoyed this pasta as a whole. Good effort.
The POV was a little odd, but I loved the story either way! Great work!


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For those who have a guest bathroom that needs a far more feminine touch, this really is probably merely a torpedo for that room. With so many special variations as possible pick, there should be work that fits you when making a determination. But nobody claims that productive bathroom remodeling is going to be an easy task.
For something somewhat different you're able to choose a Playbuzz (lovely Goosebumps It Came From Beneath The Sink Pdf #6) that is sincerely graded. One end-of the raise is only an inch or two heavy, as the idea of the oval may be the common degree for your torpedo. You should possess a greater counter place to accommodate this fashion nonetheless it is breathtaking to see and all sorts of fun showing down for your buddies. You may also uncover additional styles including rectangle or square. Some comes with while others possess a pan that resembles a semicircle a serving that is exactly the same depth throughout the jar. Both variations are simply a matter of deciding what type works best in your restroom.