Areas of superconductivity and giant proximity effects in underdoped cuprates

Phenomenological models for the antiferromagnetic (AF) versus $d$-wave superconductivity competition in cuprates are studied using conventional Monte Carlo techniques. The analysis suggests that cuprates may show a variety of different behaviors in the very underdoped regime: local coexistence or first-order transitions among the competing orders, stripes, or glassy states with nanoscale superconducting (SC) puddles. The transition from AF to SC does not seem universal. In particular, the glassy state leads to the possibility of “colossal” effects in some cuprates, analog of those in manganites. Under suitable conditions, nonsuperconducting Cu-oxides could rapidly become superconducting by the influence of weak perturbations that align the randomly oriented phases of the SC puddles in the mixed state. Consequences of these ideas for thin-film and photoemission experiments are discussed.

Figure 1

(Color online) (a) MC phase diagram for Eq. (1) without disorder at low temperatures. Instead of presenting a three-dimensional phase diagram we have chosen to present a two-dimensional cut along $V=1-J∕2$ for simplicity. Five regions are observed: AF, $d$-SC, stripes, coexisting $\mathrm{SC}+\mathrm{AF}$, coexisting $\text{stripes}+\mathrm{SC}$, and metallic. (b) MC phase diagram including temperature along path 1. (c) MC phase diagram along path 2. Lattice sizes in all cases are $8\times 8$ and $12\times 12$. (d) $n$ vs $\mu$ along Paths 1 and 2. Transitions along Path 1 appear continuous, whereas along path 2 there are indications of first-order transitions. (e) Spin structure factor $S\left(\mathbf{q}\right)$ at $(\pi ,\pi )$ and for incommensurate (IC) momenta.

Figure 2

(Color online) (a,b) Plaquette impurity schematic representation. Disorder may have several forms, but here we mimic Sr doping in single layers. ${\mathrm{Sr}}^{2+}$ replaces ${\mathrm{La}}^{3+}$ above the center of a Cu-plaquette in the Cu-oxide square lattice. Then, as hole carriers are added, a hole-attractive plaquette-centered potential should also be incorporated. Near the center of this potential, $n$ should be sufficiently reduced from 1 that, phenomenologically, tendencies to SC should be expected. To interpolate between the SC central plaquette and the AF background, a plaquette “halo” with no dominant tendency was introduced. Parameters are chosen such that the blue (black) region favors superconductivity, $(J,V,\mu )=(0.1,1.0,-1.0)$, with a surrounding white region where $(J,V,\mu )=(0.1,0.1,-0.5)$ with no order prevailing. The impurity is embedded in a background (red and dark gray), which favors the AF state, $(J,V,\mu )=(1.0,0.1,0.0)$. However, the overall conclusions found here are simple and independent of the disorder details. MC phase diagram for model Eq. (1), including quenched disorder (lattices studies are $8\times 8$ (results shown) and $12\times 12$). Shown are $T_{\mathrm{c}}$ and $T_{\mathrm{N}}$ vs number of impurities (equal to number of holes). The SC and AF regions with short-range order (dashed lines) and $T*$ as obtained from the PG (dot-dashed line) are also indicated. (c) DOS at points $a$, $b$, and $c$ of (a), with a PG.

Figure 3

(Color online) (a) MC $S(\pi ,\pi )$ and superconducting correlation at maximum distance $C_{sc}^{xy}$ [Eq. (2)] along path 1 [Fig. 1a], using an $8\times 8$ lattice and at low temperature $T=0.02$. (b) AF and SC correlations at maximum distance for the model with eight impurities and without disorder (clean). The latter is the point $J=0.6$ of path 1. The disordered case corresponds to eight plaquettes on a $12\times 12$ lattice. Typically, 5000 sweeps were used for thermalization and for measurements. With quenched disorder, many points were obtained after averaging over $N_{\mathrm{dis}}=10$ disorder realizations. The results were found to mildly depend on the disorder configuration, thus, many results were obtained with a smaller $N_{\mathrm{dis}}$.

Figure 4

(Color online) (a) MC phase diagram [for Eq. (3)] at $u_{12}=0.7$. Parameters are $r_1=-1$, $r_2=-0.85$, and $u_1=u_2=1$ but the conclusions are not dependent on coupling fine-tuning. Spin $C_{\text{spin}}\left(\mathbf{m}\right)=(1∕N){\sum }_{\mathbf{i}}⟨{\mathbf{S}}_{\mathbf{i}}\bullet {\mathbf{S}}_{\mathbf{i}+\mathbf{m}}⟩$ and SC correlations $C_{\mathrm{SC}}\left(\mathbf{m}\right)=(1∕N){\sum }_{\mathbf{i}}\mid {\Delta }_{\mathbf{i}}\mid \mid {\Delta }_{\mathbf{i}+\mathbf{m}}\mid \cos ({\Psi }_{\mathbf{i}}-{\Psi }_{\mathbf{i}+\mathbf{m}})$ were measured. The behavior of these functions at the largest (shortest) distance determine $T_{\mathrm{c}}$ and $T_{\mathrm{N}}$ $(T*)$ [same criteria as for Eq. (1)]. With disorder, the phase diagram (shown) has an intermediate “clustered” state with short-range order. $T*$ is also indicated (dashed line). Note the similarity with Fig. 2b. Inset: results at $W=0$ showing tetracriticality [magenta (dark) indicates SC-AF coexistence]. (b) AF and SC correlations at maximum distance for the model Eq. (3) without and with disorder ($W=0.0$ and 0.7, respectively). ${\rho }_1=0.5$ and $u_{12}=0.7$ were used, using a $24\times 24$ lattice. Typically, for the LG model 25 000 sweeps were used for thermalization and measurements. (c) MC phase diagram of model Eq. (3) at $u_{12}=2$. The clean case ($W=0$, solid lines) is bicritical-like, but with disorder $W=0.5$ a clustered region between SC and AF opens as well.

Figure 5

(Color online) Left: $C_{\mathrm{SC}}^{\max }$ versus ${\Delta }_{\mathrm{SC}}^{\mathrm{ext}}$ (see text) on a $24\times 24$ lattice, with $u_{12}=0.7$ and $W=0.5$, at the five points $a\text{‐}f$ indicated in Fig. 4a. A “colossal” effect is observed in $a$ and $b$ where the ${\Delta }_{\mathrm{SC}}^{\mathrm{ext}}=0$ state is “clustered.” A much milder (linear) effect occurs far from the SC phase ($e$ and $f$). MC snapshots are shown at ${\Delta }_{\mathrm{SC}}^{\mathrm{ext}}=0.0$ (right, top) and ${\Delta }_{\mathrm{SC}}^{\mathrm{ext}}=0.2$ (right, bottom), both at $T=0.1$ and ${\rho }_2=0.5$, using the same quenched-disorder configuration. The color convention is explained in the circle [colors indicate the SC phase, while intensities are proportional to $\mathrm{Re}\left({\Delta }_{\mathbf{i}}\right)$]. The AF order parameter is not shown. The multiple-color nature of the upper snapshot, reflects a SC phase randomly distributed (i.e., an overall non-SC state). However, a small external field rapidly aligns those phases, leading to a coherent state (bottom).

Figure 6

(Color online) (a) MC phase diagram [for Eq. (3)] at $u_{12}=0.7$. Parameters are $r_1=-1$, $r_2=-0.85$, $u_1=u_2=1$, and $W=0.5$ with one layer (solid colors) and two layers (dashed line). The addition of an extra layer increases the critical temperature of the superconductor as well as the Néel temperature. (b) $T_{\mathrm{c}}$ vs $N_{\mathcal{l}}$ for $u_{12}=0.7$, ${\rho }_2=0.3$, $W=0.7$, and ${24}^2\times N_l$ clusters. Shown are results with and without disorder. (c) The experimental $T_{\mathrm{c}}$ (in K) is shown for three HTS families, as indicated, up to three layers (data from Ref. 29).

Figure 7

(Color online) Schematic representation of the phase diagrams that our models show in the clean (a)–(c) and dirty (d) limits. The theory discussed in this paper shows the possible appearance of regions with local coexistence of AF and SC (a), or a first-order transition separating AF from SC (b) with the first-order character of the transition possibly continuing in the AF-disordered and SC-disordered transitions, or an intermediate striped regime (c). Possibilities (a) and (b) have already been discussed in Ref. 16, although here we do not invoke a higher-symmetry group, such as SO(5). The main result contained in this figure is the proposed phase diagram in the presence of quenched disorder (d). Shown are the glassy region, proposed to be a mixture of SC and AF clusters, and the $T*$, where local order starts upon cooling. This phase diagram has similarities to those proposed before for manganites,[4, 5] and certainly it is in excellent agreement with the experimental phase diagram of LSCO.

Figure 8

(Color online) Schematic representation of the glassy state that separates the SC and AF regions. The arrow indicates the phase of the SC order parameter.