Defects in correlated metals and superconductors

In materials with strong local Coulomb interactions, simple defects such as atomic substitutions strongly affect both macroscopic and local properties of the system. A nonmagnetic impurity, for instance, is seen to induce magnetism nearby. Even without disorder, models of such correlated systems are generally not soluble in two or three dimensions, and so few exact results are known for the properties of such impurities. Nevertheless, some simple physical ideas have emerged from experiments and approximate theories. Here the authors review what we can learn about this problem from one-dimensional (1D) antiferromagnetically correlated systems. Experiments on the high-$T_c$ cuprate normal state which probe the effect of impurities on local charge and spin degrees of freedom are discussed, and compared with theories of single impurities in correlated hosts, as well as phenomenological effective Kondo descriptions. Subsequently, theories of impurities in $d$-wave superconductors including residual quasiparticle interactions are reviewed and compared with experiments in the superconducting state. Existing data exhibit a remarkable similarity to impurity-induced magnetism in the 1D case, implying the importance of electronic correlations for the understanding of these phenomena, and suggesting that impurities may provide excellent probes of the still poorly understood ground state of the cuprates.

Figure 1

A selection of actively studied inorganic oxides. Only the chain (${\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_3$, ${\mathrm{Y}}_2\mathrm{Ba}\mathrm{Ni}{\mathrm{O}}_5$, $\mathrm{Cu}\mathrm{Ge}{\mathrm{O}}_3$), ladder ($\mathrm{Sr}{\mathrm{Cu}}_2{\mathrm{O}}_3$, ${\mathrm{Sr}}_2{\mathrm{Cu}}_3{\mathrm{O}}_5$), or planar (cuprates) unit in the molecular structure is shown.

Figure 2

(Color online) Schematic phase diagram of cuprate HTSC materials: temperature $T$ vs hole doping $n_h$. AF refers to a state with long-range antiferromagnetic order, which disappears at the Néel temperature $T_N$. Superconductivity with $d$-wave symmetry sets in below a critical temperature $T_c$.

Figure 3

(Color online) Cartoon of the staggered paramagnetic polarization induced by a nonmagnetic defect (dot at center) in a correlated 2D material. Such oscillations of the local susceptibilities are directly detectable by spin probe experiments such as NMR, since the NMR shifts of the various shells of neighbors of the defects can be resolved.

Figure 4

Cu and O orbitals in the copper-oxide planes; relevant site, hopping, and correlation energies are included. Integrating out the O variables yields a $t,t^{\prime }\text{−}J$ model for the effective Cu lattice (right side). The notation is defined in the text.

Figure 5

Schematic rendition of the main energy hole states for the three-band and $t\text{−}J$ models. The UHB (upper Hubbard band), NB (nonbonding band), and LHB (lower Hubbard band) pertain to the former while the $S$ (Zhang-Rice singlet band) and $T$ (Zhang-Rice triplet band) refer to the latter. $\Delta$ is the charge-transfer gap between the UHB and $S$ bands. In the absence of doping, the UHB is completely filled with holes.

Figure 6

Conception of a valence bond state. Each spin-1 $\vec{S}$ is written as the sum of two spin-$1∕2$ states, $\vec{\sigma }+\vec{\tau }$; valence bonds are formed between a $\sigma$ and a $\tau$ belonging to neighboring sites.

Figure 7

The spin-1 chain as a special type of two-leg ladder. Left: A two-leg spin-$\frac{1}{2}$ ladder. Dots represent the spins with couplings $J$ along the legs and $J_{\text{rung}}$ along the rungs. Right: A spin-1 chain can be viewed as a two-leg spin-$\frac{1}{2}$ ladder with diagonal couplings (dotted lines).

Figure 8

Alternating magnetization caused by a boundary in a spin-$1\text{−}2$ chain according to Monte Carlo simulations at $T=J∕15$. From 125.

Figure 9

Uniform $\left({\chi }_i^u\right)$ and staggered $\left({\chi }_i^a\right)$ components of the local susceptibility ${\chi }_i$ obtained with the DMRG method applied to the $S=\frac{1}{2}$ Heisenberg model on a chain with $L$ sites and open boundaries, using $m$ states in the iterations and studying the subspace with a total spin in the $z$ direction equal to 1. (a) $L=80$ and $m=16$, and (b) $L=33$ and $m=32$. From 256.

Figure 10

(Color online) Theoretical and experimental NMR line shapes due to defects in spin chains. Left: The NMR spectrum expected for a staggered magnetization of the spin-$\frac{1}{2}$ chain displayed in the inset, corresponding to its histogram. Right: The experimental Cu NMR spectrum in Cu chain ${\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_3$ displays a similar $\Delta H$ pattern. It is interpreted as induced by intrinsic defects. From 419.

Figure 11

The introduction of a nonmagnetic impurity in the spin-1 chain displayed in Fig. 6 breaks two singlet bonds, thereby releasing two spins $1∕2$.

Figure 12

Alternating magnetization in the case of spin-$1∕2$ and spin-1 systems with a defect. The polarization for $S=1$ is normalized to that for $S=1∕2$. At high temperature, the two polarizations are roughly identical, exponentially decaying. At low temperatures, for $S=1$, the polarization does not evolve further because $\xi$ saturates. On the other hand, the extension of the polarization for $S=1∕2$ systems continues to increase as $T$ is decreasing and exhibits a nonmonotonic behavior (see Fig. 8).

Figure 13

Spin-1 chain with a Cu spin-$\frac{1}{2}$ defect. Spin-$\frac{1}{2}$ states are formed on either side of the Cu atom and align to form a spin-1 state.

Figure 14

(Color online) NMR reconstruction of an impurity-induced magnetization in a spin chain. The left panel displays an ${}^{89}\mathrm{Y}$ NMR spectrum in pure and Zn-substituted $\mathrm{Y}{\mathrm{Ba}}_2\mathrm{Ni}{\mathrm{O}}_5$. The various satellites which appear with Zn doping allow the reconstruction site by site of the induced staggered magnetization near the Zn atoms substituting Ni in the chain (425; 116).

Figure 15

Comparison of-the-extension of the staggered induced magnetism due to Mg (425), Zn, or Cu (116) impurities in YBNiO. The correlation length measured by QMC by 233 for the pure system is shown for comparison.

Figure 16

(Color online) An oxygen inserted between two copper sites in a vacant chain row only changes the (initially) ${\mathrm{Cu}}^{+}$ into ${\mathrm{Cu}}^{2+}$. This does not transfer any charge to the $\mathrm{Cu}{\mathrm{O}}_2$ plane. When a chain segment with at least two oxygen atoms is formed, charge transfer to the plane occurs. Full chains like that shown on the right might then form an ordered pattern, with specific values of the hole content in the planes, and minimal disorder.

Figure 17

(Color online) Phase diagram of pure and Zn-substituted $\mathrm{Y}\mathrm{B}\mathrm{C}{\mathrm{O}}_{6+x}$. Note that the phase diagram of pure $\mathrm{Y}\mathrm{B}\mathrm{C}{\mathrm{O}}_{6+x}$ (full symbols) only exhibits a small range of oxygen contents, near the tetragonal-to-orthorhombic transition at $x=0.4$, for which disordered static magnetism occurs at low $T$. With Zn substitution, at a concentration of 4% (open symbols), the superconducting range is highly suppressed; the local magnetism induced by Zn drives the appearance of a large spin-glass (SG) regime. The phase diagram becomes then similar to that of Fig. 2. Note that the onset of pseudogap opening $T^{*}$ is not modified by Zn substitution. Figure from 12.

Figure 18

(Color online) ${}^{17}\mathrm{O}$ NMR shift vs $T$ for $\mathrm{Y}\mathrm{B}\mathrm{C}{\mathrm{O}}_{6.6}$ pure and substituted with 2% Zn and 2% Ni. The NMR shift for the sites far from the Zn is found to be totally independent of the substitution, with the onset of opening of the pseudogap remaining at $\approx 350\mathrm{K}$ for all samples.

Figure 19

${}^{63}\mathrm{Cu}$ NMR ${\left(T_{1}T\right)}^{-1}$ data taken in Zn-substituted $\mathrm{Y}\mathrm{B}\mathrm{C}{\mathrm{O}}_{6.6}$ display a large increase at low $T$, which is also seen in the inelastic neutron-scattering data taken at the AF wave vector $(\pi ,\pi )$ shown in the inset. Both data are dominated at low $T$ by the contribution of sites near the Zn, but display evidence for the onset of the pseudogap as a high-$T$ maximum of the dynamic susceptibility around $150\mathrm{K}$, seen in the pure samples. From 213.

Figure 20

${}^{63}\mathrm{Cu}$ nuclei far from the Zn impurities display in YBCO-1248 the same reduction of ${\left(T_{1}T\right)}^{-1}$ as that found in the pure compound below the pseudogap temperature. From 208.

Figure 21

${}^{89}\mathrm{Y}$ NMR spectrum of $\mathrm{Y}\mathrm{B}\mathrm{C}{\mathrm{O}}_{6.6}:\mathrm{Zn}$ (or equivalently $\mathrm{Y}\mathrm{B}\mathrm{C}{\mathrm{O}}_{6.6}:\mathrm{Li}$) exhibits a main line and two satellite resonances as shown in the inset. Their shifts ${}^{89}\mathrm{K}$ are plotted vs $T$ as empty and full symbols, respectively, for Zn and Li substitution. While the main line shifts (circles) retain the characteristic pseudogap $T$ variation of the pure sample (crosses), the NN satellite shifts depart markedly at low $T$, the difference of shift with the main line displaying a Curie $T^{-1}$ variation. This results from the large Curie-like spin polarization which appears on the four Cu sites around the Zn or Li, two of them being directly sensed by the ${}^{89}\mathrm{Y}$ NN to the Zn (see Fig. 22). The ${}^7\mathrm{Li}$ NMR shift (cross in a circle, right scale) directly probes as well this Curie polarization of its four Cu neighbors. Figure from 64.

Figure 22

Cu $3d_{x^2-y^2}$ and O $2p_x$, O $2p_y$ orbitals of the $\mathrm{Cu}{\mathrm{O}}_2$ plane, with a Li or Zn impurity substituted on one Cu site. The magnetization induced on each Cu site by the applied field is staggered and decreases in magnitude to reach the very small uniform value found in the pure compound. Inset: The hyperfine coupling paths linking the different nuclear-spin sites with their near-neighbor Cu orbitals (also with the on site orbitals for Cu). The Y sites are below the $\mathrm{Cu}{\mathrm{O}}_2$ plane (and above the other $\mathrm{Cu}{\mathrm{O}}_2$ plane of the bilayer).

Figure 23

(Color online) Difference $\Delta \nu$ of the ${}^{17}\mathrm{O}$ NMR linewidth between the substituted and pure $\mathrm{Y}\mathrm{B}\mathrm{C}{\mathrm{O}}_{6.6}$ samples gives a measure of the histogram width of local fields induced by Zn or Ni on ${}^{17}\mathrm{O}$ nuclei. The product $T\Delta \nu$ increases markedly at low $T$, while it should remain constant if the spin polarization had a mere Curie $1∕T$ variation. This is evidence that both the magnitude of the local fields and the width of their histogram, that is, the decay length ${\xi }_{\mathrm{imp}}$ of the spin polarization, increase sizably at low $T$. Data from 62 and 346.

Figure 24

(Color online) Spatial variation of the model spin densities which fit the NMR data on ${}^7\mathrm{Li}$, ${}^{89}\mathrm{Y}$, and ${}^{17}\mathrm{O}$ nuclei in $\mathrm{Y}\mathrm{B}\mathrm{C}{\mathrm{O}}_{6.6}$, as a function of distance to the impurity site along the (10) direction in the $\mathrm{Cu}{\mathrm{O}}_2$ plane. It can be seen that the local magnetization magnitude and decay length ${\xi }_{\mathrm{imp}}$ both decrease with increasing $T$. For a large number of sites, the local magnetization is much larger than that of the pure sample, which is the limiting large-$r$ value at $160\mathrm{K}$, and which is even smaller at $80\mathrm{K}$. From 346.

Figure 25

Doping dependence of ${}^7\mathrm{Li}$ NMR shift. Left: ${}^7\mathrm{Li}$ NMR shift vs $T^{-1}$ for a series of $\mathrm{Y}\mathrm{B}\mathrm{C}{\mathrm{O}}_y$ samples as well as for an overdoped ${\mathrm{Ca}}^{2+}$/${\mathrm{Y}}^{3+}$-substituted sample for $y=7$. The data are fitted with Curie-Weiss laws. Right bottom: The corresponding Weiss $\Theta$ values vs oxygen content $y$ are seen to increase sharply near optimal doping. The full curve represents the variation of $T_c$ in the pure samples. Right top: Effective moment parameter ${\mu }_{\mathrm{eff}}^2$ of the Curie-Weiss fit (64).

Figure 26

Temperature dependence of the spatial extension ${\xi }_{\mathrm{imp}}$ of the polarization induced by Zn or Li impurities in underdoped $\mathrm{Y}\mathrm{Ba}\mathrm{Cu}{\mathrm{O}}_{6.6}$ and optimally doped $\mathrm{Y}\mathrm{Ba}\mathrm{Cu}{\mathrm{O}}_7$. The $T$ variation is larger for the underdoped case (346).

Figure 27

Electronic relaxation rate ${\tau }^{-1}$ obtained from $T_{1}TK$, as measured for the ${}^7\mathrm{Li}$ NMR for different hole dopings and Li contents (empty symbols, 1%; full symbols, 2%) in YBaCuO. From 279.

Figure 28

Effect of Zn on the transport properties of YBCO crystals. (a) Temperature dependence of the in-plane resistivity averaged over $a$ and $b$ for a pure single crystal of $\mathrm{Y}\mathrm{B}\mathrm{C}{\mathrm{O}}_7$ and for various Zn concentrations in substituted single crystals; (b) $T$ variation of $\cot {\theta }_H$, where ${\theta }_H$ is the Hall angle, for the same samples (100).

Figure 29

(Color online) Resistivity vs $T$ for a single crystal of $\mathrm{Y}\mathrm{B}\mathrm{C}{\mathrm{O}}_{6.6}$ irradiated at $20\mathrm{K}$ by $2.5\text{−}\mathrm{MeV}$ electrons for increasing irradiation doses. The resistivity curves display at high $T$ a parallel vertical shift associated with the increase of the impurity induced residual resistivity. Therefore the defect content does not modify the high-$T$ inflection point, linked with the pseudogap $T^{*}$. For high defect contents, when $T_c$ is sufficiently reduced, low-$T$ upturns are evident, and ultimately the resistivity curve diverges exponentially (381).

Figure 30

(Color online) Decrease of $T_c$ vs $n_h\Delta {\rho }_{2\mathrm{D}}$ for various families of cuprates, for Zn impurities or electron irradiation defects. The universal relation which applies whatever the hole doping $n_h$, except for the LSCO family, establishes that the $T_c$ depression is governed by the scattering rate ${\tau }^{-1}$ (384).

Figure 31

Temperature dependence of magnetic susceptibility in the RPA at $\mathbf{q}=\mathbf{0}$ for different sites $\mathbf{R}$ in the presence of a strong scattering potential $V_1=100t$ at (0,0) and a weaker one $V_2=10t$ on nearest-neighbor sites. $\overline{\chi }$ is a weighted average of $\chi$ to account for the fact that $\mathbf{R}$ may coincide with a lattice site, the center of a bond, or of a plaquette. $\tilde{\chi }({\mathbf{q}}^{\prime },{\mathbf{R}}^{\prime })$ is the same quantity neglecting the mode-mode coupling effect described in the text. From 338.

Figure 32

Temperature dependence of Knight shift (64) for Li impurities in YBCO within the RPA approach in the presence of a strong potential at $V_1=-100t$ at (0,0) and a weaker one on nearest-neighbor sites of strength given. From 79.

Figure 33

Schematic plot of spinon density of states for (a) spin-1 impurity; (b) spin-0 impurity in model of 232. Dashed line, density of states for pure flux phase. Solid line, density of states with impurity.

Figure 34

(Color online) Results for fully self-consistent evaluation of magnetization from slave-boson equations (156). (a) Normalized magnetization $s_1$ on nearest-neighbor site as function of $T$ for values of doping $\delta$ from 0.14 to 0.30. Solid lines show fits to $s_1=C∕(T+\Theta )$. (b) Effective moment constant $C$ (filled squares) and Curie-Weiss constant $\Theta$ (open squares) extracted from fits in (a) vs $\delta$. (c) Spinon bandwidth in fully self-consistent evaluation as a function of distance from the impurity site at filling $\delta =0.15$ and $T=100\mathrm{K}$ (filled squares) and $300\mathrm{K}$ (filled circles). Bandwidth from semianalytic calculation at $\delta =0.3$ and $100\mathrm{K}$ is shown (open circles) for comparison. (d) Temperature dependence of extracted theoretical correlation lengths $\xi$ vs $T$ (compare Fig. 26).

Figure 35

Schematic picture of spin-$\frac{1}{2}$ impurity produced in case of (a) Ni and (b) Zn.

Figure 36

(Color online) Summary of LDOS results for one-impurity problem on a tight-binding lattice, band given by 331, bond order parameter ${\Delta }_d=0.1$, $\mu =0$, $V_0=10$, $\mid {\Omega }_0^{\pm }\mid \simeq 0.013$: (a) LDOS vs $\omega$ on the impurity site; (b) LDOS vs $\omega$ on nearest-neighbor site; (c), (d) LDOS map at resonance $\omega ={\Omega }_0^{+}$ and $\omega ={\Omega }_0^{-}$. Color scales in (c) and (d) are relative to the nearest-neighbor peak heights shown in (b). From 475.

Figure 37

Magnetization along (1,0) direction through impurity with $d$-wave superconductor described by tight-binding band with $t^{\prime }∕t=-0.2$, $V_{\mathrm{imp}}=100t$, $g{\mu }_{B}B∕2=0.004t$, $U∕t=0,1,1.5$. From 180.

Figure 38

(Color online) Origin of impurity band in $d$-wave superconductors. Left: Density of states $\rho \left(0\right)$ vs position in a $70\times 70$ lattice, showing interference of many impurities in band with no particle-hole symmetry. White dots represent strong potential scatterers. From 34. Right: Schematic density of states of a $d$-wave superconductor, and impurity band of width $\gamma$.

Figure 39

(Color online) Spatial variations of the staggered magnetization $M_s$ [(a), (c)], and LDOS spectra [(b), (d)] calculated for a $32\times 32$ system. The impurity site is at (16,16). Solid lines in (b) and (d) are for (16,16), dashed lines for (17,16), dash-dotted lines for (17,17), and dotted lines for (16,1). The upper [(a),(b)] and the lower panels [(c),(d)] are for $(U,V_{\mathrm{imp}})=(2,3)$ and (2.35,100), respectively. From 99.

Figure 40

Zero magnetic-field phase diagram for a single impurity of strength $ϵ=V_{\mathrm{imp}}$ in the weak-coupling approach, with single band characterized by $t^{\prime }∕t=-0.2$, $n=0.85$, as a function of $U$. Values of $S$ indicated are total values of $S$ induced by an impurity in the paramagnetic state of a $d$-wave superconductor. From 180.

Figure 41

Magnetization histogram for a $34\times 34$ system with $U=1.75$, and 1, 4, 8, and 12 impurities at $T=0.013$ and $g{\mu }_{B}B∕2=0.004$. From 180.

Figure 42

Fits to penetration depth measurements of Hardy and-coworkers (65) on crystals of near optimally doped YBCO substituted with Zn concentrations 0, 0.15%, and 0.31%, using dirty $d$-wave theory (189) with scattering parameters $\Gamma$ as indicated.

Figure 43

Imaginary part of dynamical susceptibility ${\chi }^{″}$ $(\mathbf{q}=\pi ,\pi ,\omega )$ on $\mathrm{Y}\mathrm{B}\mathrm{C}\mathrm{Zn}{\mathrm{O}}_7$ extracted from neutron scattering at $T=10.5\mathrm{K}$. Circles and squares are independent measurements in different energy ranges; open and closed symbols correspond to different methods of determining ${\chi }^{″}$ on the same sample. Results obtained at $4.5\mathrm{K}$ in pure sample at the same O content given for comparison by crosses.

Figure 44

(Color online) Dependence on temperature of the ordered magnetic moment squared deduced from neutron measurements on LSCO (253). The zero-field signal increases gradually below $T_c$ $(H=0\mathrm{T})$, and the dashed line is a guide to the eye. Also plotted is the field-induced signal (equal to the signal measured in field minus the zero-field signal) for fields of $H=5$, 10, and $14.5\mathrm{T}$.

Figure 45

$T$ dependence of ${}^{63}\mathrm{Cu}$ Knight shift in $\mathrm{Y}\mathrm{B}\mathrm{C}{\mathrm{O}}_7$ with (left) Zn or (right) Ni substituted for Cu (205).

Figure 46

(Color online) Experimental STM impurity spatial patterns at resonance and associated spectra. (a) Observed differential conductance around Zn impurity at $-1.2\mathrm{mV}$. (b) Comparison of spectra directly above Zn impurity site and at site far from impurity. (c) Observed STM intensity around Ni impurity site at $9\mathrm{mV}$ (inset: pattern at $-9\mathrm{mV}$). (d) Comparison of spectra directly above Ni impurity site and away from Ni. (e) Observed STM intensity around native defect, possibly Cu vacancy at $-0.5\mathrm{mV}$. (f) Comparison of spectra directly above native defect (thick solid line) and away from defect (thin solid line). Note that in the images (a), (c), and (f) the Cu-O bond is at 45° with respect to the horizontal.

Figure 47

$T$ variation of the ${}^7\mathrm{Li}$ NMR shift ${}^{7}K$ in YBCO (59). Top: Underdoped O6.6 sample, with Curie-Weiss fit (solid line). Bottom: Optimally doped O6.97 samples, Kondo model susceptibility fit to data above (solid line) and below $T_c$ (dashed line).

Figure 48

${}^{17}\mathrm{O}$ NMR lines in $\mathrm{Y}\mathrm{B}\mathrm{C}{\mathrm{O}}_7$ at $T=15\mathrm{K}$ for three different in-plane concentrations of Zn $x_p=1.5$, 3, and 6 % shown relative to the pure case (in gray). From 347.