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arXiv:cond-mat/9802260 · Physical Review Letters

Order from Disorder: Nonmagnetic Impurities in the Spin-Gap Phase of the Cuprates

C. Pépin, Patrick A. Lee

Published Sep 28, 1998• 44 citations
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Abstract

We solve the problem of $N$ nonmagnetic impurities in the staggered flux phase of the Heisenberg model which we assume to be a good mean-field approximation for the spin-gap phase of the cuprates. The density of states is evaluated exactly in the unitary scattering limit and is proportional to $1/\ensuremath{\omega}{ln}^{2}(|\ensuremath{\omega}|/D)$, in analogy with the 1D case of doped spin-Peierls and two-leg ladder compounds. We argue that the system exhibits a quasi-long-range order at $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ with instantaneous spin-spin correlations decreasing as ${n}_{i}/{ln}^{2}({n}_{i}{R}_{\mathrm{ij}})$ for large distances ${R}_{\mathrm{ij}}$ and we predict enhanced low energy fluctuations in neutron scattering.

DOI
10.1103/physrevlett.81.2779

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We solve the problem of NNN nonmagnetic impurities in the staggered flux phase of the Heisenberg model which we assume to be a good mean-field approximation for the spin-gap phase of the cuprates. The density of states is evaluated exactly in the unitary scattering limit and is proportional to 1/\ensuremathωln2(∣\ensuremathω∣/D)1/\ensuremath{\omega}{ln}^{2}(|\ensuremath{\omega}|/D)1/\ensuremathωln2(∣\ensuremathω∣/D), in analogy with the 1D case of doped spin-Peierls and two-leg ladder compounds. We argue that the system exhibits a quasi-long-range order at T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 with instantaneous spin-spin correlations decreasing as ni/ln2(niRij){n}_{i}/{ln}^{2}({n}_{i}{R}_{\mathrm{ij}})ni​/ln2(ni​Rij​) for large distances Rij{R}_{\mathrm{ij}}Rij​ and we predict enhanced low energy fluctuations in neutron scattering.
We report characteristics of impurity-induced staggered polarization (IISP) and antiferromagnetic long-range order (AF-LRO) in the gapped spin-1/2 Heisenberg two-leg ladder compound SrCu2O3{\mathrm{SrCu}}_{2}{\mathrm{O}}_{3}SrCu2​O3​ (Sr123). We have carried out comprehensive NMR and NQR investigations on three impurity-doped systems, $\mathrm{Sr}({\mathrm{Cu}}_{1\ensuremath{-}x}{M}_{x}{)}_{2}{\mathrm{O}}_{3} (M=\mathrm{Z}\mathrm{n},\mathrm{}\mathrm{Ni})$ with x< 0.02x<~0.02x< 0.02 and ${\mathrm{Sr}}_{1\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{Cu}}_{2}{\mathrm{O}}_{3}$ with x< 0.03.x<~0.03.x< 0.03. Either the Zn or Ni impurity that is nonmagnetic depletes a single spin on the ladders, whereas the La impurity is believed to dope electrons onto the ladders. The width of the Lorentzian Cu NMR spectrum increases with the increase in impurity content x and follows the Curie-like temperature (T) dependence as W/T.W/T.W/T. The W's for the Zn- and Ni-doped samples (M doping) are larger than for the La-doped one (La doping). The NMR spectra were fit by assuming that unpaired spin S0=1/2{S}_{0}=1/2S0​=1/2 induced next to impurity on the rung for the Zn and Ni doping (S0=1/4{(S}_{0}=1/4(S0​=1/4 for the La doping) creates the staggered spin polarization along the leg, which decreases exponentially from S0.{S}_{0}.S0​. In Sr123, an instantaneous spin-correlation length ${\ensuremath{\xi}}_{0}$ was theoretically predicted as ${\ensuremath{\xi}}_{0}/a\ensuremath{\sim}3\ensuremath{-}8,$ where a is the lattice spacing between the Cu sites along the leg. However, a correlation length ${\ensuremath{\xi}}_{s}/a$ estimated from the IISP along the leg was found to be much longer than ${\ensuremath{\xi}}_{0}/a$ in x=0.001x=0.001x=0.001 and 0.005. The notable result is that ${\ensuremath{\xi}}_{s}/a$ that was found to be T independent is scaled to mean distances DAV=1/(2x){D}_{\mathrm{AV}}=1/(2x)DAV​=1/(2x) between the Zn and Ni impurities and DAV=1/x{D}_{\mathrm{AV}}=1/xDAV​=1/x between the La impurities. When DAV=500{D}_{\mathrm{AV}}=500DAV​=500 for x=0.001x=0.001x=0.001 (Zn doping), ${\ensuremath{\xi}}_{s}/a\ensuremath{\sim}50$ is estimated. The significantly broadened NQR spectrum has provided unambiguous evidence for the AF-LRO in the Zn and Ni doping (x=0.01 and 0.02). Rather uniform AF moments at the middle Cu sites between the impurities are estimated to be about $0.04{\ensuremath{\mu}}_{B}$ at 1.4 K along the a axis. By assuming that exponential decay constants of AF moments are equivalent to ${\ensuremath{\xi}}_{s}/a$'s for the IISP, the size of an AF moment next to the impurity is deduced as ${S}_{\mathrm{AF}}\ensuremath{\sim}1/4.$ We propose that these exponential distributions of IISP and AF moments along the two-leg suggest that an interladder interaction is in a weakly coupled quasi-one-dimensional (WC-Q1D) regime. The formula of ${T}_{N}{=J}_{0}\mathrm{exp}(\ensuremath{-}{D}_{\mathrm{AV}}/({\ensuremath{\xi}}_{s}/a))$ based on the WC-Q1D model explains TN(exp)=3{T}_{N}(\mathrm{exp})=3TN​(exp)=3 K (x=0.01)(x=0.01)(x=0.01) and 5.8 K (x=0.02)(x=0.02)(x=0.02) quantitatively and predicts to be as small as TN=0.09{T}_{N}=0.09TN​=0.09 K for x=0.001x=0.001x=0.001 using J0=2000{J}_{0}=2000J0​=2000 K. On the other hand, there is no evidence of AF-LRO for the La doping (x=0.02(x=0.02(x=0.02 and 0.03) down to 1.4 K, nevertheless their ${\ensuremath{\xi}}_{s}/a$'s are almost equivalent to those in the Zn and Ni doping (x=0.01(x=0.01(x=0.01 and 0.02). We remark that the Q1D-IISP is dramatically enhanced by the interladder interaction even though so weak, once the impurity breaks up the quantum coherence in the short-range resonating valence bond (RVB) state with the gap. On the one hand, we propose that TN{T}_{N}TN​ is determined by a strength of the interladder interaction and a size of S0.{S}_{0}.S0​.
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