Magnetic properties of Sr${\mathrm{Cu}}_{2}$${\mathrm{O}}_{3}$ and ${\mathrm{Sr}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{5}$, containing two-leg and three-leg $S=\frac{1}{2}$ ladders made of antiferromagnetic Cu-O-Cu linear bonds, were investigated. The susceptibility of the two-leg ladder material, Sr${\mathrm{Cu}}_{2}$${\mathrm{O}}_{3}$, is characteristic of thermal excitation from a nonmagnetic ground state with a spin gap of about 420 K, while the susceptibility of ${\mathrm{Sr}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{5}$, containing three-leg ladders, reflects a gapless spin excitation spectrum. The temperature dependence of the nuclear-spin lattice relaxation rate, $\frac{1}{{T}_{1}}$, of $^{63}\mathrm{Cu}$ NMR also indicates the existence of a spin gap only for Sr${\mathrm{Cu}}_{2}$${\mathrm{O}}_{3}$. The spin gap we observed for Sr${\mathrm{Cu}}_{2}$${\mathrm{O}}_{3}$ confirms a recent theoretical prediction for the magnetic behavior of spin = \textonehalf{} antiferromagnetically coupled Heisenberg chains.
Magnetic properties of Sr{\mathrm{Cu}}_{2}$${\mathrm{O}}_{3} and {\mathrm{Sr}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{5}, containing two-leg and three-leg S=21 ladders made of antiferromagnetic Cu-O-Cu linear bonds, were investigated. The susceptibility of the two-leg ladder material, Sr{\mathrm{Cu}}_{2}$${\mathrm{O}}_{3}, is characteristic of thermal excitation from a nonmagnetic ground state with a spin gap of about 420 K, while the susceptibility of {\mathrm{Sr}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{5}, containing three-leg ladders, reflects a gapless spin excitation spectrum. The temperature dependence of the nuclear-spin lattice relaxation rate, T11, of 63Cu NMR also indicates the existence of a spin gap only for Sr{\mathrm{Cu}}_{2}$${\mathrm{O}}_{3}. The spin gap we observed for Sr{\mathrm{Cu}}_{2}$${\mathrm{O}}_{3} confirms a recent theoretical prediction for the magnetic behavior of spin = \textonehalf{} antiferromagnetically coupled Heisenberg chains.
We report comprehensive Cu NMR studies on single crystals of ${\mathrm{Sr}}_{14\ensuremath{-}x}{\mathrm{Ca}}_{x}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41},$ which contain simple CuO2 chains and two-leg Cu2O3 ladders. From measurements of the 63Cu NMR shift, it is clear that the spin gap in the ladders decreases with isovalent Ca substitution from $\ensuremath{\Delta}=550\ifmmode\pm\else\textpm\fi{}30$ K for Sr14Cu24O41 (Sr14) to $350\ifmmode\pm\else\textpm\fi{}30$ K, $280\ifmmode\pm\else\textpm\fi{}30$ K, and $270\ifmmode\pm\else\textpm\fi{}30$ K for x=6 (Ca6), x=9 (Ca9), and x=11.5 (Ca11.5), respectively. The exponential decrease of the nuclear spin-lattice relaxation rate 1/T1 below \ensuremath{\sim}130 K is consistent with the presence of the spin gap in the spin excitation spectrum. In the T range higher than $\ensuremath{\sim}200$ K, we observed the following dependences: 1/T1=const and the square of Gaussian spin-echo decay time, ${T}_{2G}^{2}\ensuremath{\propto}T$ which are consistent with the scaling theory for the S=1/2 one-dimensional (1D) Heisenberg model. The value of T2G/T1T is compatible with the theoretical prediction of an exchange constant along the leg ${J}_{\ensuremath{\parallel}}\ensuremath{\sim}1800$ K for Ca6 and ${J}_{\ensuremath{\parallel}}\ensuremath{\sim}1500$ K for Ca9 and Ca11.5. A notable finding is that the magnitude of the spin gap remains nearly constant and characteristics of novel 1D-like spin dynamics are maintained in the content varying from Ca9 to Ca11.5. On the other hand, the charge transport changes with increasing Ca content so that the more conductive Ca11.5 exhibits pressure-induced superconductivity exceeding 3.5 GPa. We have found that T2G2, which is proportional to the inverse spin correlation length ${\ensuremath{\xi}}^{\ensuremath{-}1}$, deviates from a linear T dependence upon cooling and is described by $A+BT\mathrm{exp}(\ensuremath{-}\ensuremath{\Delta}/T)$, regardless of the Ca substitution. We point out that the value of T2G2(T=0)=A is proportional to the finite value of ${\ensuremath{\xi}}_{0}^{\ensuremath{-}1}=\ensuremath{\Delta}{/c}_{1D},$ where ${c}_{1D}=(\ensuremath{\pi}{/2)J}_{\ensuremath{\parallel}}$ is the spin-wave velocity. From the result that the values of ${A}^{\ensuremath{-}1}\ensuremath{\sim}{\ensuremath{\xi}}_{\mathrm{eff}}$ for Ca6, Ca9, and Ca11.5 are significantly reduced compared to that for Sr14, it is suggested that ${\ensuremath{\xi}}_{\mathrm{eff}}$ is dominated at low T by an average distance d among mobile holes obeying the relation ${\ensuremath{\xi}}_{\mathrm{eff}}\ensuremath{\sim}d={\ensuremath{\xi}}_{h}.$ From an estimate of ${\ensuremath{\xi}}_{0}/a\ensuremath{\sim}$ 5.2 for Sr14, where a is the Cu-Cu distance along the leg, ${\ensuremath{\xi}}_{h}/a$ is obtained as $\ensuremath{\sim}3.5$, 2.3, and 2.0, and hole content x as $\ensuremath{\sim}0.14$, 0.22, and 0.25 per Cu2O3 ladder for Ca6, Ca9, and Ca11.5, respectively. These values were consistent with x= 0.14, 0.2, and 0.22 for Ca6, Ca9, and Ca11 estimated from the optical conductivity experiment by Osafune et al. [Phys. Rev. Lett. 78, 1980 (1997)]. The ${\mathrm{Sr}}_{14\ensuremath{-}x}{\mathrm{Ca}}_{x}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ compounds are thus hole-doped two-leg spin-ladder systems which reveal the metallic behavior dominated by the 1D-like spin dynamics at high T and accompanied by the spin gap formation at low T. For Ca11.5, as the spin gap is formed upon cooling below $\ensuremath{\sim}180$ K, the resistivity increases in the direction perpendicular to the ladder, whereas the conductivity along the ladder remains metallic, followed by the localization of mobile holes in both directions below ${T}_{L}\ensuremath{\sim}60$ K. We point out that preformed pairs are confined in each ladder and localized below $\ensuremath{\sim}60$ K at an ambient pressure.
We have performed inelastic-neutron-scattering and dc susceptibility measurements on a polycrystalline sample of (Sr0.8Ca0.2)14Cu24O41, which has a complex crystal structure comprising alternating layers containing isolated CuO2 chains and Cu2O3 two-legged ladders. The observed excitation spectrum is consistent with a spin gap for the ladder of 35 meV. At lower energy transfers we have observed a broad, nondispersive, feature centered on 10 meV that corresponds to singlet-triplet excitations in the CuO2 chains. We see some evidence in our susceptibility and neutron-scattering measurements to suggest that portions of the CuO2 chains are dimerized, or broken up into larger finite chains by nonmagnetic CuO2 units.
We present a careful analysis of the temperature dependence of the nuclear spin-lattice relaxation rate l/ T 1 in gapped quasi-1D Heisenberg antiferromagnets. It is found that in order to estimate the value of the gap correctly from l/ T 1 , the peculiar features of the dispersion curve for the triplet excitations must be taken into account. The temperature dependence of l/ T 1 due to two-magnon processes is reported for different values of the ratio r = J ⌊ / J ‖ between the superexchange constants in a 2-leg ladder. As an illustrative example, we compare our results to the experimental findings for 63 Cu l/ T 1 in the dimerized chains and 2-leg ladders contained in Sr 14 Cu 24 O 41
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 (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<0.02 and ${\mathrm{Sr}}_{1\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{Cu}}_{2}{\mathrm{O}}_{3}$ with 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. 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 induced next to impurity on the rung for the Zn and Ni doping (S0=1/4 for the La doping) creates the staggered spin polarization along the leg, which decreases exponentially from 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=0.001 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) between the Zn and Ni impurities and DAV=1/x between the La impurities. When DAV=500 for x=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 K (x=0.01) and 5.8 K (x=0.02) quantitatively and predicts to be as small as TN=0.09 K for x=0.001 using J0=2000 K. On the other hand, there is no evidence of AF-LRO for the La doping (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 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 is determined by a strength of the interladder interaction and a size of S0.
The authors report on NMR and susceptibility measurements in Sr14−xNaxCu24O41 for 0⩽x⩽2. The analysis of the susceptibility evidences that only a small fraction of Cu2+ spins are involved in the dimerization of Cu(1)O2 chains. The introduction of extra holes by Na+ doping is found to decrease the Cu(1)2+ uniform static susceptibility without affecting the amplitude of the dimerization gap. On the other hand, the amplitude of the spin gap in the Cu(2)2O3 two-leg-ladders and the resistivity are reduced by doping. The origin of the discrepancy in the amplitude of the gap estimated by means of nuclear spin-lattice relaxation and susceptibility measurements is briefly discussed.
The temperature dependence of the static spin susceptibility, χ, and NMR relaxation rate, T 1 -1 , of S =1/2 Heisenberg two-leg ladder systems have been studied based on the Majorana fermion representation of the S =1/2 Heisenberg spin Hamiltonian introduced by Shelton, Nersesyan and Tsvelik (Phys. Rev. B 53 (1996) 8521). It is shown that in the case of SrCu 2 O 3 , we can fit the experimental data of χ and T 1 -1 in terms of the massive Majorana fermion excitation with mass m =440 K to resolve the apparent discrepancy of the spin gap activation energies between χ and T 1 -1 .