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Physical review. B, Condensed matter

Spin gap and dynamics in Sr14−xCaxCu24O41 comprising hole-doped two-leg spin ladders: Cu NMR study on single crystals

K. Magishi, Shinji Matsumoto, Yoshitaka KITAOKA, Kenji Ishida, Kei Asayama, M. Uehara, Takashi Nagata, Jun Akimitsu

Published May 1, 1998• 136 citations
Publisher

Abstract

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 ${\mathrm{CuO}}_{2}$ chains and two-leg ${\mathrm{Cu}}_{2}{\mathrm{O}}_{3}$ ladders. From measurements of the ${}^{63}\mathrm{Cu}$ 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 ${\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ (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 ${1/T}_{1}$ 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: ${1/T}_{1}=\mathrm{const}$ 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 ${T}_{2G}{/T}_{1}\sqrt{T}$ 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 ${T}_{2G}^{2}$, 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 ${T}_{2G}^{2}(T=0)=A$ 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 ${\mathrm{Cu}}_{2}{\mathrm{O}}_{3}$ 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.

DOI
10.1103/physrevb.57.11533

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Physical Review Letters•Dec 19, 1994•732 citations

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Journal of Applied Physics•Jun 1, 1998•0 citations

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We have performed inelastic-neutron-scattering and dc susceptibility measurements on a polycrystalline sample of ${({\mathrm{Sr}}_{0.8}{\mathrm{Ca}}_{0.2})}_{14}$${\mathrm{Cu}}_{24}$${\mathrm{O}}_{41}$, which has a complex crystal structure comprising alternating layers containing isolated Cu${\mathrm{O}}_{2}$ chains and ${\mathrm{Cu}}_{2}$${\mathrm{O}}_{3}$ 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 Cu${\mathrm{O}}_{2}$ chains. We see some evidence in our susceptibility and neutron-scattering measurements to suggest that portions of the Cu${\mathrm{O}}_{2}$ chains are dimerized, or broken up into larger finite chains by nonmagnetic Cu${\mathrm{O}}_{2}$ units.

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Properties of undoped and doped spin-1/2 ladders at finite temperature

Isabelle D. Exius

Antiferromagnetic spin-1/2 ladders have been studied in theoretical and experimental physics for several years. The spin ladder is a good model to bridge between theory and experiment, because it is realized in the copper oxide family Sr14-x(La,Ca)xCu24O41. These copper oxides consist of alternating layers of edge-sharing chains and ladders. The insulating mother compound Sr14Cu24O41 is hole doped in the chains as well as slightly in the ladders. The hole content in the Sr14Cu24O41 ladders is reduced, if La3+ is substituted for Sr2+ resulting in a pure antiferromagnetic spin ladder given by La4Sr10Cu24O41. In this thesis such spin ladders are studied by an amalgamated approach of theoretical work and experiments. For the La4Sr10Cu24O41 ladder the temperature development is investigated using scattering techniques as well as computer modelling. At zero temperature the spin ladder has already been well described by effective models computed via perturbative continuous unitary transformations (PCUTs). These PCUT results are now combined with a mean field approach allowing for incorporation of temperature induced conditional excitations via vertex corrections. The vertex correction results in a decreasing one-triplon spectral weight upon increasing temperature. This effect is studied by inelastic neutron scattering (INS) on La4Sr10Cu24O41 crystals by measuring the scattering amplitude at various temperatures. Convincing agreement is found within the experimental and theoretical error bars. Additionally the coupling constants found in Ref. [1] could be confirmed. The importance of an inter-ladder coupling Jinter between isolated ladders in the plane is investigated by combination of the PCUT with a mean field approach. This calculation is undertaken for the square and the trellis lattice. The difference to the existing calculations [2], [3] and [4] is the partial inclusion of the hardcore interaction for neighbouring ladders. At zero temperature we reveal the effect of quantum fluctuations on the spin gap. In the case of the square lattice we find the closure of the spin gap at a critical inter-ladder coupling enlarged by 3 - 4%. In the case of the trellis lattice, the spin gap decreases also as a function of inter-ladder coupling. In this case we find an enlarged critical inter-ladder coupling up to 21%. In additon, the theoretically predicted shift of the spectral weight by Uhrig and Schmidt could experimentally be confirmed. Substituting Ca2+ for Sr2+ results in hole doped ladders, such as Sr2.5Ca11.5Cu24O41 and Sr8Ca6Cu24O41, where the hole doping depends on the substituted Ca2+ content. The magnetic spectra of these two doped ladders are measured by INS and compared with the undoped ladder La4Sr10Cu24O41. The comparison reveals new features tracing back to the presence of holes. At low temperature in both compounds scattering below the gap of the undoped ladder is found resulting in a subgap1 state at about 8meV. The subgap broadens at high temperature so much that the closing of the subgap results. Additionally for both dopings and at low and high temperature a hole-pair captured by triplons is found with a fixed energy gap at 23meV. The new features are qualitatively explained in terms of resonant valence bond theory (RVB) as well as stripe ordering theory.

•Jan 18, 2011•1 citation

Impurity-induced staggered polarization and antiferromagnetic order in spin- 12 Heisenberg two-leg ladder compound SrCu2O3: Extensive Cu NMR and NQR studies

Shigeki Ohsugi, Y. Tokunaga, Kenji Ishida +4 more

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 ${\mathrm{SrCu}}_{2}{\mathrm{O}}_{3}$ (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 ${S}_{0}=1/2$ induced next to impurity on the rung for the Zn and Ni doping ${(S}_{0}=1/4$ for the La doping) creates the staggered spin polarization along the leg, which decreases exponentially from ${S}_{0}.$ 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 ${D}_{\mathrm{AV}}=1/(2x)$ between the Zn and Ni impurities and ${D}_{\mathrm{AV}}=1/x$ between the La impurities. When ${D}_{\mathrm{AV}}=500$ 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 ${T}_{N}(\mathrm{exp})=3$ K $(x=0.01)$ and 5.8 K $(x=0.02)$ quantitatively and predicts to be as small as ${T}_{N}=0.09$ K for $x=0.001$ using ${J}_{0}=2000$ 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 ${T}_{N}$ is determined by a strength of the interladder interaction and a size of ${S}_{0}.$

Physical review. B, Condensed matter•Aug 1, 1999•37 citations

Magnetic excitations from the singlet ground state in theS=1/2 quasi-one-dimensional system Sr14−x Yx Cu24 O41

Masaaki Matsuda, K. Katsumata, Hiroshi Eisaki +4 more

Inelastic neutron scattering measurements have been performed on the S=1/2 quasi-one-dimensional system ${\mathrm{Sr}}_{14\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Y}}_{\mathit{x}}$${\mathrm{Cu}}_{24}$${\mathrm{O}}_{41}$, which has both simple chains and two-leg ladders of copper ions. Strong magnetic inelastic peaks, which originate from the simple chains, have been observed at 9--14 meV in ${\mathrm{Sr}}_{14}$${\mathrm{Cu}}_{24}$${\mathrm{O}}_{41}$. These spin gaps have been confirmed to originate from a dimerized state of the chain. The dimers are formed between spins which are separated by 2 and 4 times the distance between the nearest-neighbor copper ions in the simple chain. In addition, a weaker magnetic inelastic peak around 11 meV, which originates from a dimerized state in the ladder, has also been observed. These dimers are formed between the nearest-neighbor copper ions which are connected by the interladder coupling. We have also studied the effect of ${\mathrm{Y}}^{3+}$ substitution for ${\mathrm{Sr}}^{2+}$ site on the dimerized states. It was found that the yttrium substitution suppresses the gap energies drastically. Possible origins of the dimerized ground state and the excitations are discussed. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matter•Nov 1, 1996•99 citations
We report comprehensive Cu NMR studies on single crystals of Sr14\ensuremath−xCaxCu24O41,{\mathrm{Sr}}_{14\ensuremath{-}x}{\mathrm{Ca}}_{x}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41},Sr14\ensuremath−x​Cax​Cu24​O41​, which contain simple CuO2{\mathrm{CuO}}_{2}CuO2​ chains and two-leg Cu2O3{\mathrm{Cu}}_{2}{\mathrm{O}}_{3}Cu2​O3​ ladders. From measurements of the 63Cu{}^{63}\mathrm{Cu}63Cu NMR shift, it is clear that the spin gap in the ladders decreases with isovalent Ca substitution from \ensuremathΔ=550\ifmmode±\else\textpm\fi30\ensuremath{\Delta}=550\ifmmode\pm\else\textpm\fi{}30\ensuremathΔ=550\ifmmode±\else\textpm\fi30 K for Sr14Cu24O41{\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}Sr14​Cu24​O41​ (Sr14) to 350\ifmmode±\else\textpm\fi30350\ifmmode\pm\else\textpm\fi{}30350\ifmmode±\else\textpm\fi30 K, 280\ifmmode±\else\textpm\fi30280\ifmmode\pm\else\textpm\fi{}30280\ifmmode±\else\textpm\fi30 K, and 270\ifmmode±\else\textpm\fi30270\ifmmode\pm\else\textpm\fi{}30270\ifmmode±\else\textpm\fi30 K for x=6x=6x=6 (Ca6), x=9x=9x=9 (Ca9), and x=11.5x=11.5x=11.5 (Ca11.5), respectively. The exponential decrease of the nuclear spin-lattice relaxation rate 1/T1{1/T}_{1}1/T1​ below \ensuremath{\sim}130 K is consistent with the presence of the spin gap in the spin excitation spectrum. In the TTT range higher than \ensuremath∼200\ensuremath{\sim}200\ensuremath∼200 K, we observed the following dependences: 1/T1=const{1/T}_{1}=\mathrm{const}1/T1​=const and the square of Gaussian spin-echo decay time, T2G2\ensuremath∝T{T}_{2G}^{2}\ensuremath{\propto}TT2G2​\ensuremath∝T which are consistent with the scaling theory for the S=1/2S=1/2S=1/2 one-dimensional (1D) Heisenberg model. The value of T2G/T1T{T}_{2G}{/T}_{1}\sqrt{T}T2G​/T1​T​ is compatible with the theoretical prediction of an exchange constant along the leg J\ensuremath∥\ensuremath∼1800{J}_{\ensuremath{\parallel}}\ensuremath{\sim}1800J\ensuremath∥​\ensuremath∼1800 K for Ca6 and J\ensuremath∥\ensuremath∼1500{J}_{\ensuremath{\parallel}}\ensuremath{\sim}1500J\ensuremath∥​\ensuremath∼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{T}_{2G}^{2}T2G2​, which is proportional to the inverse spin correlation length \ensuremathξ\ensuremath−1{\ensuremath{\xi}}^{\ensuremath{-}1}\ensuremathξ\ensuremath−1, deviates from a linear TTT dependence upon cooling and is described by A+BTexp(\ensuremath−\ensuremathΔ/T)A+BT\mathrm{exp}(\ensuremath{-}\ensuremath{\Delta}/T)A+BTexp(\ensuremath−\ensuremathΔ/T), regardless of the Ca substitution. We point out that the value of T2G2(T=0)=A{T}_{2G}^{2}(T=0)=AT2G2​(T=0)=A is proportional to the finite value of \ensuremathξ0\ensuremath−1=\ensuremathΔ/c1D,{\ensuremath{\xi}}_{0}^{\ensuremath{-}1}=\ensuremath{\Delta}{/c}_{1D},\ensuremathξ0\ensuremath−1​=\ensuremathΔ/c1D​, where c1D=(\ensuremathπ/2)J\ensuremath∥{c}_{1D}=(\ensuremath{\pi}{/2)J}_{\ensuremath{\parallel}}c1D​=(\ensuremathπ/2)J\ensuremath∥​ is the spin-wave velocity. From the result that the values of A\ensuremath−1\ensuremath∼\ensuremathξeff{A}^{\ensuremath{-}1}\ensuremath{\sim}{\ensuremath{\xi}}_{\mathrm{eff}}A\ensuremath−1\ensuremath∼\ensuremathξeff​ for Ca6, Ca9, and Ca11.5 are significantly reduced compared to that for Sr14, it is suggested that \ensuremathξeff{\ensuremath{\xi}}_{\mathrm{eff}}\ensuremathξeff​ is dominated at low TTT by an average distance ddd among mobile holes obeying the relation \ensuremathξeff\ensuremath∼d=\ensuremathξh.{\ensuremath{\xi}}_{\mathrm{eff}}\ensuremath{\sim}d={\ensuremath{\xi}}_{h}.\ensuremathξeff​\ensuremath∼d=\ensuremathξh​. From an estimate of \ensuremathξ0/a\ensuremath∼{\ensuremath{\xi}}_{0}/a\ensuremath{\sim}\ensuremathξ0​/a\ensuremath∼ 5.2 for Sr14, where aaa is the Cu-Cu distance along the leg, \ensuremathξh/a{\ensuremath{\xi}}_{h}/a\ensuremathξh​/a is obtained as \ensuremath∼3.5\ensuremath{\sim}3.5\ensuremath∼3.5, 2.3, and 2.0, and hole content xxx as \ensuremath∼0.14\ensuremath{\sim}0.14\ensuremath∼0.14, 0.22, and 0.25 per Cu2O3{\mathrm{Cu}}_{2}{\mathrm{O}}_{3}Cu2​O3​ ladder for Ca6, Ca9, and Ca11.5, respectively. These values were consistent with x=x=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 Sr14\ensuremath−xCaxCu24O41{\mathrm{Sr}}_{14\ensuremath{-}x}{\mathrm{Ca}}_{x}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}Sr14\ensuremath−x​Cax​Cu24​O41​ compounds are thus hole-doped two-leg spin-ladder systems which reveal the metallic behavior dominated by the 1D-like spin dynamics at high TTT and accompanied by the spin gap formation at low TTT. For Ca11.5, as the spin gap is formed upon cooling below \ensuremath∼180\ensuremath{\sim}180\ensuremath∼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 TL\ensuremath∼60{T}_{L}\ensuremath{\sim}60TL​\ensuremath∼60 K. We point out that preformed pairs are confined in each ladder and localized below \ensuremath∼60\ensuremath{\sim}60\ensuremath∼60 K at an ambient pressure.
Magnetic properties of SrCu2{\mathrm{Cu}}_{2}Cu2​O3{\mathrm{O}}_{3}O3​ and Sr2{\mathrm{Sr}}_{2}Sr2​Cu3{\mathrm{Cu}}_{3}Cu3​O5{\mathrm{O}}_{5}O5​, containing two-leg and three-leg S=12S=\frac{1}{2}S=21​ ladders made of antiferromagnetic Cu-O-Cu linear bonds, were investigated. The susceptibility of the two-leg ladder material, SrCu2{\mathrm{Cu}}_{2}Cu2​O3{\mathrm{O}}_{3}O3​, is characteristic of thermal excitation from a nonmagnetic ground state with a spin gap of about 420 K, while the susceptibility of Sr2{\mathrm{Sr}}_{2}Sr2​Cu3{\mathrm{Cu}}_{3}Cu3​O5{\mathrm{O}}_{5}O5​, containing three-leg ladders, reflects a gapless spin excitation spectrum. The temperature dependence of the nuclear-spin lattice relaxation rate, 1T1\frac{1}{{T}_{1}}T1​1​, of 63Cu^{63}\mathrm{Cu}63Cu NMR also indicates the existence of a spin gap only for SrCu2{\mathrm{Cu}}_{2}Cu2​O3{\mathrm{O}}_{3}O3​. The spin gap we observed for SrCu2{\mathrm{Cu}}_{2}Cu2​O3{\mathrm{O}}_{3}O3​ confirms a recent theoretical prediction for the magnetic behavior of spin = \textonehalf{} antiferromagnetically coupled Heisenberg chains.
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.
We have performed inelastic-neutron-scattering and dc susceptibility measurements on a polycrystalline sample of (Sr0.8Ca0.2)14{({\mathrm{Sr}}_{0.8}{\mathrm{Ca}}_{0.2})}_{14}(Sr0.8​Ca0.2​)14​Cu24{\mathrm{Cu}}_{24}Cu24​O41{\mathrm{O}}_{41}O41​, which has a complex crystal structure comprising alternating layers containing isolated CuO2{\mathrm{O}}_{2}O2​ chains and Cu2{\mathrm{Cu}}_{2}Cu2​O3{\mathrm{O}}_{3}O3​ 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{\mathrm{O}}_{2}O2​ chains. We see some evidence in our susceptibility and neutron-scattering measurements to suggest that portions of the CuO2{\mathrm{O}}_{2}O2​ chains are dimerized, or broken up into larger finite chains by nonmagnetic CuO2{\mathrm{O}}_{2}O2​ units.
Antiferromagnetic spin-1/2 ladders have been studied in theoretical and experimental physics for several years. The spin ladder is a good model to bridge between theory and experiment, because it is realized in the copper oxide family Sr14-x(La,Ca)xCu24O41. These copper oxides consist of alternating layers of edge-sharing chains and ladders. The insulating mother compound Sr14Cu24O41 is hole doped in the chains as well as slightly in the ladders. The hole content in the Sr14Cu24O41 ladders is reduced, if La3+ is substituted for Sr2+ resulting in a pure antiferromagnetic spin ladder given by La4Sr10Cu24O41. In this thesis such spin ladders are studied by an amalgamated approach of theoretical work and experiments. For the La4Sr10Cu24O41 ladder the temperature development is investigated using scattering techniques as well as computer modelling. At zero temperature the spin ladder has already been well described by effective models computed via perturbative continuous unitary transformations (PCUTs). These PCUT results are now combined with a mean field approach allowing for incorporation of temperature induced conditional excitations via vertex corrections. The vertex correction results in a decreasing one-triplon spectral weight upon increasing temperature. This effect is studied by inelastic neutron scattering (INS) on La4Sr10Cu24O41 crystals by measuring the scattering amplitude at various temperatures. Convincing agreement is found within the experimental and theoretical error bars. Additionally the coupling constants found in Ref. [1] could be confirmed. The importance of an inter-ladder coupling Jinter between isolated ladders in the plane is investigated by combination of the PCUT with a mean field approach. This calculation is undertaken for the square and the trellis lattice. The difference to the existing calculations [2], [3] and [4] is the partial inclusion of the hardcore interaction for neighbouring ladders. At zero temperature we reveal the effect of quantum fluctuations on the spin gap. In the case of the square lattice we find the closure of the spin gap at a critical inter-ladder coupling enlarged by 3 - 4%. In the case of the trellis lattice, the spin gap decreases also as a function of inter-ladder coupling. In this case we find an enlarged critical inter-ladder coupling up to 21%. In additon, the theoretically predicted shift of the spectral weight by Uhrig and Schmidt could experimentally be confirmed. Substituting Ca2+ for Sr2+ results in hole doped ladders, such as Sr2.5Ca11.5Cu24O41 and Sr8Ca6Cu24O41, where the hole doping depends on the substituted Ca2+ content. The magnetic spectra of these two doped ladders are measured by INS and compared with the undoped ladder La4Sr10Cu24O41. The comparison reveals new features tracing back to the presence of holes. At low temperature in both compounds scattering below the gap of the undoped ladder is found resulting in a subgap1 state at about 8meV. The subgap broadens at high temperature so much that the closing of the subgap results. Additionally for both dopings and at low and high temperature a hole-pair captured by triplons is found with a fixed energy gap at 23meV. The new features are qualitatively explained in terms of resonant valence bond theory (RVB) as well as stripe ordering theory.
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​.
Inelastic neutron scattering measurements have been performed on the S=1/2 quasi-one-dimensional system ${\mathrm{Sr}}_{14\mathrm{\ensuremath{-}}\mathit{x}}$Yx{\mathrm{Y}}_{\mathit{x}}Yx​Cu24{\mathrm{Cu}}_{24}Cu24​O41{\mathrm{O}}_{41}O41​, which has both simple chains and two-leg ladders of copper ions. Strong magnetic inelastic peaks, which originate from the simple chains, have been observed at 9--14 meV in Sr14{\mathrm{Sr}}_{14}Sr14​Cu24{\mathrm{Cu}}_{24}Cu24​O41{\mathrm{O}}_{41}O41​. These spin gaps have been confirmed to originate from a dimerized state of the chain. The dimers are formed between spins which are separated by 2 and 4 times the distance between the nearest-neighbor copper ions in the simple chain. In addition, a weaker magnetic inelastic peak around 11 meV, which originates from a dimerized state in the ladder, has also been observed. These dimers are formed between the nearest-neighbor copper ions which are connected by the interladder coupling. We have also studied the effect of Y3+{\mathrm{Y}}^{3+}Y3+ substitution for Sr2+{\mathrm{Sr}}^{2+}Sr2+ site on the dimerized states. It was found that the yttrium substitution suppresses the gap energies drastically. Possible origins of the dimerized ground state and the excitations are discussed. \textcopyright{} 1996 The American Physical Society.