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arXiv:cond-mat/9508037 · Physical review. B, Condensed matter

Accurate determination of the exchange constant in Sr2 CuO3 from recent theoretical results

Sebastian Eggert

Published Mar 1, 1996• 41 citations
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Abstract

Data from susceptibility measurements [Ami et al., Phys. Rev. B 51, 5994 (1995)] on ${\mathrm{Sr}}_{2}$${\mathrm{CuO}}_{3}$ are compared with recent theoretical predictions [Eggert et al., Phys. Rev. Lett. 73, 332 (1994)] for the magnetic susceptibility of the antiferromagnetic spin-1/2 Heisenberg chain. The experimental data fully confirms the theoretical predictions and in turn we establish that ${\mathrm{Sr}}_{2}$${\mathrm{CuO}}_{3}$ behaves almost perfectly like a one-dimensional antiferromagnet with an exchange coupling of J=${1700}_{\mathrm{\ensuremath{-}}100}^{+150}$ K. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.5116

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Extraction of antiferromagnetic (AF) exchange constants J and J\ensuremath{'} in two-leg spin-1/2 nearest-neighbor (NN) Heisenberg ladders from experimental magnetic spin susceptibility \ensuremath{\chi}(T) data is studied, where J\ensuremath{'} is the NN exchange constant in the rungs and J is that in the legs. Two low-T (T\ensuremath{\ll}J/${\mathit{k}}_{\mathit{B}}$) approximations of Troyer, Tsunetsugu, and W\"urtz [Phys. Rev. B 50, 13 515 (1994)] for \ensuremath{\chi}(T) of the isolated spin-1/2 two-leg ladder with J\ensuremath{'}/J=1 are shown to be accurate to \ensuremath{\sim}10% when extrapolated to T\ensuremath{\sim}J\ensuremath{'}/${\mathit{k}}_{\mathit{B}}$. The variations in the magnetic excitation dispersion relation parameters [which enter the low-T expression of Troyer et al. for \ensuremath{\chi}(T)] for J\ensuremath{'}/J1 are inferred from previous theoretical results. Next, the values of the maxima in \ensuremath{\chi}(T)\ensuremath{\equiv}${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$ from many numerical calculations in the literature for various nonfrustrated spin-1/2 NN Heisenberg antiferromagnets are considered. The values of the product ${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$${\mathit{z}}_{\mathrm{eff}}$ are found to depend systematically and almost uniqely on an effective magnetic coordination number ${\mathit{z}}_{\mathrm{eff}}$, defined in the text, but are nearly independent of the dimensionality of the spin lattice and of the magnitude of any gap in the magnetic excitation spectrum. This observation allows accurate bounds on the exchange coupling constants in arbitrary quasi-low-dimensional spin-1/2 antiferromagnets to be inferred from experimental ${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$ values. Finally, a mean-field-type expression for \ensuremath{\chi}(T) for arbitrary quasi-low-dimensional spin-1/2 Heisenberg antiferromagnets is derived, using the above ${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$ phenomenology as input, which allows the influence of intersystem coupling on \ensuremath{\chi}(T) to be evaluated. These results, and the \ensuremath{\chi}(T) calculations of Barnes and Riera [Phys. Rev. B 50, 6817 (1994)] for isolated two-leg spin-1/2 Heisenberg ladders, are used to analyze the experimental \ensuremath{\chi}(T) data of Azuma et al. [Phys. Rev. Lett. 73, 3463 (1994)] for the spin-1/2 two-leg ladder compound ${\mathrm{SrCu}}_{2}$${\mathrm{O}}_{3}$. The analyses together suggest that J\ensuremath{'}/J\ensuremath{\sim}0.5, contrary to the expectation that J\ensuremath{'}/J\ensuremath{\approxeq}1, and that J is very large (\ensuremath{\sim}2000 K), similar to the value of J in the linear-chain cuprate ${\mathrm{Sr}}_{2}$${\mathrm{CuO}}_{3}$. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matter•Nov 1, 1996•74 citations
Data from susceptibility measurements [Ami et al., Phys. Rev. B 51, 5994 (1995)] on {\mathrm{Sr}}_{2}$${\mathrm{CuO}}_{3} are compared with recent theoretical predictions [Eggert et al., Phys. Rev. Lett. 73, 332 (1994)] for the magnetic susceptibility of the antiferromagnetic spin-1/2 Heisenberg chain. The experimental data fully confirms the theoretical predictions and in turn we establish that {\mathrm{Sr}}_{2}$${\mathrm{CuO}}_{3} behaves almost perfectly like a one-dimensional antiferromagnet with an exchange coupling of J=1700\ensuremath−100+150{1700}_{\mathrm{\ensuremath{-}}100}^{+150}1700\ensuremath−100+150​ K. \textcopyright{} 1996 The American Physical Society.
The NMR relaxation data on Sr2CuO3{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}Sr2​CuO3​ [Phys. Rev. Lett. 76, 4612 (1996)] are reexamined and compared with the analytic theory of the dynamic susceptibility in the S=1/2S=1/2S=1/2 antiferromagnetic Heisenberg chain including multiplicative logarithmic corrections [Phys. Rev. Lett. 78, 539 (1997)]. Comparisons of the spin-lattice and the Gaussian spin-echo decay rates (1/T1{1/T}_{1}1/T1​ and 1/T2G{1/T}_{2G}1/T2G​) and their ratio all show good quantitative agreement. Our results demonstrate the importance of the logarithmic corrections in the analysis of experimental data for quasi-one-dimensional systems and indicate that the dynamics of Sr2CuO3{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}Sr2​CuO3​ is well described by an S=1/2S=1/2S=1/2 one-dimensional Heisenberg model with a nearest-neighbor exchange.
Magnetic susceptibility is measured on the one-dimensional spin 1/2 antiferromagnets Sr2CuO3{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}Sr2​CuO3​ and SrCuO2\mathrm{Sr}{\mathrm{CuO}}_{2}SrCuO2​ using single crystals. Large, nearly contamination-free single crystals enable us to measure the intrinsic spin susceptibility over a wide temperature range between 5 and 800 K. The results are in excellent agreement with the recent calculation by Eggert, Affleck, and Takahashi, and the exchange interaction energy JJJ is estimated to be $2200\ifmmode\pm\else\textpm\fi{}200$ and $2100\ifmmode\pm\else\textpm\fi{}200\mathrm{K}$, respectively. In Sr2CuO3{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}Sr2​CuO3​, an isotropic susceptibility drop is observed below about 20 K, which is also consistent with the result of this rigorous calculation.
A comprehensive theoretical and experimental study is presented of the magnetic susceptibility versus temperature χ(T) of spin S = 1/2 two- and three-leg Heisenberg ladders and ladder oxide compounds. Extensive quantum Monte Carlo simulations of χ(T) were carried out for both isolated and coupled two-leg ladders with spatially anisotropic intraladder exchange. Accurate fits to these and related literature QMC data were obtained. We have also calculated the one- and two-magnon dispersion relations and the dynamical spin structure factor for anisotropic isolated 2 x 12 ladders. The exchange constants in the two-leg ladder compound SrCu2O3 are estimated from LDA+U calculations. We report the detailed crystal structure of SrCu2O3 and of the three-leg ladder compound Sr2Cu3O5. New experimental χ(T) data are reported for the two-leg ladder cuprates SrCu2O3 and LaCuO_{2.5}, and for the (nominally) two-leg ladder vanadates CaV2O5 and MgV2O5. The new and literature χ(T) data for these compounds and for Sr2Cu3O5 are modeled using our QMC χ(T) simulation fits, and the exchange coupling constants between the spins-1/2 are thereby estimated for each material. The surpisingly strong spatial anisotropy of the bilinear intraladder exchange constants in the cuprate compounds is discussed together with the results of other experiments sensitive to this anisotropy. Recent theoretical predictions are discussed including those which indicate that a four-spin cyclic exchange interaction within a Cu4 plaquette is important to determining the magnetic properties and which can significantly influence the exchange interactions estimated from χ(T) data assuming the presence of only bilinear exchange.
The magnetic susceptibility ${\ensuremath{\chi}}^{*}(t)$ and specific heat C(t)C(t)C(t) versus temperature t of the spin S=1/2S=1/2S=1/2 antiferromagnetic (AF) alternating-exchange (J1{(J}_{1}(J1​ and J2){J}_{2})J2​) Heisenberg chain are studied for the entire range $0<~\ensuremath{\alpha}<~1$ of the alternation parameter $\ensuremath{\alpha}\ensuremath{\equiv}{J}_{2}{/J}_{1}{(J}_{1},$ J2> 0,{J}_{2}>~0,J2​> 0, J2< J1,{J}_{2}<~{J}_{1},J2​< J1​, t=kBT/J1,{t=k}_{\mathrm{B}}{T/J}_{1},t=kB​T/J1​, ${\ensuremath{\chi}}^{*}=\ensuremath{\chi}{J}_{1}{/Ng}^{2}{\ensuremath{\mu}}_{B}^{2}).$ For the uniform chain $(\ensuremath{\alpha}=1),$ the high-accuracy ${\ensuremath{\chi}}^{*}(t)$ and C(t)C(t)C(t) Bethe ansatz data of Kl\"umper and Johnston (unpublished) are shown to agree very well at low t with the respective exact theoretical low-ttt logarithmic correction predictions of Lukyanov [Nucl. Phys. B 522,522,522, 533 (1998)]. Accurate $(\ensuremath{\sim}{10}^{\ensuremath{-}7})$ independent empirical fits to the respective data are obtained over t ranges spanning 25 orders of magnitude, $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}25}<~t<~5,$ which contain extrapolations to the respective exact t=0t=0t=0 limits. The infinite temperature entropy calculated using our C(t)C(t)C(t) fit function is within 8 parts in 108{10}^{8}108 of the exact value ln2.\mathrm{ln}2.ln2. Quantum Monte Carlo (QMC) simulations and transfer-matrix density-matrix renormalization group (TMRG) calculations of ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ are presented for 0.002< t< 100.002<~t<~100.002< t< 10 and $0.05<~\ensuremath{\alpha}<~1,$ and an accurate $(2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4})$ two-dimensional $(\ensuremath{\alpha},t)$ fit to the combined data is obtained for 0.01< t< 100.01<~t<~100.01< t< 10 and $0<~\ensuremath{\alpha}<~1.$ From the low-ttt TMRG data, the spin gap $\ensuremath{\Delta}(\ensuremath{\alpha})$ is extracted for $0.8<~\ensuremath{\alpha}<~0.995$ and compared with previous results, and a fit function is formulated for $0<~\ensuremath{\alpha}<~1$ by combining these data with literature data. We infer from our data that the asymptotic critical regime near the uniform chain limit is only entered for $\ensuremath{\alpha}\ensuremath{\gtrsim}0.99.$ We examine in detail the theoretical predictions of Bulaevskii [Sov. Phys. Solid State 11,11,11, 921 (1969)], for ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ and compare them with our results. To illustrate the application and utility of our theoretical results, we model our experimental $\ensuremath{\chi}(T)$ and specific heat Cp(T){C}_{\mathrm{p}}(T)Cp​(T) data for NaV2O5{\mathrm{NaV}}_{2}{\mathrm{O}}_{5}NaV2​O5​ single crystals in detail. The $\ensuremath{\chi}(T)$ data above the spin dimerization temperature ${T}_{\mathrm{c}}\ensuremath{\approx}34\mathrm{K}$ are not in quantitative agreement with the prediction for the S=1/2S=1/2S=1/2 uniform Heisenberg chain, but can be explained if there is a moderate ferromagnetic interchain coupling and/or if J changes with T. Fitting the $\ensuremath{\chi}(T)$ data using our ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ fit function, we obtain the sample-dependent spin gap and range $\ensuremath{\Delta}{(T=0)/k}_{\mathrm{B}}=103(2)\mathrm{K},$ alternation parameter $\ensuremath{\delta}(0)\ensuremath{\equiv}(1\ensuremath{-}\ensuremath{\alpha})/(1+\ensuremath{\alpha})=0.034(6)$ and average exchange constant J(0)/kB=640(80)K.{J(0)/k}_{\mathrm{B}}=640(80)\mathrm{K}.J(0)/kB​=640(80)K. The $\ensuremath{\delta}(T)$ and $\ensuremath{\Delta}(T)$ are derived from the data. A spin pseudogap with magnitude $\ensuremath{\approx}0.4\ensuremath{\Delta}(0)$ is consistently found just above Tc,{T}_{\mathrm{c}},Tc​, which decreases with increasing temperature. From our Cp(T){C}_{\mathrm{p}}(T)Cp​(T) measurements on two crystals, we infer that the magnetic specific heat at low temperatures $T\ensuremath{\lesssim}15\mathrm{K}$ is too small to be resolved experimentally, and that the spin entropy at Tc{T}_{\mathrm{c}}Tc​ is too small to account for the entropy of the transition. A quantitative analysis indicates that at Tc,{T}_{\mathrm{c}},Tc​, at least 77% of the entropy change due to the transition at Tc{T}_{\mathrm{c}}Tc​ and associated order parameter fluctuations arise from the lattice and/or charge degrees of freedom and less than 23% from the spin degrees of freedom.
Finite-size effects in a spin-1/21/21/2 one-dimensional (1D) Heisenberg antiferromagnet (HAF) are studied experimentally and theoretically. The calculated susceptibility is compared with that observed in the quasi-1D HAF ${\mathrm{SrCu}}_{1\ensuremath{-}x}{\mathrm{Mg}}_{x}{\mathrm{O}}_{2}$. Excellent agreement between theory and experiment is obtained without any adjustable parameters above about 0.02J0.02J0.02J, where JJJ is the dominant exchange interaction constant.
Extraction of antiferromagnetic (AF) exchange constants J and J\ensuremath{'} in two-leg spin-1/2 nearest-neighbor (NN) Heisenberg ladders from experimental magnetic spin susceptibility \ensuremath{\chi}(T) data is studied, where J\ensuremath{'} is the NN exchange constant in the rungs and J is that in the legs. Two low-T (T\ensuremath{\ll}J/kB{\mathit{k}}_{\mathit{B}}kB​) approximations of Troyer, Tsunetsugu, and W\"urtz [Phys. Rev. B 50, 13 515 (1994)] for \ensuremath{\chi}(T) of the isolated spin-1/2 two-leg ladder with J\ensuremath{'}/J=1 are shown to be accurate to \ensuremath{\sim}10% when extrapolated to T\ensuremath{\sim}J\ensuremath{'}/kB{\mathit{k}}_{\mathit{B}}kB​. The variations in the magnetic excitation dispersion relation parameters [which enter the low-T expression of Troyer et al. for \ensuremath{\chi}(T)] for J\ensuremath{'}/J1 are inferred from previous theoretical results. Next, the values of the maxima in \ensuremath{\chi}(T)\ensuremath{\equiv}${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$ from many numerical calculations in the literature for various nonfrustrated spin-1/2 NN Heisenberg antiferromagnets are considered. The values of the product ${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$zeff{\mathit{z}}_{\mathrm{eff}}zeff​ are found to depend systematically and almost uniqely on an effective magnetic coordination number zeff{\mathit{z}}_{\mathrm{eff}}zeff​, defined in the text, but are nearly independent of the dimensionality of the spin lattice and of the magnitude of any gap in the magnetic excitation spectrum. This observation allows accurate bounds on the exchange coupling constants in arbitrary quasi-low-dimensional spin-1/2 antiferromagnets to be inferred from experimental ${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$ values. Finally, a mean-field-type expression for \ensuremath{\chi}(T) for arbitrary quasi-low-dimensional spin-1/2 Heisenberg antiferromagnets is derived, using the above ${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$ phenomenology as input, which allows the influence of intersystem coupling on \ensuremath{\chi}(T) to be evaluated. These results, and the \ensuremath{\chi}(T) calculations of Barnes and Riera [Phys. Rev. B 50, 6817 (1994)] for isolated two-leg spin-1/2 Heisenberg ladders, are used to analyze the experimental \ensuremath{\chi}(T) data of Azuma et al. [Phys. Rev. Lett. 73, 3463 (1994)] for the spin-1/2 two-leg ladder compound SrCu2{\mathrm{SrCu}}_{2}SrCu2​O3{\mathrm{O}}_{3}O3​. The analyses together suggest that J\ensuremath{'}/J\ensuremath{\sim}0.5, contrary to the expectation that J\ensuremath{'}/J\ensuremath{\approxeq}1, and that J is very large (\ensuremath{\sim}2000 K), similar to the value of J in the linear-chain cuprate Sr2{\mathrm{Sr}}_{2}Sr2​CuO3{\mathrm{CuO}}_{3}CuO3​. \textcopyright{} 1996 The American Physical Society.