Bond disorder and spinon heat transport in the $S=\frac{1}{2}$ Heisenberg spin chain compound Sr${}_2$CuO${}_3$: From clean to dirty limits

We investigate the effect of disorder on the heat transport properties of the $S=\frac{1}{2}$ Heisenberg spin chain compound Sr${}_2$CuO${}_3$ upon chemically substituting Sr by increasing concentrations of Ca. As Ca occupies sites outside but near the Cu–O–Cu spin chains, bond disorder, i.e., a spatial variation of the exchange interaction $J$, is expected to be realized in these chains. We observe that the magnetic heat conductivity (${\kappa }_{\mathrm{mag}}$) due to spinons propagating in the chains is gradually but strongly suppressed with increasing amounts of Ca, where the doping dependence can be understood in terms of increased scattering of spinons due to Ca-induced disorder. This is also reflected in the spinon mean-free path which can be separated in a doping-independent but temperature-dependent scattering length due to spinon-phonon scattering, and a temperature-independent but doping-dependent spinon-defect scattering length. The latter spans from very large ($>$1300 lattice spacings) to very short ($\sim$12 lattice spacings) and scales with the average distance between two neighboring Ca atoms. Thus, the Ca-induced disorder acts as an effective defect within the spin chain, and the doping scheme allows us to cover the whole doping regime between the clean and dirty limits. Interestingly, at maximum impurity levels, we observe in Ca-doped Sr${}_2$CuO${}_3$ an almost linear increase of ${\kappa }_{\mathrm{mag}}$ at temperatures above 100 K which reflects the intrinsic low-temperature behavior of heat transport in a Heisenberg spin chain. These findings are quite different from that observed for the Ca-doped double spin chain compound, SrCuO${}_2$, where the effect of Ca seems to saturate already at intermediate doping levels.

Figure 1

Heat conductivities ${\kappa }_b$ measured parallel to the chains and ${\kappa }_c$ measured perpendicular to the chains for the undoped Sr${}_2$CuO${}_3$ compound as a function of temperature (data from Ref. [5]). Inset: Ratio of heat conductivities ${\kappa }_b$ and ${\kappa }_c$ for Sr${}_2$CuO${}_3$ and the doped compounds to give an idea of the magnitude and temperature dependence of the anisotropy in heat conduction. The curves for the doped compounds are normalized to the value of ${\kappa }_b$/${\kappa }_c$ for the pure compound at 300 K to illustrate the doping dependence of this ratio more clearly. The actual value of ${\kappa }_b$/${\kappa }_c$ at room temperature is always large ($\sim$4) and ranges from $\sim$3.8 for the pure compound to $\sim$4.7 for $x=0.5$.

Figure 2

Heat conductivity ${\kappa }_b$ as a function of temperature measured parallel to the chains. Inset: A semilog plot of ${\kappa }_b$ versus temperature to better illustrate the changes upon introducing disorder.

Figure 3

Heat conductivity $\kappa$${}_c$ as a function of temperature measured perpendicular to the chains. Inset: A semilog plot of ${\kappa }_c$ versus temperature to better illustrate the changes upon introducing disorder. The solid black curves are fits to the curves according to the Callaway model.

Figure 4

Magnetic heat conductivity ${\kappa }_{\mathrm{mag}}$ as a function of temperatures for the undoped and the Ca-doped compounds. Inset: A semilog plot of ${\kappa }_{\mathrm{mag}}$ versus temperature to more clearly show the increase of ${\kappa }_{\mathrm{mag}}$ with increasing temperature for higher concentrations of Ca. Solid black lines are ${\kappa }_{\mathrm{mag}}$ recalculated using the fits for $l_{\mathrm{mag}}$ and Eq. (2). The gray regions, shown exemplarily for $x=0.05$ and $x=0.5$, depict the uncertainties associated with the extraction of $\kappa$${}_{\text{mag}}$.

Figure 5

Mean-free path of spinons as a function of temperature $l_{\mathrm{mag}}$($T$) and the fits according to Eq. (3). The grey regions around the curve for $x=0.05$ and $x=0.5$ depict the uncertainty associated with the calculation of $l_{\mathrm{mag}}$. Inset shows a semilog plot of $l_{\mathrm{mag}}$ versus temperature.

Figure 6

Magnetic heat conductivity ${\kappa }_{\mathrm{mag}}$ as a function of temperature for $x=0.1$ and $x$ = 0.5 compounds. Solid lines are ${\kappa }_{\mathrm{mag}}$ recalculated using the fits for $l_{\mathrm{mag}}$ and Eq. (2). Gray regions around the curve indicate the uncertainties in ${\kappa }_{\mathrm{mag}}$.

Figure 7

Spinon-defect scattering length $l_0$ plotted against the inverse of Ca concentration ($x$) (lower $x$ axis) and the mean distance between two Ca atoms (upper $x$ axis) for SrCuO${}_2$ (open symbols) and Sr${}_2$CuO${}_3$ (filled symbols); the solid lines are linear fits to the data. The $y$ axis on the right gives an idea of the mean-free path in terms of the number of lattice spacings between two Cu sites ($d_0$).