Thermal transport in antiferromagnetic spin-chain materials

We study the problem of heat transport in one-dimensional (1D) spin-chain systems weakly coupled to three-dimensional phonons and impurities. We consider the limit of fast spin excitations and slow phonons, applicable to a number of compounds of the cuprate family, such as ${\mathrm{Sr}}_2{\mathrm{CuO}}_3$, where the superexchange $J$ is much larger than the Debye energy ${\Theta }_D$. In this case the umklapp scattering among the spin excitations is strongly suppressed for all relevant temperatures. We argue that the leading scattering mechanism for the spin excitations at not too low temperatures is the normal (as opposed to the umklapp) spin-phonon scattering in which the nonequilibrium momentum is transferred from the spin subsystem to phonons where it quickly relaxes through the phonon bath. Because of the lower dimensionality of the spin excitations it is only the momentum along the chains that is conserved in such a scattering. We find that this effect leads to a particular momentum and temperature dependence of the spin-phonon relaxation rate valid for the broad class of low-dimensional spin systems. Subsequently we demonstrate that the spin-phonon relaxation mechanism is insufficient for the low-energy, long-wavelength 1D spin-chain excitations, which make the thermal conductivity diverge. We complete our consideration by taking into account the impurity scattering, which in 1D cuts off the quasiballistic spin excitations and renders the thermal conductivity finite. Altogether, these effects yield the following spin-boson thermal conductivity behavior: ${\kappa }_s\propto T^2$ at low temperatures, ${\kappa }_s\propto T^{-1}$ at intermediate temperatures, and ${\kappa }_s=\mathrm{const}$ at higher temperatures $T\sim {\Theta }_D$. The saturation at higher temperatures is of rather nontrivial origin and we provide a detailed discussion of it. Our results compare very well with the existing experimental data for ${\mathrm{Sr}}_2{\mathrm{CuO}}_3$. Using our microscopic insight into the problem we propose further experiments and predict an unusual impurity concentration dependence for a number of quantities.

Figure 1

The scattering diagrams of the spin boson on the phonon for the processes contributing to the collision integral (a) $S^{\left(1\right)}$ and (b) $S^{\left(2\right)}$. Solid lines are spin bosons, wavy lines are phonons (Ref. 51).

Figure 2

Second-order diagrams for the spin-boson scattering on impurities. Vertices represent interactions with impurities at times ${\tau }^{\prime }$ and ${\tau }^{″}$. Shaded ellipses emphasize that there is a summation over virtual states with any number of spin-boson excitations. This is due to the fact that the interaction Hamiltonian is exponential in the bosonic field. Formally, the ellipses correspond to $⟨e^{i\sqrt{2\pi }\tilde{\Phi }(0,{\tau }^{\prime })}{e}^{i\sqrt{2\pi }\tilde{\Phi }(0,{\tau }^{″})}⟩$ (Ref. 51).

Figure 3

The sketch of the spin-phonon and spin-impurity relaxation rates ${\tau }_{\mathrm{sp}}^{-1}$ and ${\tau }_{\mathrm{imp}}^{-1}$ as a function of (a) temperature and (b) momentum. In (a) ${\tilde{\Theta }}_D={\Theta }_D∕4$.

Figure 4

(Color online) Spin thermal conductivity ${\kappa }_s$ normalized to its maximal value vs reduced temperature $T∕T_m$. Diamonds are the experimental data for ${\mathrm{Sr}}_2{\mathrm{CuO}}_3$, Ref. 5. $T_m=79.4\mathrm{K}$ and ${\kappa }_s^{\max }=36.7\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ for this material. Shaded area schematically represents the range where the phonon background subtraction creates a large uncertainty in the data (Ref. 5). Solid line is the result of this work, Eq. (51), for ${\Theta }_D=11.6T_m$. Arrows mark the saturation value ${\kappa }_s^{\infty }$ for the solid line at $T⪢{\tilde{\Theta }}_D$ and the spin-phonon scattering crossover scale ${\tilde{\Theta }}_D$. Dashed line corresponds to ${\kappa }_s\left(T\right)$, Eq. (51), for ${\Theta }_D\to \infty$. (These results were recently presented in Ref. 39.)

Figure 5

(Color online) Spin thermal conductivity ${\kappa }_s$ normalized to its value at $\infty$ vs reduced temperature $T∕T^{*}$ for various values of $z={\tilde{\Theta }}_D∕T^{*}$ ($z$ decreases from the top to the bottom curve).

Figure 6

(Color online) $k$-dependent mean free path $\ell (k,T)\text{vs}x=vk∕T$ for three representative temperatures $T=T_m$ (solid), $T_m∕2$ (dot-dashed), and $2T_m$ (dashed). Arrows mark the cutoff length ${\ell }_{\mathrm{imp}}\left(T\right)$ for each temperature. The approximate boundary of the bosonic thermal population is marked by the shaded area.

Figure 7

(Color online) The temperature dependence of the cutoff length ${\ell }_{\mathrm{imp}}$ ($\propto T$, dashed) and the thermal length ${\ell }_{\mathrm{th}}\left(T\right)=\ell (T∕v,T)$ (dash-dotted). In the phonon-dominated regime ${\ell }_{\mathrm{th}}\left(T\right)\propto T^{-5}$ for $T<{\tilde{\Theta }}_D$ and $\propto T^{-3}$ for $T>{\tilde{\Theta }}_D$. The effective mean free path ${\ell }_{\mathrm{eff}}\left(T\right)$, Eq. (75), and the approximate effective mean free path ${\ell }_{\mathrm{eff}}^{\mathrm{app}}\left(T\right)$, Eq. (76), are also shown (solid and dotted, respectively).

Figure 8

(Color online) The effective mean free path Eq. (75) compared to the experimental results of Refs. 4, 5. Asymptotic fits $\propto T$ for $T<T_m$ and $\propto 1∕T+C∕T^2$ for $T>T_m$ are shown by the dashed lines. The shaded area schematically represents the experimental error bars due to the phonon subtraction.

Figure 9

This diagram shows the regions in the momentum space allowed by momentum and energy conservation for the spin boson scattered from the state $k$ to the state $k^{\prime }$. The processes in which the original spin boson decays into another spin boson and emits the phonon are represented by the light gray area. For them $\mid k^{\prime }\mid <\mid k\mid$ as dictated by energy conservation. The dark gray areas correspond to the processes of phonon absorption by the spin boson. For them $\mid k^{\prime }\mid >\mid k\mid$. Lines $OA$ and $OB$ are bisectors $k^{\prime }=\pm k$.

Figure 10

The transformation of the integration path in the complex plane for Eq. (B17) is shown. The thick gray lines represent the branch cuts. The unit circle is transformed into a new contour circumventing one of the branch cuts. This new contour is shown by the dashed line.