Thermal conductivity of insulating ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$${\mathrm{YCu}}_2$${\mathrm{O}}_8$ and superconducting ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$${\mathrm{CaCu}}_2$${\mathrm{O}}_8$: Failure of the phonon-gas picture

The ab-plane thermal conductivity κ(T) of insulating ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{YCu}}_2$ ${\mathrm{O}}_8$ and superconducting ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{CaCu}}_2$ ${\mathrm{O}}_8$ is measured from T=10 to 300 K on single-crystal samples. Metallic ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{CaCu}}_2$ ${\mathrm{O}}_8$ has a significantly higher κ than ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{YCu}}_2$ ${\mathrm{O}}_8$; the difference Δκ agrees well in magnitude with a Wiedemann-Franz estimate of the electronic contribution to κ in ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{CaCu}}_2$ ${\mathrm{O}}_8$. The shape of κ(T) in insulating ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{YCu}}_2$ ${\mathrm{O}}_8$ differs from normal insulators described by the Peierls-Boltzmann theory. Assuming that atomic vibrations are the main heat carrier, and noting that κ is more similar to that of silica glass than to a normal insulating crystal like CuO, we suggest that the ‘‘phonon’’ mean free path is sufficiently short that the Peierls-Boltzmann theory is not applicable. This is consistent with evidence from neutron scattering that phonons are poorly defined. Our data support the idea that the peak in κ(T) observed below ${\mathit{T}}_{\mathit{c}}$ in superconducting samples originates from electronic rather than vibrational heat currents.