We analyze, both theoretically and numerically, the temperature dependent thermal conductivity k{appa} of two-dimensional nanowires with surface roughness. Although each sample is characterized by three independent parameters - the diameter (width) of the wire, the correlation length and strength of the surface corrugation - our theory predicts that there exists a universal regime where k{appa} is a function of a single combination of all three model parameters. Numerical simulations of propagation of acoustic phonons across thin wires confirm this universality and predict a d 1/2 dependence of k{appa} on the diameter d.
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