▎ 摘　　要

Transient Joule heating of graphene or nanowires, supported by substrates, is an important and complex heat transfer problem to be solved in the context of nanoscale electronic devices. We formulate here an analytical model that allows the examination of the effects of input power, heat transfer through substrate and the end contacts on the transient temperature evolution. An exact analytical solution is obtained here by the superposition of the Joule heating, the substrate heat transfer and the temperatures at the ends of contacts, within the Fourier formalism, through Laplace transformations. The transient solution explicitly relates temperature profile to the input power, size and the thermal properties of the element, and the heat transfer properties of the substrate. The transient solution naturally reduces to steady state solution. The solutions can be useful for the thermal design of devices made of graphene or nanowires or electrically heated filaments. Experimental steady-state temperature distribution data, obtained by 2D phonon band spectroscopy, are used to confirm the accuracy of the solution. It is found that the time to achieve steady-state temperatures in graphene is of the order of nanoseconds and is a strong function of substrate heat transfer. Further, the maximum temperature and the gradient in steady state profile temperature profile are greatly affected by heat transfer to the substrate. (C) 2015 Elsevier Ltd. All rights reserved.