We propose a model for the thermal conductivity of 2D-samples when the mean free paths due to the phonon–phonon interactions exceed the sample dimensions. The physical mechanisms that ensure the stationary heat flux and the stationary nonequilibrium temperature distribution are examined. A recursive equation is derived to quantify the contribution of phonon scattering to the net heat flux in the pure elastic isotropic diffusive regime, as a function of the number of scattering at the boundaries. As the length to width ratio L/W increases above unity, our model shows an increase in the phonon mean free path compared to that predicted by the Casimir model in which the lateral walls are assumed to be black bodies. The present model also reveals that the multiple phonon interactions with the lateral walls lead to a stationary, but non-linear temperature distribution. The dependence of the heat flux and the thermal conductivity on the sample dimensions is then shown to be non-monotonic. This infers that the location of the thermometers on the sample influences experimental measurements of the thermal conductivity.

中文翻译：

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扩散声子边界散射对热流的影响

当由于声子-声子相互作用而产生的平均自由程超过样品尺寸时，我们提出了二维样品导热系数的模型。研究了确保稳态热通量和稳态非平衡温度分布的物理机制。推导了一个递归方程，以量化声子散射对纯弹性各向同性扩散状态下净热通量的贡献，它是边界处散射数量的函数。长宽比L / W如果增加到大于1，则我们的模型显示，与假设侧壁为黑体的卡西米尔模型所预测的结果相比，声子平均自由程增加了。本模型还揭示了与侧壁的多个声子相互作用导致了稳定的但非线性的温度分布。然后显示出热通量和热导率对样品尺寸的依赖性是非单调的。这表明温度计在样品上的位置会影响热导率的实验测量。