Stan Gheorghe, Mays Ebony, Yoo Hui Jae, King Sean W
Material Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, MD, 20899, USA.
Logic Technology Development, Intel Corporation, Hillsboro, OR 97124, USA.
Exp Mech. 2018;58(7). doi: 10.1007/s11340-018-0421-4.
In the last two decades, significant progress has been made on developing new nanoscale mechanical property measurement techniques including instrumented indentation and atomic force microscopy based techniques. The changes in the tip-sample contact mechanics during measurements uniquely modify the displacement and force sensed by a measurement sensor and much effort is dedicated to correctly retrieve the sample mechanical properties from the measured signal. It turns out that in many cases, for the sake of simplicity, a simple contact mechanics model is adopted by overlooking the complexity of the actual contact geometry. In this work, a newly developed matrix formulation is used to solve the stress and strain equations for samples with edge geometries. Such sample geometries are often encountered in today's nanoscale integrated electronics in the form of high-aspect-ratio fins with widths in the range of tens of nanometers. In the matrix formulation, the fin geometries can be easily modeled as adjacent overlapped half-spaces and the contact problem can be solved by a numerical implementation of the conjugate gradient method. This method is very versatile in terms of contact geometry and contact interaction, either non-adhesive or adhesive. The discussion will incorporate a few model examples that are relevant for the nanoscale mechanics investigated by intermittent contact resonance AFM (ICR-AFM) on low-k dielectric fins of high-aspect-ratio. In such ICR-AFM measurements, distinct dependence of the contact stiffness was observed as a function of the applied force and distance from the edges of the fins. These dependences were correctly predicted by the model and used to retrieve the mechanical changes undergone by fins during fabrication and processing.
在过去二十年中，在开发新的纳米级机械性能测量技术方面取得了重大进展，包括仪器化压痕和基于原子力显微镜的技术。测量过程中探针与样品接触力学的变化独特地改变了测量传感器感知的位移和力，并且人们致力于从测量信号中正确获取样品的机械性能。事实证明，在许多情况下，为了简单起见，采用了简单的接触力学模型，而忽略了实际接触几何形状的复杂性。在这项工作中，使用新开发的矩阵公式来求解具有边缘几何形状的样品的应力和应变方程。这种样品几何形状在当今的纳米级集成电子器件中经常以高宽比鳍片的形式出现，其宽度在几十纳米范围内。在矩阵公式中，鳍片几何形状可以很容易地建模为相邻重叠的半空间，并且可以通过共轭梯度法的数值实现来解决接触问题。该方法在接触几何形状和接触相互作用方面非常通用，无论是非粘性还是粘性。讨论将纳入一些与通过间歇接触共振原子力显微镜（ICR-AFM）对高宽比低k介质鳍片进行的纳米级力学研究相关的模型示例。在这种ICR-AFM测量中，观察到接触刚度与施加力和距鳍片边缘的距离有关的明显依赖性。这些依赖性由模型正确预测，并用于获取鳍片在制造和加工过程中经历的机械变化。
