各向异性多层结构中垂直热边界导纳的格林函数方法
A Green's-function method for vertical thermal boundary conductance in anisotropic multilayers
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- University of Pittsburgh（匹兹堡大学）
- Brigham Young University（杨百翰大学）
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中文总结 AI 辅助
提出格林函数边界积分方程方法，高效求解各向异性多层结构中的深度依赖垂直热边界导纳，速度快且内存占用低，适用于大纵横比场景。
中文摘要 AI 辅助
垂直热界面存在于工程材料和天然材料中。其垂直热边界导纳可能与水平方向的热边界导纳不同，因此需要专门的表征方法。然而，由于缺乏能够同时处理各向异性、多层结构以及深度依赖的垂直热边界导纳的高效正演解，当前的热测量技术仅在受限几何结构（如两种块体介质）中解析垂直热边界导纳。我们提出了一种格林函数边界积分方程（GBIE）方法，该方法将传递矩阵格林函数与仅含界面的积分方程耦合，用于求解深度依赖的$G_v(z)$，支持两侧不同的正交各向异性多层结构（$k_x\neq k_y\neq k_z$）以及水平导纳$G_h$。对于薄膜厚度从$1\\,μ\mathrm{m}$到$100\\,\mathrm{nm}$的各向异性薄膜-衬底多层结构，GBIE与三维有限元法（FEM）预测的平均归一化相位和幅度误差在百分之一以内，同时在单核测试中运行速度快$29\text{--}210\times$，峰值内存减少因子为$120\text{--}450$；JIT编译的JAX实现匹配的16核比较中速度提升高达$4100\times$。GBIE还再现了覆盖埋入界面的连续薄膜（代表热反射测量）以及具有深度依赖$G_v(z)$的有限深度界面。GBIE可处理横向尺寸与薄膜厚度之比超过$10^{5}$的情况，此时体积有限元法可能变得计算上不可行。这些结果建立了从微电子器件侧壁到多晶固体晶界等系统中垂直界面热输运的高效正演解。
英文摘要
Vertical thermal interfaces occur in both engineered and natural materials. Their vertical thermal boundary conductance can differ from the horizontal counterpart, requiring dedicated characterization. Yet current thermal metrology resolves vertical thermal boundary conductance only in restricted geometries such as two bulk media, for lack of an efficient forward solution that admits anisotropy, multilayers, and depth-dependent vertical thermal boundary conductance together. We present a Green's-function boundary integral equation (GBIE) method that couples transfer-matrix Green's functions to an interface-only integral equation for depth-dependent $G_v(z)$, supporting dissimilar orthotropic multilayers ($k_x\neq k_y\neq k_z$) on either side and horizontal conductance $G_h$. For anisotropic film-on-substrate multilayers with films from $1~μ\mathrm{m}$ to $100~\mathrm{nm}$, the GBIE agrees with three-dimensional finite element method (FEM) predictions to within one percent mean normalized phase and amplitude error, while running $29\text{--}210\times$ faster and reducing peak memory by factors of $120\text{--}450$ in single-core tests; a JIT-compiled JAX implementation reaches up to $4100\times$ on a matched 16-core comparison. The GBIE further reproduces a continuous film over a buried interface, representative of a thermoreflectance measurement, and a finite-depth interface with depth-dependent $G_v(z)$. The GBIE accommodates lateral-to-film-thickness ratios above $10^{5}$, where volumetric FEM can become computationally prohibitive. These results establish an efficient forward solution for vertical-interface heat transport in systems ranging from microelectronic device sidewalls to grain boundaries in polycrystalline solids.
