<p>The morphology evolution and position movement of the void induced by electromigration is of great significance to the reliability, performance, and surface nanotechnology of metal interconnects. In this paper, the multi-phase-field model, adaptive mesh technique and finite element method are used to study the void crossing grain boundary behavior in metal interconnects with face-centered cubic metal (110) orientation under electromigration-induced anisotropic surface diffusion. The results indicate that the process of void crossing grain boundary depends on the anisotropic strength <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="339_2025_8363_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(Q\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>Q</mi> </math></EquationSource> </InlineEquation>, the electric field strength <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="339_2025_8363_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\chi\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation>, the misorientation of the fast interface diffusion direction concerning the electric field in left-side grain <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="339_2025_8363_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\(\theta_{0L}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>θ</mi> <mrow> <mn>0</mn> <mi>L</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and the initial aspect ratio of void <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="339_2025_8363_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>. And the effects of the four governing parameters are discussed in detail. When <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="339_2025_8363_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">\(Q \ge 17.5\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mo>≥</mo> <mn>17.5</mn> </mrow> </math></EquationSource> </InlineEquation>, the circular void can split into two small voids after breaking away from the grain boundary and entering the right-side grain, and increasing <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="339_2025_8363_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(Q\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>Q</mi> </math></EquationSource> </InlineEquation> will accelerate the void crossing process. With the increase of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="339_2025_8363_Article_IEq7.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\chi\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="339_2025_8363_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\(\theta_{0L}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>θ</mi> <mrow> <mn>0</mn> <mi>L</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="339_2025_8363_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>, the void can split into two voids during crossing the grain boundary, and the two small voids have two evolution trends of merging and non-merging. When <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="339_2025_8363_Article_IEq10.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\chi\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation> is small, the time of void crossing grain boundary is significantly dependent on <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="339_2025_8363_Article_IEq11.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(Q\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>Q</mi> </math></EquationSource> </InlineEquation>. On the contrary, the time is significantly dependent on <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="339_2025_8363_Article_IEq12.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\chi\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation>. Increasing <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="339_2025_8363_Article_IEq13.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\chi\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="339_2025_8363_Article_IEq14.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\(\theta_{0L}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>θ</mi> <mrow> <mn>0</mn> <mi>L</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> will accelerate the void crossing process, while increasing <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="339_2025_8363_Article_IEq15.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> will hinder it. The results obtained in this paper are of great significance to the dynamics of void in interconnects and the fabrication of nanostructures with the required characteristics and sizes.</p>

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Multi-phase-field simulation of the void crossing grain boundary under electromigration-induced anisotropic surface diffusion in (110)-oriented interconnects

  • Yuanzhe Guo,
  • Peizhen Huang

摘要

The morphology evolution and position movement of the void induced by electromigration is of great significance to the reliability, performance, and surface nanotechnology of metal interconnects. In this paper, the multi-phase-field model, adaptive mesh technique and finite element method are used to study the void crossing grain boundary behavior in metal interconnects with face-centered cubic metal (110) orientation under electromigration-induced anisotropic surface diffusion. The results indicate that the process of void crossing grain boundary depends on the anisotropic strength \(Q\) Q , the electric field strength \(\chi\) χ , the misorientation of the fast interface diffusion direction concerning the electric field in left-side grain \(\theta_{0L}\) θ 0 L and the initial aspect ratio of void \(\lambda\) λ . And the effects of the four governing parameters are discussed in detail. When \(Q \ge 17.5\) Q 17.5 , the circular void can split into two small voids after breaking away from the grain boundary and entering the right-side grain, and increasing \(Q\) Q will accelerate the void crossing process. With the increase of \(\chi\) χ , \(\theta_{0L}\) θ 0 L or \(\lambda\) λ , the void can split into two voids during crossing the grain boundary, and the two small voids have two evolution trends of merging and non-merging. When \(\chi\) χ is small, the time of void crossing grain boundary is significantly dependent on \(Q\) Q . On the contrary, the time is significantly dependent on \(\chi\) χ . Increasing \(\chi\) χ or \(\theta_{0L}\) θ 0 L will accelerate the void crossing process, while increasing \(\lambda\) λ will hinder it. The results obtained in this paper are of great significance to the dynamics of void in interconnects and the fabrication of nanostructures with the required characteristics and sizes.