Niobium hydride formation in superconducting qubit thin films

The formation of nonsuperconducting hydrides in 160–170-nm-thick films of niobium is examined. We identify six elastically distinct orientation relationships <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>ɛ</a:mi><a:mtext>−</a:mtext><a:mrow><a:msub><a:mi>Nb</a:mi><a:mn>4</a:mn></a:msub><a:msub><a:mi mathvariant="normal">H</a:mi><a:mn>3</a:mn></a:msub></a:mrow></a:mrow></a:math> takes within the matrix, solid-solution <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>α</c:mi><c:mtext>−</c:mtext><c:msub><c:mi>NbH</c:mi><c:mi>x</c:mi></c:msub></c:mrow></c:math> phase. We employ a phase field model to assess the impact of elastic energy induced by the strain of phase transformation on the morphology and transformation dynamics of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>ɛ</d:mi></d:math> precipitates within a thin film. We consider the dimensions of the thin film, crystallographic growth direction, and diffusion rates to predict the timescale of hydride evolution. Leveraging the finite element method, we predict two-dimensional and three-dimensional equilibrium shapes of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>ɛ</e:mi></e:math>-hydrides within a bulk sample and in a thin film that has a traction-free surface. Our results suggest that niobium hydrides migrate to the free surface of the film. Precipitates which reach the free surface coarsen, while precipitates within the film dissolve. Precipitates in both two dimensions and three dimensions experience a repulsive interaction force at the free surface, that is attractive in the bulk, shown in experiment and theory of previous studies.