Formation of non-uniformity of coating thickness when sprayed with annular magnetrons on planetary installations

The urgency of the problem of ensuring uniform thickness of thin-film coatings applied by magnetron sputtering is due to the increasing requirements for the quality and functional characteristics of products in microelectronics, optics and other high-tech industries. Uneven coating thickness can lead to deterioration of the operational properties of the products, such as electrical conductivity, optical transparency or mechanical strength. In existing studies, mathematical models based on the Lambert-Knudsen law of cosines have been proposed that describe the deposition process for simple geometric configurations, for example, for parallel and coaxial targets and substrates. However, these models do not take into account the complex motion of the substrate in three-dimensional space, which is typical for planetary sputtering systems. Other papers note that analytical calculations for such systems are very difficult, and it is recommended to focus on optimizing the configuration of magnetron emitters. In this article, the task is to develop a mathematical model that makes it possible to predict the distribution of coating thickness on parts of complex geometry when using a planetary-type installation with dual spraying. The case of applying a titanium coating to a double-sided fiberglass part with a copper sublayer is specifically considered. The main difficulty lies in taking into account the relative location of the target and the substrate in three-dimensional space, as well as in describing the kinematics of the planetary mechanism, which includes rotation both around its own axis and around the central axis of the system. It is necessary to determine the growth rate of the coating at each point of the surface of the part, taking into account the distance to the target, the angles of spraying and condensation, as well as the geometric parameters of the system. This requires converting the coordinates of the substrate and target points, calculating the deposition spray angles, and integrating the deposition rate over all target points. Special attention is paid to eliminating physically incorrect scenarios, such as deposition on the back of the substrate. In this article, a comprehensive mathematical approach is being developed to predict the thickness distribution of thin-film coatings applied by magnetron sputtering on planetary-type installations. The methodology is based on analytical calculations based on the fundamental physical principles of the spraying process. The method is based on the Lambert-Knudsen law of cosines, according to which the rate of coating growth at a surface point is proportional to the product of the cosines of the emission angles (φ) and condensation (ε), and is inversely proportional to the square of the distance (p) from the source of the sprayed material to the point under consideration. A coordinate transformation model has been developed to account for the complex motion of the substrate in the planetary system. Using the transfer and rotation matrices, the transition from the coordinate system of the part to the coordinate system of the magnetron was carried out. This allows you to determine the spatial position of any point on the surface of the part relative to the target at any given time. Based on the coordinate transformation, the distance between the point on the target and the point on the part, the spray angle φ between the spray vector and the normal to the target plane, and the condensation angle ε between the spray vector and the normal to the part plane are calculated. The key element of the technique is the integration of the deposition rate over all target points and over time. This approach allows us to take into account the contribution of all areas of the target to the formation of the coating and obtain a thickness distribution over the entire surface of the part. The presented method is highly versatile and can be adapted to various configurations of magnetron systems and types of substrates, which makes it a valuable tool for designing and optimizing magnetron sputtering processes. Solving this problem will allow not only to predict the thickness of the coating at the design stage of the technological process, but also to optimize the deposition parameters to achieve maximum uniformity, which will reduce the number of experimental tests and increase production efficiency.