Two-dimensional wave propagation in cylindrical single-crystal solid–liquid bodies

Abstract

The significant difference in the ultrasonic velocity of liquid and solid semiconductors has led to an interest in ultrasonic characterization of anisotropic solid–liquid interfaces during semiconductor single-crystal growth. However, the large solid:liquid velocity ratio could result in potentially severe ray bending at the interface and so detailed ray path and wavefront analyses are needed both to design the sensing approach and to analyze its results. A ray tracing algorithm has been developed and used to analyze two-dimensional wave propagation on the transverse cross-sectional and the diametral planes of cylindrical single-crystal solid–liquid bodies. For a prescribed point source position and an initial ray angle, the refraction and reflection angles at an interface are determined using the Christoffel equation, Snell’s law, and the group velocity. The ray path between prescribed source and receiver positions is then determined using the shooting method. Wavefronts are obtained by connecting points on ray paths with the same traveling time. Ultrasonic time of flights are obtained by integrating the ultrasonic slowness along the ray path. Ray paths are indeed found to be severely bent, particularly at interfaces with crystal planes of greatest group velocity and largest interfacial curvature. It indicates the need for precise ray path calculations if time-of-flight data are to be used for interface reconstruction. On the transverse plane, the simulated time-of-flight data identify simple strategies for determining the position and radius of the solid–liquid interface. Time-of-flight data obtained on the diametral plane with the source and receiver points having the same axial coordinate are found to easily identify the interfacial region. The curvature of convex interfaces can be directly determined from diametral plane data using the time of flight of doubly refracted ray paths or from the discontinuity in the time of flight for rays that propagate in the homogeneous solid and liquid regions adjacent to the interface. The convexity of concave interfaces can be determined from diametral plane time-of-flight measurements for the interface diffracted (i.e., creeping) ray provided the crystal orientation exhibits weak velocity anisotropy; for other orientations, additional measurements with unequal source/receiver axial coordinates are needed.